阿尔弗雷德·塔斯基：语言哲学和逻辑哲学

阿尔弗雷德·塔斯基：语言哲学和逻辑哲学

\nHistoryofAnalyticPhilosophySeriesEditor:MichaelBeaneyTitlesinclude:StewartCandlishTHERUSSELL/BRADLEYDISPUTEANDITSSIGNIFICANCEFORTWENTIETH-CENTURYPHILOSOPHYAnnalisaColivaMOOREANDWITTGENSTEINScepticism,CertaintyandCommonSenseGregoryLandiniFREGE’SNOTATIONSWhatTheyAreandHowTheyMeanSandraLapointeBOLZANO’STHEORETICALPHILOSOPHYAnIntroductionOmarW.NasimBERTRANDRUSSELLANDTHEEDWARDIANPHILOSOPHERSConstructingtheWorldDouglasPattersonALFREDTARSKIPhilosophyofLanguageandLogicGrahamStevensTHETHEORYOFDESCRIPTIONSNunoVenturinha(editor)WITTGENSTEINAFTERHISNACHLASSPierreWagner(editor)CARNAP’SLOGICALSYNTAXOFLANGUAGEForthcoming:AndrewAranaandCarlosAlvarez(editors)ANALYTICPHILOSOPHYANDTHEFOUNDATIONSOFMATHEMATICSRosalindCareyRUSSELLONMEANINGTheEmergenceofScientificPhilosophyfromthe1920stothe1940sGiusseppinaD’OroREASONSANDCAUSESCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01CausalismandNon-CausalisminthePhilosophyofActionGeorgeDukeDUMMETTONABSTRACTOBJECTS10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“FM”—2011/11/29—17:16—PAGEi—#1\nSébastienGandonRUSSELL’SUNKNOWNLOGICISMAStudyintheHistoryandPhilosophyofMathematicsAnssiKorhonenLOGICASUNIVERSALSCIENCERussell’sEarlyLogicismandItsPhilosophicalContextConsueloPretiTHEMETAPHYSICALBASISOFETHICSTheEarlyPhilosophicalDevelopmentofG.E.MooreSandraLapointe(translator)FranzPrihonskyTHENEWANTI-KANTErichReck(editor)THEHISTORICTURNINANALYTICPHILOSOPHYMariavanderSchaarG.F.STOUT:ONTHEPSYCHOLOGICALORIGINOFANALYTICPHILOSOPHYPierreWagner(editor)CARNAP’SIDEALOFEXPLICATIONANDNATURALISMHistoryofAnalyticPhilosophySeriesStandingOrderISBN978–0–230–55409–2(hardcover)SeriesStandingOrderISBN978–0–230–55410–8(paperback)outsideNorthAmericaonlyYoucanreceivefuturetitlesinthisseriesastheyarepublishedbyplacingastandingorder.Pleasecontactyourbookselleror,incaseofdifficulty,writetousattheaddressbelowwithyournameandaddress,thetitleoftheseriesandtheISBNquotedabove.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01CustomerServicesDepartment,MacmillanDistributionLtd,Houndmills,Basingstoke,HampshireRG216XS,EnglandAlsobyDouglasPattersonNEWESSAYSONTARSKIANDPHILOSOPHY(editor)10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“FM”—2011/11/29—17:16—PAGEii—#2\nAlfredTarski:PhilosophyofLanguageandLogicDouglasPattersonVisitingScholar,UniversityofPittsburgh,Pittsburgh,PACopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-0110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“FM”—2011/11/29—17:16—PAGEiii—#3\n©DouglasPatterson2012Allrightsreserved.Noreproduction,copyortransmissionofthispublicationmaybemadewithoutwrittenpermission.Noportionofthispublicationmaybereproduced,copiedortransmittedsavewithwrittenpermissionorinaccordancewiththeprovisionsoftheCopyright,DesignsandPatentsAct1988,orunderthetermsofanylicencepermittinglimitedcopyingissuedbytheCopyrightLicensingAgency,SaffronHouse,6–10KirbyStreet,LondonEC1N8TS.Anypersonwhodoesanyunauthorizedactinrelationtothispublicationmaybeliabletocriminalprosecutionandcivilclaimsfordamages.TheauthorhasassertedhisrighttobeidentifiedastheauthorofthisworkinaccordancewiththeCopyright,DesignsandPatentsAct1988.Firstpublished2012byPALGRAVEMACMILLANPalgraveMacmillanintheUKisanimprintofMacmillanPublishersLimited,registeredinEngland,companynumber785998,ofHoundmills,Basingstoke,HampshireRG216XS.PalgraveMacmillanintheUSisadivisionofStMartin’sPressLLC,175FifthAvenue,NewYork,NY10010.PalgraveMacmillanistheglobalacademicimprintoftheabovecompaniesandhascompaniesandrepresentativesthroughouttheworld.Palgrave®andMacmillan®areregisteredtrademarksintheUnitedStates,theUnitedKingdom,EuropeandothercountriesISBN:978–0–230–22121–5Thisbookisprintedonpapersuitableforrecyclingandmadefromfullymanagedandsustainedforestsources.Logging,pulpingandmanufacturingprocessesareexpectedtoconformtotheenvironmentalregulationsofthecountryoforigin.AcataloguerecordforthisbookisavailablefromtheBritishLibrary.AcatalogrecordforthisbookisavailablefromtheLibraryofCongress.1098765432121201918171615141312PrintedandboundinGreatBritainbyCPIAntonyRowe,ChippenhamandEastbourneCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-0110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“FM”—2011/11/29—17:16—PAGEiv—#4\nContentsSeriesEditor’sForewordviii0Introduction10.1Expressiveandrepresentationalsemantics10.2Thereceivedview40.3Themes71IntuitionisticFormalism121.1WhatwasIntuitionisticFormalism?121.1.1Apuzzleaboutconceptsanddefinitions121.1.2Tarski,Le´sniewskiandIntuitionisticFormalism161.1.3Formalism191.2Le´sniewski201.2.1Le´sniewski’searlywork201.2.2Le´sniewski’slaterwork251.3Kotarbi´nski311.4Tarskiincontext371.4.1Theaxiomaticmethod371.4.2Monismvstolerance411.4.3Fivedoctrines431.4.4Tarski’sproject492TarskiasIntuitionisticFormalist532.1Theearlymetamathematicalworks532.1.1Axiomatizingconsequence532.1.2Relativizationtoadeductivescience552.2Explicitdefinition622.2.1Definingdefinition632.2.2Twoconceptionsofdefinition652.2.3Padoa’smethod672.3Categoricityandcompletenessofterms70Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-012.3.1Provablemonotransformability702.3.2Absolutemonotransformability762.4Theoryandconcept803Semantics843.1Philosophicalresistance85v10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“FM”—2011/11/29—17:16—PAGEv—#5\nviContents3.1.1Thequantifier863.1.2Paradox893.2Mathematicalacceptance913.3IntuitionisticFormalismin“OnDefinableSets”943.3.1Theintuitivenotionofdefinability953.3.2Definingdefinablesetsvsdefining“Defines”1004Truth1084.1ConventionT1094.1.1Terminologicalnotes1094.1.2TruthintheLvov–Warsawschool1114.1.3Semanticconceptsinamathematicaltheory1144.1.4T-sentences1174.2Tarski’sdefinitions1224.2.1Truthforthelanguageofthecalculusofclasses1224.2.2Higherorderandpolyadicity1244.2.3Domainrelativizationandconsequence1284.3EvaluatingTarski’saccount1294.3.1Familiarquestions1294.3.2TarskiandefinitionsandTarski’s“theory”1334.3.3Reductionandphysicalism1384.3.4Correspondenceanddeflationism1405IndefinabilityandInconsistency1445.1Indefinability1455.1.1Indefinabilitybefore19311455.1.2TheoremI:textualissues1475.1.3TheoremIandIntuitionisticFormalism1555.1.4Axiomaticsemantics1585.2Inconsistencyineverydaylanguage1605.2.1InconsistentKotarbi´nskianconventions1625.2.2TarskiafterKotarbi´nski1666Transitions:1933–19351696.1The1935postscript1706.2Carnaponanalyticityandtruth1746.3Theestablishmentofscientificsemantics179Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-017LogicalConsequence1817.1Tarski’sdefinition1827.1.1Synopsis1827.1.2ObjectionstoTarski’saccount1857.2ConsequenceinLogicalSyntax18710.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“FM”—2011/11/29—17:16—PAGEvi—#6\nContentsvii7.2.1L-consequenceandconditionF1877.2.2TractarianismintheViennacircle1917.3Theovergenerationproblemanddomainvariation1947.3.1Domainvariation1947.3.2ConsequenceinGödel’scompletenesstheorem1987.3.3Tarski’sfixeddomain2017.4Themodalityproblemand“Tarski’sFallacy”2037.4.1Modalities2047.4.2Consequenceandtruth2067.4.3Tarski’s“must”2087.5Theformalityproblemandthelogicalconstants2097.5.1Constantandconsequence2097.5.2Anachronisticreadings2117.5.3Carnaponformality2137.5.4Theω-ruleandGödelsentences2147.5.5Antitractarianismandthenatureoflogic2157.6EvaluatingTarski’saccount2197.6.1Theanalyticproblem2197.6.2Eliminatingtransformationrules2217.6.3Epistemicandgeneralityconceptionsoflogic2238Conclusion2278.1Paris1935andthereceptionofsemantics2278.2Finalremarks232Notes234Bibliography249Index260Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-0110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“FM”—2011/11/29—17:16—PAGEvii—#7\nSeriesEditor’sForewordDuringthefirsthalfofthetwentiethcenturyanalyticphilosophygradu-allyestablisheditselfasthedominanttraditionintheEnglish-speakingworld,andoverthelastfewdecadesithastakenfirmrootinmanyotherpartsoftheworld.Therehasbeenincreasingdebateoverjustwhat‘ana-lyticphilosophy’means,asthemovementhasramifiedintothecomplextraditionthatweknowtoday,buttheinfluenceoftheconcerns,ideasandmethodsofearlyanalyticphilosophyoncontemporarythoughtisindisputable.Allthishasledtogreaterself-consciousnessamongana-lyticphilosophersaboutthenatureandoriginsoftheirtradition,andscholarlyinterestinitshistoricaldevelopmentandphilosophicalfoun-dationshasblossomedinrecentyears.Theresultisthathistoryofanalyticphilosophyisnowrecognizedasamajorfieldofphilosophyinitsownright.Themainaimoftheseriesinwhichthepresentbookappears–thefirstseriesofitskind–istocreateavenueforworkonthehistoryofanalyticphilosophy,consolidatingtheareaasamajorfieldofphilosophyandpromotingfurtherresearchanddebate.The‘historyofanalyticphilos-ophy’isunderstoodbroadly,ascoveringtheperiodfromthelastthreedecadesofthenineteenthcenturytothestartofthetwenty-firstcentury–beginningwiththeworkofFrege,Russell,MooreandWittgenstein,whoaregenerallyregardedasitsmainfounders,andtheinfluencesuponthem–andgoingrightuptothemostrecentdevelopments.Inallowingthe‘history’toextendtothepresent,theaimistoencourageengagementwithcontemporarydebatesinphilosophy–forexample,inshowinghowtheconcernsofearlyanalyticphilosophyrelatetocurrentconcerns.Infocusingonanalyticphilosophy,theaimisnottoexcludecomparisonswithother–earlierorcontemporary–traditions,orconsid-erationoffiguresorthemesthatsomemightregardasmarginaltotheanalytictraditionbutwhichalsothrowlightonanalyticphilosophy.Indeed,afurtheraimoftheseriesistodeepenourunderstandingoftheCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01broadercontextinwhichanalyticphilosophydeveloped,bylooking,forexample,attherootsofanalyticphilosophyinneo-KantianismorBritishidealism,ortheconnectionsbetweenanalyticphilosophyandphe-nomenology,ordiscussingtheworkofphilosopherswhowereimportantinthedevelopmentofanalyticphilosophybutwhoarenowoftenforgotten.viii10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“FM”—2011/11/29—17:16—PAGEviii—#8\nSeriesEditor’sForewordixInthisbookDouglasPattersonprovidesthefirstfull-lengthaccountofAlfredTarski’sphilosophy.TarskiwasborninWarsawin1901andgainedhisdoctorateinlogicattheUniversityofWarsawin1924,supervisedbyStanislawLe´sniewski.Inthe1920sand1930shepublishedextensivelyonlogicandsettheory,andasarepresentativeoftheso-calledLvov–WarsawSchool,maintainedcloselinkswithGödel,CarnapandothermembersoftheViennaCircle.WhenNaziGermanyinvadedPolandon1September1939,TarskiwasataconferenceintheUnitedStatesandwasunabletoreturnhome.Hestayedtherethroughoutthewaryears,teachingatHarvard,NewYorkandPrincetonbeforeeventuallybeinggivenapermanentpostattheUniversityofCaliforniaatBerkeleyin1945,whereheremaineduntilhisdeathin1983.TarskiwasthusoneofthemanylogiciansandphilosophersfromCentralEuropewhomovedtotheUnitedStatesasaresultoftheriseofNazisminGermany,andwhosestoryispartofthebroaderstoryofthedevelopmentofanalyticphilosophyinNorthAmerica,astheideasofthePolishlogiciansandthelogicalpositiviststookrootinnewsoil.Tarski’stwomostfamouspapersare‘TheConceptofTruthinFormal-izedLanguages’,firstpublishedinPolishin1933(andinGermanin1935)and‘OntheConceptofLogicalConsequence’,publishedinbothPolishandGermanin1936.InthefirstpaperTarskioffersadefinitionoftruthforformallanguagesbyintroducingthenotionofsatisfactionandappealingtotherecursivestructureofformallanguages.Tarskifirstformulateshis‘T-schema’here(famouslyexemplifiedinthestatementthat‘Snowiswhiteistrue’ifandonlyifsnowiswhite),andalsooffersasolutiontotheLiarparadoxbyinsistingthattruthforalanguagecanonlybedefinedinametalanguage.InthesecondpaperTarskioffersacorrespondingsemanticdefinitionoflogicalconsequencebyutiliz-ingtheideaoftruth-under-an-interpretationintroducedinhisearlierpaper.Thesetwopapershavebeenseenasestablishingthefoundationsoftruth-conditionalsemantics,withTarskialsoregardedasakeyfigureinthedevelopmentofmodeltheory.AsPattersonarguesinthisbook,however,Tarski’sactualviewsarebothmorecomplexandmoreintriguingthantheyhavestandardlybeentakentobe.Pattersonbeginsbydistinguishingwhathecallsthe‘expres-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01sive’conceptionofmeaningfromthe‘representational’conceptionofmeaning.Ontheexpressiveconception,languageexpressesthoughtsandthenotionsofassertionandjustificationhaveprimacy.Ontherep-resentationalconception,languagerepresentsthingsintheworldandthenotionsofreferenceandtruthhaveprimacy.Theformerfindsitsnaturalhomeintheproof-theoreticconceptionoflogic,whilethelatter10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“FM”—2011/11/29—17:16—PAGEix—#9\nxSeriesEditor’sForewordisreflectedinthemodel-theoreticconceptionoflogic.PattersonagreesthatTarski’stwopapersmademajorcontributionstorepresentationalsemanticsandmodeltheory,butbycarefullyexaminingthedevelop-mentofTarski’sworkfromthelate1920stothemid-1930s,heshowsthatthatworkwasoriginallymotivatedbytheexpressiveconceptionofmeaning.Tarskihimselfreferredtohisearlierviewas‘intuitionisticformalism’,aviewthatheinheritedfromLe´sniewski.Itwasonlyoncehehadconvincedhimselfthatlogicalconsequencecouldbedefinedsemanticallythatheabandonedhisearlierview.Pattersonremarksinhisintroductionthat“Tarskimaywellbethemostenigmaticfigureinthehistoryofanalyticphilosophy”.Patterson’sdetailedaccountofthecrucialperiodinTarski’sintellectualdevelopmentnotonlyshedsagreatdealoflightonTarski’sevolvingideasonthefundamentalnotionsoflogicandsemanticsbutalsoinauguratesanewerainourunderstandingofTarski’sworkanditscontributiontoanalyticphilosophy.MichaelBeaneySeptember2011Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-0110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“FM”—2011/11/29—17:16—PAGEx—#10\n0Introduction0.1ExpressiveandrepresentationalsemanticsThisbooktellsthestoryofthebirthoftruth-conditionalsemanticsfromanearlierconceptionofmeaning.Ifwethinkoflanguageasstandingbetweenmindandworld,therearetwosimplewaysinwhichtothinkofitasmeaningful:intermsofitsrelationtothemind,orintermsofitsrelationtotheworld.Onemaythusconceiveofmeaningintermsoftheexpressionofthought,orintermsoftherepresentationofthings.Callthetwoconceptionsexpressiveandrepresentationalsemantics.Ontheexpressiveconceptionthefunctionoflanguageistoexpressthoughts,whichmaythemselvesberepresentational.Ontherepresentationalcon-ceptionlanguageisconceivedofasrepresentationalinitsownright,andtheexpressionofthoughtisaderivativefunction.Centraltotheexpres-siveconceptionoflanguagearethenotionsofassertionandjustificationsinceourbasicnotionsareasubject’ssayingsomethingandtheirreasonsfordoingso;centraltotherepresentationalconceptionarethenotionsofreferenceandtruthbeingaboutthingsandaccuratelyrepresentinghowtheyare.Bothviewsaddressthetopicofinference.Theexpressiveconceptionoffersanatural,intuitiveconnectiontothebasicideaofanargumentassomethingthatgivesoneareasontobelieveonethinggiventhatonebelievesothers:inavalidargument,justificationforbelievingtheCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01premisesistransmittedtotheconclusionandbecomesjustificationforbelievingit.Therepresentationalconception,sinceitsfocusisonaccu-racyinwhatstatementsareabout,givesrisetotheideathatanargumentisvalidjustincaseifitspremisesaretrue,soisitsconclusion.Continuinginthesamebroadbrush-strokes,theexpressiveconceptionsitsnaturallywiththeconceptionofinferenceasderivationofoneclaimfromothers110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“INTRO”—2011/11/28—10:13—PAGE1—#1\n2AlfredTarski:PhilosophyofLanguageandLogicinaccordwithintuitivelyvalidrules—aconceptionthatbecomes,inmorerefinedstudiesoflogic,prooftheory.Therepresentationalconcep-tionnaturallyleadstotheideaofaninferenceasvalidjustincaseallmodelsofthepremisesarealsomodelsoftheconclusion,anideaatthefoundationofmodel-theoreticstudiesoflogic.Theinteractionofthetwoconceptionsintheirrefinedformsthengivesustwoofthecentralresultsofthe20thcentury,Gödel’scompletenessandincompletenesstheorems.Neitherconceptionofmeaningissufficientofitself;likewise,neithertheproof-theoreticnorthemodel-theoreticconceptionoflogicofitselfcapturesthenotionofonething’sfollowingfromanother.Indeed,whenitcomestologic,bothderivationandsemanticconsequencearearguablyincompleteoreveninaccuraterepresentationsofwhatitisforoneclaimtofollowfromothers.Derivationalconceptionsofconsequenceseemlimitedtothefinite,andthereisn’tanythingparticularlyexplicitinthe“therefore”ofordinarydeductiveargumentationthatconcernsrulesofproof,whilesemanticconceptionsofconsequenceseemtolosetheepistemiccastofthissame“therefore”,sinceonethingmightbethecaseifotherswereeveniftheirbeingsogaveone,eveninprinciple,noreasonatalltothinkthatitwasso.Therelationsamongtheintuitivenotionsofdeductivevalidity,ofderivation,andofsemanticconsequencewillbeamatterforfurtherinvestigationaswegoalong.However,whatevertheconsequencesofthatinvestigationmightbe,therefinementofthesetwotraditionsintoprooftheoryandmodeltheorywasamajorachievement.AlfredTarskiisrememberedbystudentsoflanguageforcontributionsthatmaderigoroustreatmentsofthecentralconceptsofthesecondofthesetwotraditionspossible.In“TheConceptofTruthinFormalizedLanguages”(CTFL)andrelatedpapersintheearly1930sTarskidevelopedthetoolsandtechniquesthatarestillatthecenteroflogic,thephilos-ophyoflanguageand,toalesserextent,linguisticsemantics.Indeed,Tarski’sachievementsinthisregardweresoinfluentialthattodayitisforgottenthatrepresentationalsemanticsashedevelopeditwasdevisedasacontributiontoacertainprojectmotivatedbytheexpressivecon-ception.StandardreportsonTarski’sworksimplytreatitasanobviousearlycontributiontorepresentationalsemantics.LurkinginthetextofCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01theclassicpapers,however,isadifferentproject,onetowhichrepre-sentationalsemanticswasatfirstintendedasasmallcontribution.Littleisknownaboutthisprojecttodayotherthanthatatoneplace[Tarski,1983a,62]Tarskireferstotheviewhewasworkingwithas“IntuitionisticFormalism”.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“INTRO”—2011/11/28—10:13—PAGE2—#2\nIntroduction3Inthisworkwewillexaminethedevelopmentoftruth-conditional,representationalsemanticsbyTarskifromthelate1920suntilthemid-1930s.Threebasicfociofattentionwillallowustofillinthesurroundingpicturefromwhichrepresentationalsemanticsisusuallyanachronisti-callyabstracted.First,wewillexaminewhatIntuitionisticFormalismwas,whereitcamefrom,andhowTarskiconceivedofhimselfascon-tributingtoit.Second,wewilllookatthedevelopmentofsemanticsfromitsbeginningswithinIntuitionisticFormalismtoitsreplacementofitin1935.Finally,wewilltraceTarski’streatmentoflogicalconse-quencethroughitsdevelopmentinhisthinking,foritwaswhenTarskirealizedthatsemanticsashehaddevelopeditcouldreplacetheconcep-tionofinferencerequiredbyIntuitionisticFormalismthatthelatterviewitselflostitsholdonhim.Ourstory,summarized,goeslikethis.Tarskioriginallybecameinter-estedinthequestionofhowourthoughtsandideas,underacertainconceptionofwhatthoseare,couldadequatelybeexpressedinanaxiomatictheoryor“deductivescience”,atopicthatwascentraltohisadvisorStanislawLe´sniewski’swork.AsIntuitionisticFormalismwascon-ceivedofbyLe´sniewskiasaconceptionofthefunctionandsignificanceofanaxiomatictheory,Tarskisethimselfthetaskofexploringhowthebasicconceptsusedinthoughtaboutaxiomatictheories—consequence,truth,referenceandrelatednotionssuchascompletenessandcategoric-ity(inseveralsenses)—couldthemselvesbecaptured,toIntuitionisticFormaliststandards,withinanaxiomatictheory.Inparticularduetohisinterestinearlyworkinwhatwewouldnowthinkofasmodeltheory,Tarskisetouttodevelopawayofcapturingthesemanticnotionsoftruth,satisfactionandreferencewithinsuchatheorytoIntuitionisticFormal-iststandards.Theresultofthisprojectwasthenow-familiarmethodofdefiningtruthbyrecursiononsatisfaction.Logicalconsequence,ontheotherhand,wasmuchslowertodevelopinTarski’shands.AsidefromaskepticalfootnotefollowingGödel[Tarski,1983a,252],aslateas1934Tarskitreatedlogicalconsequencederivation-ally,intermsofarecognizedsetofapparentlyvalidrulesforassertingsentencesgiventhatothershadbeenasserted.Onlywhenherealizedthatthenotionofamodelcouldbedefinedintermsofsemantics,andCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01consequenceinturndefinedintermsofthat,didTarskiseethatseman-ticscouldactuallystandonitsownasatreatmentoflanguage,andatthatpointhemovedonfromtheprojectthathadoriginallymotivatedhim:after1936IntuitionisticFormalistconcernsdisappearfromhisworkand,inparticular,1944’s“TheSemanticConceptionofTruth”,though10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“INTRO”—2011/11/28—10:13—PAGE3—#3\n4AlfredTarski:PhilosophyofLanguageandLogicitappearstosummarizetheworkofthe1930s,leavesoutthethemescharacteristicofIntuitionisticFormalism.0.2ThereceivedviewTarskimaywellbethemostenigmaticfigureinthehistoryofanalyticphilosophy.Althoughhisimportanceiswidelyaccepted,hisworkisrarelyactuallyreadandinterpretationsandassessmentsofhisviewsdif-ferwildly.Itwasthiswayfromthebeginning.PopperfirstlearnedofTarski’sinnovationswhenheaskedTarskitoexplainhiswork:andhedidsoinalectureofperhapstwentyminutesonabench(anunforgottenbench)intheVolksgarteninVienna.HealsoallowedmetoseethesequenceofproofsheetsoftheGermantranslationofhisgreatpaperontheconceptoftruth,whichwerethenjustbeingsenttohim…NowordscandescribehowmuchIlearnedfromallthis,andnowordscanexpressmygratitudeforit.AlthoughTarskiwasonlyalittleolderthanI,andalthoughwewere,inthosedays,ontermsofconsiderableintimacy,IlookeduponhimastheonemanwhomIcouldtrulyregardasmyteacherinphilosophy.Ihaveneverlearnedsomuchfromanybodyelse[Popper,1974,399].Carnaprecallshisowninitiation:WhenTarskitoldmeforthefirsttimethathehadconstructedadefi-nitionoftruth,Iassmedthathehadinmindasyntacticaldefinitionoflogicaltruthorprovability.Iwassurprisedwhenhesaidthathemeanttruthinthecustomarysense,includingcontingentfactualtruth.SinceIwasonlythinkingintermsofasyntacticalmetalan-guage,Iwonderedhowitwaspossibletostatethetruth-conditionforasimplesentencelike“thistableisblack”.Tarskireplied:“Thisissimple:thesentence‘thistableisblack’istrueifandonlyifthistableisblack”…WhenImetTarskiagaininViennaIurgedhimtodeliverapaperonsemanticsandonhisdefinitionoftruthattheInternationalCongressforScientificPhilosophytobeheldinParisinSeptember.ItoldhimthatallthoseinterestedinscientificphilosophyandtheCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01analysisoflanguagewouldwelcomethisnewinstrumentwithenthu-siasm,andwouldbeeagertoapplyitintheirownphilosophicalwork.[Carnap,1963,60–1].However,Tarskiwasskeptical,andrightlyso.Ontheonehand,that“thistableisblack”istrueifandonlyifthistableisblackcaneasilyseemsomethinglessthanapenetratinginsight;ontheotherhand,as10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“INTRO”—2011/11/28—10:13—PAGE4—#4\nIntroduction5Carnap’sreferencetosyntaxindicates,therewasatthetimeagreatdealofskepticismoroutrighthostilitytowardthetreatmentoflanguageintermsofword–worldrelations.ThelatterreactiondominatedthetheCongress,where“therewasvehementoppositionevenonthesideofourphilosophicalfriends”[Carnap,1963,61].Posterityhasn’tbeenmoreaccommodating,thoughthesourcesofresistancehaveshifted.ConsiderthesefamiliarassessmentsofTarski’sviews:Theconcernofphilosophyispreciselytodiscoverwhattheintuitivenotionoftruthis.Asaphilosophicalaccountoftruth,Tarski’stheoryfailsasbadlyasitispossibleforanaccounttofail[Putnam,1994,333].MyclaimisthatTarski’sanalysisiswrong,thathisaccountoflog-icaltruthandlogicalconsequencedoesnotcapture,orevencomeclosetocapturing,anypretheoreticconceptionofitslogicalproperties[Etchemendy,1990,6].Whatprovokedthesedisparatereactions?ThestandardconceptionofTarski’sviewsisrelativelyeasytosummarize.In“TheConceptofTruthinFormalizedLanguages”[Tarski,1983a](CTFL)Tarskisetshimselfthetaskof“definingtruth”andmaintainsthatagooddefinitionoftruthwouldbesomethingthat,addedtotheformalsyntaxofalanguage,resultsinimplicationofasentenceoftheform“sistrueifandonlyifp”(aswenowsay,a“T-sentence”)foreachsentenceofthelanguage,wherewhatissubstitutedfor“p”isortranslatess.Ifthisisthestandard,Tarskinotes,thenifalanguagehadafinitenumberofsentencesonecouldsimplylisttheT-sentencesandbedonewithit[Tarski,1983a,188].ThetechnicalachievementcomesinTarski’srecognitionthatifalanguagehasaninfinitenumberofsentencesandonedemandsafinitenumberofaxiomsinone’ssyntactic-cum-semantictheory,someuseof“therecursivemethod”[Tarski,1983a,189]isrequired.Furthermore,sincecomplexsentencesaren’tnecessarilybuiltoutofsentences(e.g.∃xFxisbuiltbyconcatenatingthequantifierwiththeopen“Fx”)thebasicrecursivedefinitionhastoconcernsomethingotherthanclosedsentences,butthenproduce,forclosedsentences,adefinitionwiththeT-sentencesasconsequences.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01ThedetailsherearefamiliarandsoIreferthereadertostandardpresentationssuchas[Soames,1999].Thebasicsareencounteredinanyadvancedundergraduatelogicclass.ConsideralanguageL,withnames,predicatesandfunctorsandvariablesofthefirstandperhapshigherordersandquantifierstobindthem,plussomestockofsententialconnectives.Syntactically,onesetsoutrulesforformingasentence10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“INTRO”—2011/11/28—10:13—PAGE5—#5\n6AlfredTarski:PhilosophyofLanguageandLogic(openorclosed)bystringingtogethernames,functors,predicates,vari-ables,quantifiersandconnectives;somenotationalbookkeepingisdonewithdeviceslikeparentheses,thedotsofthePeanoschool,ororderalone,asinPolishnotation,todeterminethingslikethescopeofaquantifierorthemainconnectiveofasentence.Therulessortstringsintoformulasandnon-formulas.Sincetheyarerecursive(e.g.onecanconjoinaconjunctionwithsomethingelseorapplyafunctortosome-thingthatcontainsfunctors)theresultisaninfinitesetofformulae,someopenandsomeclosed,whereanopenformulacontainsatleastonevariableboundbynoquantifier.Semantically,byTarski’stechniques,oneassignsdenotationstonamesandsetsoforderedn-tuplestopredicatesandtofunctors.Theformern-tuplesrepresentobjectsasstandingintherelationexpressedbythepredicateandthelatterhaveadistinguishedelement(thelast,say)thatexpressesthereferenceofthefunctorwhenitisappliedtothefirstn−1membersofthen-tupleasarguments.Truth-functionsareassignedtothesententialconnectivesintheobviousway.Variablesareenumeratedandobjectsinthedomainofdiscoursearetakentoformsequences—enumerationsofobjectstocorrespondtoenumerationsofvariables.AsentenceswitharelationsymbolRandnoboundvariablesissatisfiedbyasequencejustincasethen-tupleofobjectswithindicesthatmatchthoseofthefreevariablesinsisoneofthen-tuplesassignedtoR.Ifthesentenceinvolvesauniversalquantifierbindingvariablexiitissatisfiedbyasequenceifandonlyifeverysentencethatdiffersfromthatsequenceatmostthexthplacesatisfiestheopensentencefromwhichitwasformedibyaddingthequantifier[Tarski,1983a,193].Closedsentencesbeingthosewithnofreevariables,theyaresatisfiedbyeverysequenceiftheyaresatisfiedbyany,andsoatruesentenceisdefinedasonesatisfiedbyeverysequence[Tarski,1983a,194–5].Giventhatreferenceandsatis-factionhavethemselvesbeenintroducedbytranslationalenumeration(e.g.“‘Frankreich’refersinGermantoFrance”)theresultisanaccountthatimpliestheT-sentencesasdesired.Asthestandardconceptionnotes,Tarskiwasmotivatedbytheanti-nomyoftheliarandrelatedsemanticparadoxesandtookthemtobetheprimaryobstacletoarigoroustreatmentoftheconceptoftruth.TheseCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01leadhimtosaythatnolanguagecancontainapredicateapplyingtoexactlythesetof(Gödelcodesof)itsowntruthsandthatthereforethedefinitionoftruthforalanguagecanonlybegivenina“richer”meta-language.Theresult,onthestandardreading,isthe“Tarskihierarchy”inwhichthesemanticsforalanguagemustbegiveninmetalanguagethatcannotbetranslatedintoit,thesemanticsofwhichinturnmust10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“INTRO”—2011/11/28—10:13—PAGE6—#6\nIntroduction7bestatedinayethigher“meta-metalanguage”,andsoon.Sincenatu-rallanguagesappeartocontainself-applyingsemanticvocabularyandthisapparentlyviolatestherestrictionsimposedbythehierarchy,Tarskicondemnsthemas“inconsistent”andrejectsthemasunfitforrigorousstudy.Continuingwiththestandardnarrative,in1935–6Tarskiturnedhisattentiontotherelationoflogicalconsequence.Herehedefinesamodelofasetofsentencesasasequencethatsatisfiesallofthemandthendefinesanargument(
,s)asvalidjustincaseeverymodelof
isamodelofs.TherearesomeodditiesaboutTarski’spresentation,butitiswidelyheldtobemoreorlessthefirststatementoftheusualcontemporaryconceptionofmodel-theoreticconsequence.Aswehavenoted,reactionshavebeenextreme.NeuraththoughtthatTarskiwasrehabilitatingmetaphysicsandrejectedtheideathatthetruthofasentencecouldbeunderstoodintermsofworldlyconditionsratherthanitsconfirmationwithinasystemofscientificstatements[Mancosu,2008].LaterthinkershavequestionedwhetherTarskireallyprovideda“reduction”ofsemanticnotionstophysicalisticallyacceptableterms[Field,1972].InterpretershaveoftenobjectedtotheapparentrestrictionofTarski’s“definition”oftruthtoasinglelanguage.OthershavepilloriedTarski’sremarksonnaturallanguage.Tarski’sresponsetotheparadoxes,moregenerally,isthemainfoilforcontemporaryworkontheparadoxes,workthatattemptstoshowthat,contrarytohispronouncements,lan-guagesthatcontaintheirowntruth-predicatescanbeconstructed.AsforTarski’saccountofconsequence,thoughithasfounddefenders,manyagreewithEtchemendy’sassessmentquotedabove.0.3ThemesSomuchforthereceivedview.Forthosekeepingscore,IwillmostlydisagreewiththeusualobjectionstoTarski’sviewsontruthandparadoxandmostlyagreewiththeobjectionstoTarski’saccountofconsequence.Thedetailswillemergeasweprogress.ButkeepinmindthatmygoalhereistoexplainwhereTarski’sviewscamefromandhowtheyinformwhathewroteatthetimeratherthantodefendeverythingthathesaid.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Theoverallstoryhereisadevelopmentalone.Tarskibeginsinthelate1920sworkingwithinaconceptionoflanguageandlogicinheritedfromhismentorsStanislawLe´sniewskiandTadeuszKotarbi´nski.Hesetshimselfthetaskofworkingout“IntuitionisticFormalist”accountsofbasicmetatheoreticalnotions—inthefirstcase,theconsequencerelationconceivedofasdeterminedbyasetofprimitivelyacceptedrulesof10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“INTRO”—2011/11/28—10:13—PAGE7—#7\n8AlfredTarski:PhilosophyofLanguageandLogicinference[Tarski,1983c,63].Sometimein1929heturnshisattentiontosemanticnotionsanddevelopsthebasictechniquesdescribedabove,atfirstimperfectlyin[Tarski,1983f]andtheninfullformin[Tarski,1983a].Duringthisperiodhedoesn’tthinktoapplythetechniquestothecon-sequencerelation,nordoeshenoticeotherobviousapplications,e.g.toPadoa’smethodforprovingtheindependenceofasetofaxioms.TheseomissionsareimportantbecausetheyshowtheextenttowhichTarskiisstillthinkingsquarelyintermsofIntuitionisticFormalism.Sometimein1934–5Tarskirealizedthathecouldreplacetheappealtoprimitivelyvalidinferenceruleswithanotionofconsequencedefinedintermsofhissemantics.Theresultistheaccountof[Tarski,2002]andwiththattheprojectcametoanend.Asabitofapreview,herearesomethemesthatwewillbeexploringaswemovealong.1.IntuitionisticFormalism:WhatwasitandhowdiditinformTarski’sworkfrom1926–1936?2.Definition:WhatwasTarski’sconceptionofdefinition?Howdohisdefinitionsrelatetowhatphilosophersofferwhenthey“analyze”concepts?3.Language:WhatwereTarski’sviewsonLanguage?WhatwastheimplicitaccountofmeaningandcommunicationwithwhichTarskiworked?4.Semantics:WhatmotivatedTarski’sprojectofdefiningsemantictermsinmathematicaltheories?Howdidheunderstandwhathewasdoingatthetime?5.Compositionality:IntuitionisticFormalismnotonlylackedanyartic-ulateaccountofhowthemeaningsofacomplexexpressionweredeterminedbythoseofitsparts—somethingrequiredofanyrea-sonabletreatmentoflanguage—butinTarski’shandsitinvolvedanaccountofmeaningthatwasincompatiblewithcompositionality.WewillseehowTarski’ssemanticsbeginstosupplythis.Oneoftheaspectsofthisshiftisthemovefroma“holist”accountofmeaningonwhichtheoriesdeterminedthemeaningoftheirsentencesandexpressions,toan“atomist”onewhichthedeterminationrunsfromCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01lexicalexpressionstosentences.6.Consequence:IntuitionisticFormalisminvolvedaconceptionoflogi-calconsequenceonwhichitwasamatterofprimitivelyvalidrulesofinference.Tarski’ssuccessorconceptiondetachesconsequencefromthisrequirement.TheresultisthatTarskimovesfromaconceptionofconsequencethatplacesitsepistemicaspectsattheforefrontto10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“INTRO”—2011/11/28—10:13—PAGE8—#8\nIntroduction9onethatstressestheindependenceoflogicfromthesemanticsofnon-logicalterms.Callthecontrastonebetween“epistemic”and“generality”conceptionsofconsequence.Whetherthisislaudableissomethingwewillconsideratlength.7.LogicalMonismandPluralism:TarskibeganhisworkacceptingthatSimpleTypeTheory(STT),conceivedofintermsofthe“theoryofsemanticalcategories”heattributedtoLe´sniewskiandothers,waslogic.UndertheinfluenceofCarnapinparticular,butalsopayingattentiontootherdevelopments,hemovedtoamorepluralistor,asCarnapwouldputit,“tolerant”attitude.8.TheStatusofMathematics:TarskiinheritedaliberalPolishattitudeaboutthestatusofmathematics.Thoughhenowheretakesthelogi-cistprogramasamotivation,agooddealofhisviewsassumethatmathematics,totheextentthatitcanbedevelopedinSTT,islogic.Later,especiallybeginningwith[Tarski,2002]amoreopen-mindedattitudeaboutlogicandmathematicscametothefore.Those,then,aresomeofthethemesthatwewillexplore.Thatsaid,Iowethereadersomewordsaboutwhatwillnotbecoveredinthisbook.OurtopicisTarski’sevolvingsetofviewsaboutlogicandlanguage,andinparticular“formalaxiomatics”intheperiod1926–1936.Tarski’smathe-maticalworkfromtheperiodwillnotbecovered.1NorwillIhavespacetodiscussTarski’sviewsafter1936,withtheexceptionof“TheSemanticConceptionofTruth”,publishedin1944butlargelyaresponsetothedebatethatarosefollowinghispresentationof“TheEstablishmentofScientificSemantics”(ESS)and“OntheConceptofFollowingLogically”(CFL)attheParisUnityofScienceCongressin1935.2Asforothersources,inordertostaywithinthepublisher’slengthlimitIhavesetmyselfthebasicruleofstickingtofiguresandworksTarskihimselfmentions,withonlyafewexceptionswhentheseseemedespe-ciallypertinent.Missing,inparticular,arethediscussionsofRussellandFregethatonemightexpectinaworkonthehistoryofanalyticphilos-ophy.TarskiwasinapositiontohavereceivedatleastsomeFregefromLe´sniewski,butFregeplayednodirectroleinhisthought.RussellseemsCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01relevantatthemanyplaceswhereTarskimentionsPrincipiaMathematica,butsinceTarskimentionsitonlytosetitasideinfavorofSTT,Ifoundlit-tleofuseinbringingRussellintoourdiscussion.Wittgensteinwillfigureintoourstoryonlyindirectly,asfilteredthroughCarnapandtheViennaCircle.Onecomparisonseemedtoointerestingtometoleaveout,andsoitoccursinfoonotes:thatofTarski’sworkwithcontemporaneouswritingsofhisunfortunatelyneglectedcolleagueKazimerzAjdukiewicz.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“INTRO”—2011/11/28—10:13—PAGE9—#9\n10AlfredTarski:PhilosophyofLanguageandLogicTherearealsosometopicsthatstudentsofTarskiinparticularmightexpecttoseediscussedmorethantheywillbeinthisbook.First,Igivethedistinctionbetweenlanguageandmetalanguagelessattentionthanitoftenreceivesintheliteraturesinceitisperfectlyclearandandtherearen’tanysignificantissuesconcerningit;interestedreaderscanconsult[Wole´nski,1989]and[Betti,2004,280ff]onLe´sniewski’sdevelopmentofthedistinction.Second,IspendonlyafewwordsonthenominalismofLe´sniewskiandKotarbi´nski,despitethefactthatTarskiinprivatedis-cussionsexpressedsympathyforitthroughouthislife,asiswellknown;here,too,interestedreaderscanbeginwith[Wole´nski,1989].OthershavewrittenonthetopictoaconsiderableextentbutinmyresearchIcametotheconclusionthatitisoflittleimportanceforourmainthemes.Thisbookissomethingofanexerciseinphilosophicaldetectivework.Tarskiisknownforhavingbeenreticentinhisexpressionofphilosoph-icalcommitments.Evenintheovertlyphilosophicallymotivatedworkshisviewsaredifficulttodiscern.ThemethodIhavefollowedisanobvi-ousone:figureoutwhattheviewsofhisteacherswereandthenfindsufficientsignsofthoseviewsinwhatTarskididsaythatwecanfeelcomfortablefillinginthegapswiththoseviews.Theproofisintheinterpretationsofferedoffamiliarpassages.Itrytomakeagoodcaseformyreadinginchapters4,5and7.Afinalwordontextualinterpretationisinorder.Meticulousashemayhavebeeninlogicandmathematics,Tarskimakesnoefforttoimposeorderonhisuseofphilosophicallyloadedterminology.Theeagerinter-preterhopingtomakesomethingofTarski’svaryingusesof,inGerman,Sinn,BedeutungandInhalt,forinstance,isquicklydisappointed.Thoughtheworkofthe1930spurportstobeconcernedwith“concepts”by1944,innote4toSCT[Tarski,1986e,697],Tarskisaysthathehasatendencytoavoidthewordin“exactdiscussion”,sinceonitsprevalentusageitsometimesreferstoextensions,sometimestomeanings,andsome-timestowordsthemselves.Hemusthavehadhisearlierselfinmindinthispassage,becausealloftheseusagescanbefoundintheworkofthe1930s.Despitethefactthathisviewssometimesdemanddistinc-tionswithintherangeofcoverageoftheseandotherphilosophicallyfraughtterms,invariouspassagesTarskiusesoneortheotherofthemCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01indifferently;e.g.inoneplace[Tarski,1983k,311][Tarski,1986b,647]Tarskitalksaboutanaxiomsystemdeterminingthe“senses”(Sinn,sin-gular:theEnglishtranslationsubstitutestheplural)ofthe“concepts”(Begriffe)it“contains”;severallineslater“concepts”istradedinfor“terms”(Zeichen).Thesedifficultiesarethencompoundedbyvariousissueswiththetranslations—forexampletheonejustnoted.Oftenthis10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“INTRO”—2011/11/28—10:13—PAGE10—#10\nIntroduction11doesn’tmatter,butinsomecasesitdoes,asintheattempttodeterminewhetherTarskiheldthattherewasoneconceptoftruththatappliedtomanylanguages,orratherforeachlanguageitsownconceptoftruth.WhiletryingtobeasfaithfulaspossibletothetextsIsometimesallowmyconfidenceintheoverallreadingtotrumpconsiderationsofword-ing.Thisisafamiliardilemmafacingthehistorianofphilosophy.IhopethatIhavehandleditwell.3Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-0110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“INTRO”—2011/11/28—10:13—PAGE11—#11\n1IntuitionisticFormalism1.1WhatwasIntuitionisticFormalism?1.1.1ApuzzleaboutconceptsanddefinitionsTarskibegins“TheConceptofTruthinFormalizedLanguages”thus:Thepresentarticleisalmostwhollydevotedtoasingleproblem—thedefinitionoftruth.Itstaskistoconstruct—withreferencetoagivenlanguage—amateriallyadequateandformallycorrectdefinitionoftheterm‘truesentence’[Tarski,1983a,152].Despitethisdesiretoconstructadefinition,Tarskialsoinsiststhathewillnotbe“analyzing”themeaningoftheword“true”:Athoroughanalysisofthemeaningcurrentineverydaylifeoftheterm‘true’isnotintendedhere.Everyreaderpossessesingreaterorlessdegreeanintuitiveknowledgeoftheconceptoftruthandhecanfinddetaileddiscussionsonitinworksonthetheoryofknowledge[Tarski,1983a,153].Thisraisesaquestion.Ifthedefinitionisn’tintendedtoprovide“athoroughanalysisofthemeaningoftheterm‘true’”,whatisitfor?Furthermore,noticethatwhatthispassagesaysisthatTarskiwillnotbeanalyzingthemeaningof“true”ortheconceptoftruth;hisworkCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01assumesthatthisanalysisisalreadyavailableandcanbefound“inworksonthetheoryofknowledge”—inparticular,welearninafoot-note,Kotarbi´nski’sElementy[Kotarbi´nski,1966].SowhateverTarskiisdoingin“constructingamateriallyadequateandformallycorrectdefi-nitionoftruth”,heisnotengagedin“analysis”ofthesortundertakenbyphilosophers.1210.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE12—#1\nIntuitionisticFormalism13ThatTarskihasnointentionofdoingwhatphilosophersdowhenthey“define”or“analyze”conceptsisborneoutbythedefinitionsheactuallygives.TheneteffectofDefinitions22and23oftheworkiscertainlynottoincreaseourconfidenceinsomeantecedentgraspwemighthavetakenourselvestohaveontheconceptoftruth:Definition22.Thesequencefsatisfiesthesententialfunctionxifandonlyiffisaninfinitesequenceofclassesandxisasententialfunctionandiffandxaresuchthateither(α)thereexistnaturalnumberskandlsuchthatx=ιk,landfk⊆fl;(β)thereisasententialfunctionysuchthatx=y¯andfdoesnotsatisfythefunctiony;(γ)therearesententialfunctionsyandzsuchthatx=y+zandfeithersatisfiesyorsatisfiesz;orfinally(δ)
thereisanaturalnumberkandasententialfunctionysuchthatx=kyandeveryinfinitesequenceofclasseswhichdiffersfromfinatmostthek-thplacesatisfiesthefunctiony[Tarski,1983a,193].Definition23.xisatruesentence—insymbolsx∈Tr—ifandonlyifx∈Sandeveryinfinitesequenceofclassessatisfiesx[Tarski,1983a,195].ThisishardlysomethingofwhichTarski’sreadershad“intuitive”knowl-edgebeforehandanditcan’tbefounddiscussedinworksonthe“theoryofknowledge”priortoTarski’swork.Italso,famously,tiesthedefinedtermtothespecificfeaturesofTarski’s“languageofthecalculusofclasses”;yetsurelyitwasnopartoftheconceptoftruthofwhichreaderscouldhavehadintuitiveknowledgethatsentencesofacertainfirst-ordertheoryofthesubsetrelationaretrueasadegeneratecaseofthesatisfac-tionrelation.Whateverthisdefinitionisfor,itcan’tinanyfamiliarsensebean“analysis”oftheconceptoftruth.Soinwhatsense,then,istheworkconcernedwith“theconceptoftruthinformalizedlanguages”?IntroducingthedefinitionoftruthisnottheonlytopicthatmovesTarskitoinsistthatthe“intuitivemeaning”or“conceptexpressedby”somedefiniendumisinsomewayperfectlyclearasapreludetogivingadefinitionthatseemsafarcryfromsomethingthatanalyzesthissup-posedlyclearconcept.Tarski’s1930paperondefinability,inthesenseofthedefinitionofasetbyapredicate,ringsasimilartheme:…wecantrytodefinethesenseofthefollowingphrase:‘AfiniteCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01sequenceofobjectssatisfiesagivensententialfunction.’Thesuccess-fulaccomplishmentofthistaskraisesdifficultieswhicharegreaterthanwouldappearatfirstsight.However,inwhateverformandtowhateverdegreewedosucceedinsolvingthisproblem,theintu-itivemeaningoftheabovephraseseemsclearandunambiguous(1983,117).10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE13—#2\n14AlfredTarski:PhilosophyofLanguageandLogicThisthemeofrenderingintuitionsprecise,relatedtouseofthephrase“intuitivemeaning”alsooccursearlierinthesamearticle:Theproblemsetinthisarticlebelongsinprincipletothetypeofproblemswhichfrequentlyoccurinthecourseofmathematicalinves-tigations.Ourinterestisdirectedtowardsatermofwhichwecangiveanaccountthatismoreorlesspreciseinitsintuitivecontent(con-tenuintuitif),butthesignificance(signification)ofwhichhasnotatpresentbeenrigorouslyestablished,atleastinmathematics.Wethenseektoconstructadefinitionofthistermwhich,whilesatisfyingtherequirementsofmethodologicalrigour,willalsorenderadequatelyandprecisely(avecjustesseetprécision)theactualmeaning(signification“trouvée”)oftheterm…Ishallbeginbypresentingtothereaderthecontent(contenu)ofthisterm,especiallyasitisnowunderstoodinmetamathematics.TheremarksIamabouttomakearenotatallnecessaryforthecon-siderationsthatwillfollow—anymorethanempiricalknowledgeoflinesandsurfacesisnecessaryforamathematicaltheoryofgeome-try.Theseremarkswillallowustograspmoreeasilytheconstructionsexplainedinthefollowingsectionand,aboveall,tojudgewhetherornottheyconveytheactualmeaning(signification“trouvée”)oftheterm[Tarski,1983f,112]([Tarski,1986g,519]).Thepassagehasmanynoteworthyfeatures.Inthefirstparagraphwelearnthat“definablesetofrealnumbers”hasanintuitivecontent,a“foundmeaning”(Woodgerrenders“trouvée”as“actual”andunhelp-fullydropsthescare-quotes)whichmightbe“rigorouslyestablished”inmathematicsbutsofarhasnotbeen.Thepictureappearstobeoneinwhichameaninggraspedoutsideofarigorousaccounthastobeimportedintoorexpressedinaformalism.Onthissametopic,thesubsequentsen-tencetellsusthatthetaskathandisto“renderadequatelyandprecisely”this“found”meaning,againsuggestingthattheissueathandishowtoensurethataformalismadequatelycapturesacertainknownmeaning.However,aswithCTFL,thedefinitionTarskiactuallygives[Tarski,1983f,128]doesn’tseemtohaveanythingtodowiththeintuitivemeaningofthedefinedterm.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Thequotationtellsusthattheformaltheorythatisconstructedinordertoexpressintuitionsisindependentoftheminthatitcanbeunderstoodevenbysomeonewholackstheintuitionsorlacksknowl-edgethatthetheoryissupposedtoexpress.Itisan“external”purposeofaformaltreatmentofsomethingtocaptureanintuitivecontent;theresultingbitofmathematicscanbeunderstoodindependently,butour10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE14—#3\nIntuitionisticFormalism15purposeinsettingitoutistoexpressourintuitions.ThisalsogivesusabitmoreinformationabouthowTarskiisseeingtherelationshipbetweenan“intuitivecontent”anditsformalexpression.Tarskiannouncesthathewillgiveusa“presentation”ofthecontentof“definableset”.Presum-ablysuchapresentationissomethinglikeaconceptualanalysis—Tarskiwillmakesomeclaimsaboutthe“found”meaningoftheterm.Thisanalysiswillthenguidetheformalprojectofdefinition.ThisstructureisthesameastheonewehavealreadyseeninCTFL:thereisaconceptthatissupposedtobesomehowencompassedbyadefinition,thedefi-nitionitself,andbetweenthemthereisaconceptualanalysisofamoretraditionalsortthatinformsorguidesthesettingoutofthedefinition.Sotheintuitivenotionsareimportant,buttheformalsystemandtheintuitivenotionsarerelativelyindependent.Ontheonehand,asnoted,thedefinitionandtheformaltreatmentofwhichitisapartcanbeunderstoodevenbysomeonewholackstheintuitivenotionthattheformaltreatmentcaptures.Ontheother,intuitivecontentisindepen-dentenoughofits“rigorousexpression”thatitexistsandcanbegraspedwithoutit.Although“Everyreaderpossessesingreaterorlessdegreeintuitiveknowledgeoftheconceptoftruth”nevertheless:In§1colloquiallanguageistheobjectofourinvestigations.Theresultsareentirelynegative.Withrespecttothislanguagenotonlydoesthedefinitionoftruthseemtobeimpossible,buteventhecon-sistentuseofthisconceptinconformitywiththelawsoflogic[Tarski,1983a,153].ItissignificantherethatTarskisaysnotwhatheisoftenpresentedassaying[Soames,1999,51],thattheordinaryconceptoftruthitselfissomehow“inconsistent”orotherwiseincoherent,butratherthatitcan-notbe“usedinconformitywiththelawsoflogic”inapplicationto“everyday”language.SinceadefinitionofthesortTarskiseekscannotbeprovidedforcolloquiallanguage,but“everyreader”possesses“intuitiveknowledge”oftheconceptoftruth,graspoftheconceptisindependentofthesortofdefinitionTarskiwantstoprovide.Clearly,therelationbetweenTarski’sdefinitionsofsemantictermsand“conceptualanalysis”oftheusualsortisn’t,asonemightexpect,thatCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01thedefinitionsimplycapturesananalysisconceivedofassomethinglikeasetofclaimsthatanyonewhograspstheconceptmustaccept.Summingupwhatwehaveseensofar,onTarski’sviewconceptsareindependentofthetermsthatexpressthemandare“intuitively”grasped.Definitionswithinrigorousmathematicaltreatmentsaresup-posedtoaidtheexpressionofintuitivelygraspedconceptsorrenderthe10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE15—#4\n16AlfredTarski:PhilosophyofLanguageandLogic“foundmeaning”ofordinaryterms.Betweenaconceptandadefinitionofatermthatexpressesitwithinaparticulartheoryliessomethinglikeconceptualanalysistraditionallyconceived.Finally,conceptsandrigorousdefinitionsareindependentenoughthateachcanbegraspedwithouttheother;onecanunderstandaformalismintendedtoexpressaconceptwithoutgraspingtheconceptorthefactthattheformalismissupposedtocaptureit,andonecangrasptheconceptwithoutaccesstoanydefinitionofTarski’ssort.LurkinginthebackgroundofTarski’sproject,then,appearstobearatherarticulateviewofconcepts,mean-ings,definitions,formalism,ordinarylanguage,andrelatedtopics—asetofviewsinthephilosophyoflogic,languageandtosomeextentmind.InordertounderstandTarski’sworkweneedtoknowwhattheseviewswere.1.1.2Tarski,LesniewskiandIntuitionisticFormalism´IntheintroductiontoanearlyarticleontheconsequencerelationTarskiexpressesallegiancetoadoctrineheattributestoLe´sniewski:Inconclusionitshouldbenotedthatnoparticularphilosophicalstandpointregardingthefoundationsofmathematicsispresupposedinthepresentwork.Onlyincidentally,thereforeImaymentionthatmypersonalattitudetowardsthisquestionagreesinprinciplewiththatwhichhasfoundemphaticexpressioninthewritingsofS.Le´sniewskiandwhichIwouldcallIntuitionisticFormalism[Tarski,1983c,62].ThemaincontentionofthesefirsttwochapterswillbethattheaspectsofTarski’sviewjustenumeratedareaspectsofIntuitionisticFormalismasheconceivesofit.Notethatthispassagesharessomethingwiththepassagefrom“OnDefinableSets”thatwediscussedintheprevioussubsection:Tarskiinsiststhatcertainmore“philosophical”viewsarenotstrictlyrelevanttoourunderstandingofcertainformalwork,yetfindsthephilo-sophicalviewsworthmentioninganyway.Unfortunately,theresultofthisisthatTarskitellsusalmostnothingaboutwhatthe“IntuitionisticFormalist”believes.WethusturnforilluminationtothepassagefromLe´sniewski’sCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01GrundzügeeinesneuenSystemsderGrundlagenderMathematikthatTarskicites.Itbearsmentioningthatthecitedpassageisprecededby15pagesofformalsyntaxinLe´sniewski’smeticulousversionleadinguptothestate-mentofthe“directives”(rulesofinferenceanddefinition)forSS5,oneofthemanyversionsof“Protothetic”,Le´sniewski’ssystemofsententiallogic,discussedinthearticle:10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE16—#5\nIntuitionisticFormalism17PerhapsIshouldaddthatformanymonthsIspentagreatdealoftimeworkingsystematicallytowardstheformulationofthesesystemsofPrototheticbymeansofaclearformlationoftheirdirectivesusingthevariousauxiliarytermswhosemeanings(Bedeutungen)Ihavefixedintheterminologicalexplanationsgivenabove.Havingnopredilec-tionforvarious‘mathematicalgames’thatconsistinwritingoutaccordingtooneoranotherconventionalrulevariousmoreorlesspicturesqueformulaewhichneednotbemeaningfuloreven—assomeofthe‘mathematicalgamers’mightprefer—whichshouldnecessarilybemeaningless,Iwouldnothavetakenthetroubletosystematizeandtooftencheckquitescrupulouslythedirectivesofmysystem,hadInotimputedtoitsthesesacertainspecificandcompletelydeterminedsense(Sinn),invirtueofwhichitsaxioms,definitions,andfinaldirec-tives(asencodedforSS5),haveformeanirresistibleintuitivevalidity(intuitiveGeltung).Iseenocontradiction,therefore,insayingthatIadvocatearatherradical‘formalism’intheconstructionofmysys-temeventhoughIamanobdurate‘intuitionist’.Havingendeavoredtoexpresssomeofmythoughtsonvariousparticulartopicsbyrepre-sentingthemasaseriesofpropositionsmeaningful(sinnvollerSätze)invariousdeductivetheories,andtoderiveonepropositionfromoth-ersinawaythatwouldharmonizewiththewayIfinallyconsideredintuitivelybinding(welcheIch“intuitiv”alsfürmichbindendbetra-chte),Iknownomethodmoreeffectiveforacquaintingthereaderwithmylogicalintuitions(“logischenIntuitionen”)thanthemethodofformalizinganydeductivetheorytobesetforth.Bynomeansdotheoriesundertheinfluenceofsuchaformalizationceasetoconsistofgenuinelymeaningfulpropositionswhichformeareintuitivelyvalid[Le´sniewski,1992e,487][Le´sniewski,1929,78].Note,first,thatLe´sniewskispeaksofSätzebeingmeaningful“in”adeduc-tivetheory.Aswewillsee,thisideaofasentencebeingmeaningful“inatheory”isfarfromidle.Second,likeTarski’stranslatorWoodger,Le´sniewski’stranslatorO’Neilhasabadhabitofdroppingscare-quotes.Le´sniewskifindstheappealtointuitionproblematic,butalsofindsthathehasnootherwaytosaywhathewantstosay.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Le´sniewskiherecontrastsacertainparodyofformalism(acaricatureheassociatedwithHilbert;seee.g.[Le´sniewski,1992h,195])withhisownapproachwhich,thoughitinvolvesformalization,aimsattheexpressionof“intuitivelyvalid”thoughts.Twocentralelementsoftheconceptionexpressedinthepassagearetheusageof“formulas”inaccordwith“conventionalrules”andtheexpressionofthoughts.Lesniewski10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE17—#6\n18AlfredTarski:PhilosophyofLanguageandLogicinsiststhatthepointoftheuseofformulasinaccordwithrulesistoexpressthoughts,andalsothatdesigningtherulessothattheyade-quatelyexpressthethoughtsisapainstakinglabor.The“formulas”inadeductivesystem,then,areassignedtworoles:theystandinrelation-shipstootherformulasinaccordwithconventionalrulesand,atleastwhenthingsgowell,theyexpressthoughts.Sofar,then,thisplacesLe´sniewskisquarelywithinthe“expressivesemantics”campdiscussedintheintroduction:asidefromlanguage–languagerelations,Le´sniewski’saccountofwhatIntuitionisticFormalismisaboutherementionsonlylanguage–mindrelationsasthelocusoflinguisticsignificance.ThisissofarconsonantwithwhatwehavefoundinTarski.ForLe´sniewskiadeductivescienceconsistsofsymbolsgovernedbyrulesorconven-tions,buttheformulas,usedinaccordwiththerules,aresupposedtoexpressthoughts—justasforTarski,ontheonehand,wehavesomethingintuitiveandclearthatissupposedtobeexpressedwithinadeductivescience,andontheotherhandwehavethedeductivescienceitself,whichmaybequitefastidiouslyconstructed,andconstructedinawaythatseems,atleasttotheuninitiated,tolacktheovertclaritythattheintuitiveconceptsalreadyhave.Onethingtonoticeinthepassageistheconceptionofinferenceitoffers.Le´sniewskisaysthatthe“intuitionistic”aspectofhis“formal-ism”consistsinpartinthefactthathe“deducesonesentencefromothersentencesinamannerthatharmonizeswiththeinferencesheconsidersintuitivelybinding”.ThismeansthatacentralplankofIntu-itionisticFormalism,inadditiontotheintuitive“validity”(Geltung—thisbeingacommontermatthetimeforthestatusoftheoremsamongthose,e.g.[Carnap,1929],whohadobjectionstosimplycallingsuchthings“true”)ofaxiomsanddefinition—istheintuitivevalidity(intheusualsense)oftheinferencesamongthesentencesofadeductivethe-ory.Le´sniewskidemandsthatderivationsbe“intuitivelybinding”.Ruledouthere,then,isaconceptionofvalidityorlogicalconsequencethatdoesn’tinvolvesuchanepistemicaspect.AcentralaspectofthedoctrinetowhichTarskiswearsfealtyin1930,then,isanepistemicconceptionofconsequence.Aswewillseeinsubsequentchapters,TarskiintroduceshistreatmentCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01ofsemanticconceptsasacontributiontoIntuitionisticFormalism.ButsincetheintroductionofthesemanticnotionstakesplacewithintheframeworkofIntuitionisticFormalism,ittakesTarskiagooddealoftime—allthewayuntil1935—toventuretoreplaceIntuitionisticFor-malism’sepistemicconceptionofinferencewithageneralistconceptionbasedinhissemantics.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE18—#7\nIntuitionisticFormalism191.1.3FormalismSinceIntuitionisticFormalismwassupposedlyakindofformalism,weshouldremindourselvesofwhattherelevantauthorstowhomLe´sniewskiandTarskialludeorreferhadtosay.TurningtoHilberthim-self,wefindremarkslikethisone,fromtheprogrammatic“MathematicalProblems”of1900,totheeffectthatthemeaningofanexpressionissomehowdeterminedbytheaxiomsinwhichitfigures:1Whenweareengagedininvestigatingthefoundationsofascience,wemustsetupasystemofaxiomswhichcontainsanexactandcompletedescriptionoftherelationssubsistingbetweentheelementaryideasofthatscience.Theaxiomssosetupareatthesametimethedefinitionsofthoseelementaryideas;andnostatementwithintherealmofthesciencewhosefoundationswearetestingisheldtobecorrectunlessitcanbederivedfromthoseaxiomsbymeansofafinitenumberoflogicalsteps.[Ewald,1996,1104]Detlefsen,inhissurvey,characterizestheformalistpositionasfollows:Themathematician,onthisview,isfreetostipulateofaconceptshe’sintroducingthatithaveexclusivelythepropertiesprovidedforbytheaxiomssheusestointroduceit.Thereisnocontentbelongingtoaconceptintroducedinthiswayexceptthatwhichisprovidedforbytheintroducingaxioms.Theaxiomsusedtointroduceaconceptthusguaranteedlytakeonakindofcompleteness—whatmightbecalledconstitutivecompletenessbecause,together,theyconstitutethe‘content’(i.e.theroleinreasoning)oftheconcept.Conceptsarethusidentifiedwiththerolestheyoccupyinmath-ematicalthinking(denken).Theydonothavetohaveanintuitivecontenttobesignificant.Hilbertthusbelievedthattheinferen-tialrolesofconceptsaredeterminednotbycontentsgivenpriortotheaxiomswhichintroducethem,butbythoseintroducingaxiomsthemselves.Indeed,allreasoningconcerningaconceptisrestrictedtothatwhichisprovidedforbyitsintroducingaxioms[Detlefsen,2008,294–5].Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01WhatonecannotfindinthisrenditionofformalismisanythinglikeLe´sniewski’sinsistenceontheexpressionof“intuitivelyvalid”thoughtsandthelike;indeed,ascharacterizedbyDetlefsen,thewholepointoftheviewistobeindependentofsuchthings.2OnecanseethisviewinactioninalaterworkbyCarnapthatcametobeimportanttoTarski,LogicalSyntax:10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE19—#8\n20AlfredTarski:PhilosophyofLanguageandLogicUptonow,inconstructingalanguage,theprocedurehasusuallybeen,firsttoassignameaningtothefundamentalmathematico-logicalsymbols,andthentoconsiderwhatsentencesandinferencesareseentobelogicallycorrectinaccordancewiththismeaning.Sincetheassignmentofthemeaningisexpressedinwords,andis,incon-sequence,inexact,noconclusionarrivedatinthiswaycanverywellbeotherwisethaninexactandambiguous.Theconnectionwillonlybecomeclearwhenapproachedfromtheoppositedirection:letanypostulatesandanyrulesofinferencebechosenarbitrarily;thenthischoice,whateveritmaybe,willdeterminewhatmeaningistobeassignedtothefundamentallogicalsymbols[Carnap,2002,xv].ThisisquitetheoppositeofLe´sniewski’sview;ratherthantheintuitivecontentthatliesoutsideofthedeductivesystembeingtheimportantthing,thedifficultyofdeterminingitrigorouslyisforCarnapinthepas-sagepreciselyareasontogetridofit.However,weshouldrememberherethatinpassageswehavealreadylookedatTarskisayssomethingsimilar:theintuitivecontentofhisdefinedtermsisofinterest,buthisdefinitionsandthetheoriestowhichtheyareaddedcanalsobeunderstoodinde-pendentlyofit.Nevertheless,forTarskiandLe´sniewskiextra-systemicintuitivemeaningdrivestheconstructionofadeductivetheory.1.2Lesniewski´1.2.1Lesniewski’searlywork´WhenweturntoLe´sniewski’searly(1911–1914)papers,wefindsomemuchmorearticulatecommentsonaviewwhichisinagreementwiththeremarksinthepassageTarskicites.Anappealtotheearlyworksisproblematic,however,sinceLe´sniewskilaterrepudiatedthem[Le´sniewski,1992h,197–8].Nevertheless,torepudiateone’sjuveniliaisnottogiveupeveryviewexpressedinit,andonethingthatremainedinthelaterperiod,Iwillargue,isIntuitionisticFormalism.3AfterlookingattheevidencefromtheearlyworkIwillconfirmthisview,andbolstermyattributionofittoTarski,intwoways:firstbychasingdownitstracesinLe´sniewski’slaterpapers,andsecondbyfindingexactlythesameviewinCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01themainworkofTarski’sothermentor,TadeuszKotarbi´nski,publishedin1929—theyearinwhichLe´sniewskipublishedthepassageTarskicites,andinwhichTarskididthebulkoftheworkonhisstrategyfordefiningtruth[Hodges,2008,120–5].ThemostillustrativepassagefromLe´sniewski’searlypaperscomesfromhis1911discussionof“existentialpropositions”,thetopicofhis10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE20—#9\nIntuitionisticFormalism21dissertation[Wole´nski,1989,8].Havingarguedforaseriesofratherradicalclaims(e.g.thatallpositiveexistentialsareanalytic)hesug-geststhathisviewsappearmoreparadoxicalthantheyarebecausepeopleareconfusedabouthowtoexpressthoughtsclearly:itissim-plyaconfusion,forinstance,tothinkthat“peopleexist”istherightwaytoputthethoughtoftenexpressedbyit;betterforthispurposeis“somebeingsarepeople”[Le´sniewski,1992b,15].Sincesayingthiscom-mitshimtoviewsabouttherightwaytoexpressthoughts,Le´sniewskicontinues:Ihavediscussed,inthissection,theadequateandinadequaterep-resentationofvariouscontentsbymeansofvariouspropositions.Anobviousquestionmightthereforearise,whatismycriterionofadequacyorinadequacyoftherepresentationofgivencontentsinaproposition…Thesymbolicfunctions(funkcjesymboliczne)ofcomplexlinguisticconstructions,e.g.,propositions,dependonthesymbolicfunctionsoftheelementsoftheseconstructions,thatisindividualwords,andontheirmutualrelationship.Intheunplannedprocessofdevel-opmentoflanguage,thesymbolicfunctionofpropositionscandependinsomeparticularcasesonidenticalsymbolicfunctionsandonidenticalrelationshipbetweenspecificwords—inquitedif-ferentways.Theplannedconstructionofcomplexlinguisticformscannot,forrepresentingvariouscontents(tresci´)inthesystemoftheoreticalpropositions,beconfinedwithinthepossibleresultsoftheunplannedevolutionoflanguage.Suchconstructioncallsfortheformationofcertaingeneralconventional-normativeschemata(schematówkonwencjonalno-normatywnych)toembodythedependenceofthesymbolicfunctionofpropositionsonthesymbolicfunctionsoftheirelements,andonthemutualrelationshipbetweentheseele-ments.Toascertainwhetherthegivencontenthasbeenrepresentedadequatelyorinadequately(adekwatnoscilubnicadekwatno´sci´)inaproposition,onehastoanalyzeindividuallyhowthespeaker’srep-resentationalintentions(intencjisymbolizatorskichmówika˛cego)relatetotheabove-mentionedschemas.TheseschemasshouldindicateinCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01whatwaythesymbolicfunctionofapropositionshouldbecon-ditionedbythesymbolicfunctionsoftheparticularwordsandbytheirmutualrelationship.ToexpressthisdependenceIhaveadoptedthefollowingnormativeschema:everypropositionistorepresent(symbolizowac´)thepossessing,bytheobjectdenotedbythesub-ject,ofthepropertiesconnotedbythepredicate.Myassertionthata10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE21—#10\n22AlfredTarski:PhilosophyofLanguageandLogicgivenpropositiondoesnotadequatelyrepresentthegivencontentsisalwaysbasedonthefactthatthepossessionbyanobjectrepresentedbythesubjectofapropositionofcertainpropertiesconnotedbythepredicatedoesnotimplythesamecontentthatthespeakerintendedtorepresent…Thesemioticanalysisoftheadequacyorinadequacyofcertainpropositionsinrelationtothecontentswhichtheyrep-resentisthenultimatelybasedonaphenomenologicalanalysisofthespeaker’srepresentationalintentions[Le´sniewski,1992b,16–17][Le´sniewski,1911,342–3].4Hereasin1929thefunctionoflanguageistorepresentthemental“contents”speakersintendtoconvey.Le´sniewskirefersinthepassageto“phenomenologicalanalysis”ofthespeaker’srepresentationalinten-tionsandearlierinthepapertoHusserl[Le´sniewski,1992b,6]and,followingTwardowski(e.g.[Twardowski,1977,8–10]),5conceivesofintuitivecontentthroughoutthepaperintermsofthenotionofcon-notationfromtraditionallogic,withreferencetoMill.6Themainpointforusisthattheadequacyofconventionshereisamatterofproperlyansweringtoaspeaker’s“referentialintentions”.Thepointoflanguageistoexpressthoughts,butnaturallanguagedoesthispoorlyandthepurposeofdeductivesciencesistodoitwell.Asystemoflanguageneedstoberigorouslycompositionalinthesensethatthecontentexpressedbyanon-lexicalexpressionmustalwaysbeuniquelydeterminedbythecontentsassociatedwithitspartsandbythewaytheyareassembled.The“symbolicfunction”ofexpressionsinsuchasystemisdeterminedby“conventional–normative”schemas:rulesassociatingthoughtsandthought-constituentsthataretobeexpressedbytheexpressionsofthesystemwiththeseexpressions.Thissoundsgood,butitraisesaproblemforLe´sniewski,onethatnei-therhenorTarskieversolvedoreverreallyevenaddressed:theproposedaccountofcompositionalityforcontentsdidn’texist.7Thereishereasimplepictureofcontentsofprimitiveterms,construedassetsofprop-ertiesinaccordwithtraditionallogicaldoctrinesaboutconnotation,andthenhand-wavingabouthowthecontentsofcomplexexpressionshavetobedeterminedbythecontentsofsimpleonesplustheirmannerofCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01composition.Onecanseehowthismightgoinafewtoycases(e.g.“redball”),butitisunclearwhatthestorywassupposedtobeforanythingmorecomplicated.ThisweaknessinIntuitionisticFormalismwasneverrepaired.WhenTarskireplacedintutionisticformalismwithhisseman-tics,oneresultwasthatanaccountofmeaningwithnostoryaboutsemanticcompositionalitywasreplacedwithanotherthathadone,but10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE22—#11\nIntuitionisticFormalism23thattherewasthisdifferenceseemsnottohavebeensomethingthatmotivatedTarski.ClearlytheviewLe´sniewskiadherestoin1911isretainedinthepas-sageTarskicitesandinthematerialsurroundingit,despiteLe´sniewski’srepudiationoftheratheroutdatedandidiosyncraticviewsoflogicandmeaningpresentinthearticle.“IntuitionisticFormalism”isthentheviewthatthepointofformalsystemsistheclearexpressionofthought,andthatsuchsystemsaretobejudged“adequate”ornotbasedonhowwelltheirexpressions,usedinaccordwith“conventional–normativeschemata”,systematicallyaccordwiththerepresentationalintentionsoftheir“speakers”.OnecanfindelementsofthisviewthroughoutLe´sniewski’swork.Letustracetheappearanceofitsmainelements—theconceptionofmeaningintermsofrepresentationalintentionswithrespecttointuitivelygraspedthoughts,theconceptionofthe“conventional–normativeschemata”governingtheuseoflanguageintermsofthis,andtheconceptionofintuitivethoughtintermsofthetraditionaldoctrineofconnotationanddenotation—throughhiswork.8Wewillfirstlookatexpressionsoftheviewintheearlypapers,andthenfinditsappearanceinthematurework.ThecontentexpressedbyatermasconceivedbyLe´sniewskiisaconno-tation.Wecanseethisinhisaccountofsynonymyat[Le´sniewski,1992a,21–22]:twosubject-predicatepropositionsaresynonymouswhenboththeirsubjectsandtheirpredicateshavethesameconnotation.Wecanalsoseeitinhisremarksonnon-denotingexpressionsthatarenever-thelessmeaningful:“‘hippocentaur’doesnotdenoteanyobject…Thisdoesnotmean,ofcourse,thatthisworddoesnotconnoteanything—onthecontrary,itcanconnotesomequitestrictlydefinedproperties…andthusitcanpossessastrictlydefined‘meaning”’[Le´sniewski,1992a,32].AtonepointintheearlyworkLe´sniewskiwritesthatthefullformofthedefinitionofanexpressionisillustratedby“Iemploytheexpres-sion‘W’insuchawaythatifitdenotesanythingatall,itdenotesonlythoseobjectswhichpossessthepropertiesc1,c2,c3,…,cn”[Le´sniewski,1992c,66].Noteheretheconceptionofadefinitionasaconventionthataccordswithaspeaker’sreferentialintentions.Le´sniewskiisquiteseriousinthinkingoflanguageasfunctioningCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01entirelytoexpressthoughtsratherthan,forinstance,toshapeordeter-minethem;witnessthispassagefromasubsequentarticleinwhichtherulesoflanguagearecomparedtorailwaysignals:Ihavemorethanoncepointedoutthatasystemoflinguisticsymbols,justasanyothersystemofsymbols,e.g.,thesystemofrailwaysignals,10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE23—#12\n24AlfredTarski:PhilosophyofLanguageandLogicrequirestheexistenceofcertainrulesforconstructingthesymbolsandkeysforreadingthem.Ihaverepeatedlystressedthatthefunctionsofvariouscomplexlinguisticstructures,e.g.,thoseofpropositions,shoulddepend,inacorrectlyconstructedpreciselanguage,uponthefunctionsortheorderofparticularwords—onthebasesofcertainpattensdeterminedbygeneralnormativeconventionstheknowledgeofwhichpermitsthecorrectsymbolizationofanobjectinagivenlanguageorthedecodingofasymbolforagivenlanguage[Le´sniewski,1992c,56].Notetheuse,hereandelsewhere,of“decode”(odcyfrowanie[Le´sniewski,1913,324])forwhatthereaderdoesinunderstandingasymbolinaccordwithconventional-normativeschemata.Thispassageisfollowedbythestatementoffour“Conventions”inaccordwithwhichLe´sniewskiwillusecertainexpressions.ThesemostlydeterminetheroleofnegationandallowtheconstructionofLe´sniewski’sargument,suchasitis,fortheprincipleofnon-contradiction:giventhefourconventions,allcon-tradictionsarefalse.Themainpointforusinthisissimplythata“Convention”isaruleinaccordwithwhichoneintendstouseasymbol:apointthatwillberelevantwhenwegettoTarski’s“Convention”T.Theconventional–normativeschemataofadeductivescienceare,then,setupbyLe´sniewskitoexpresshisthoughts.Le´sniewski’saccountofthelatter,however,accordsthemnoparticularjustificatorystatus.Considertheseremarksfromanearlydiscussionofhismereologicalconceptionofsettheory:Thepsychic‘sources’ofmyaxiomsaremyintuitions,whichsimplymeans,thatIbelieveinthetruthofmyaxioms,butIamunabletosaywhyIbelieve,sinceIamnotacquaintedwiththetheoryofcausal-ity.Myaxiomsdonothavealogical‘source’,whichsimplymeansthattheseaxiomsdonothaveproofswithinmysystem[Le´sniewski,1992d,130].AppealstointuitionbyLe´sniewskiarenottobeunderstoodashavinganyjustificatoryforce;thewholepointoftheaxiomsissimplytoexpressclearlytheauthor’sthoughts.9ThiswillbeimportantwhenweturnlatertothequestionofwhethertheIntuitionisticFormalistfollowedHilbertCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01intakingconsistencytobesufficientfortruth.Withthesebasicviewsonmeaninginmind,wecanturntoLe´sniewski’sconceptionof“science”asembodiedinadeductivetheory:Scienceis,then,asystemoflinguisticsymbols.Toconstructandtounderstandlinguisticsymbols,itisnecessary,asinthecaseofany10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE24—#13\nIntuitionisticFormalism25othersymbols,tohavecertainrulesofconstructingthesesymbolsandcertainkeystodecipherthem.Therulesonwhichtheconstructionoflinguisticsymbolsarebasedandthekeyswhichdecipherthemare,ontheonehand,precisedefinitionsofvariousexpressionsand,ontheother,differentkindsofconventionsconcerninglinguisticsym-bols…Linguisticconventionsare,therefore,thenecessaryconditionofanyscientificprocedureandtheindispensiblekeytounderstandingscience[Le´sniewski,1992a,35].AsLe´sniewskimentionselsewhere,suchconventionscanamounttodef-initionsornot[Le´sniewski,1992c,75–6]:eitherwaytheyarerulesthatdeterminetheintuitivemeaningsofsymbols.Apassagefromthetransitional“FoundationsoftheGeneralTheoryofSets”of1916alreadystatestheIntuitionisticFormalistviewwell:Onthequestionconcerningthewayofusingexpressions,ImentionthatofthemathematicaltermswhichIuse,theexpression‘part’istheonlyoneIdonotdefine,supposingthatthistermwillnotcausemisunderstandings—consideringthatitsintuitivecharacteracquiresconsiderableclarityinthelightofAxiomsIandII[Le´sniewski,1992d,131].AxiomsIandIIstatethatparthoodisanti-symmetricandtransi-tive,respectively.Unlikethepureformalists,Le´sniewskiholdsthattheintuitivecontentoftheprimitiveexpressionmatters.Onefunc-tionoftheaxiomsistoexpressthiscontentsufficientlytowardoffmisunderstandings.1.2.2Lesniewski’slaterwork´AfterLe´sniewski’srelativelysilentperiodofteachingandworkingdili-gentlyatWarsawheresumedpublicationwith“OntheFoundationsofMathematics”(1927–1931)andrepudiatedtheearlywork.Hisviewsonthepointofsettingoutadeductivetheoryrecededintothebackground,buttracesofthemcanstillbefoundinremarksthroughouttheremain-derofhiswritings,includingthepassageTarskicites.OnethingthattheselaterremarksmakeclearisthatthoughLe´sniewskimadesomeref-erencetoBrentanoandHusserlinhisearlydiscussionsof“referentialCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01intentions”andthelike,noheavytheoryof“mentalacts”liesbehindhisviews;hisinterestisreallyapracticaloneinbeingunderstoodtohavesaidexactlywhatheintendedtosay.Le´sniewskiretakesthephilosophicalstageafterhisabsencewithatrenchantcriticismofknownsystemsofmathematicallogic,criticizingthemajorcontributionsoftheperiod—Frege’sproposedmodificationto10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE25—#14\n26AlfredTarski:PhilosophyofLanguageandLogichisownsystemintheappendixtotheGrundgesetze,Principia,Zermeloandothers—fortheirlackofintuitivebacking.Considerthispassage:Theproblemofthe‘antinomies’,underthepowerfulinfluenceoftheresearchesofBertrandRussell,hasbecomethecentralproblemintheintellectualendeavorsofanumberofprominentmathematicians.Theseendeavorswereoftensignificantlyremotefromthehistorical,intuitivebasisfromwhichthe‘antinomies’developed.Theyencour-agedthedisappearanceofthefeelingforthedistinctionbetweenthemathematicalsciences,conceivedofasdeductivetheories,whichservetocapturevariousrealitiesoftheworldinthemostexactlawspossible,andsuchnon-contradictorydeductivesystems,whichindeedensurethepossibilityofobtaining,ontheirbasis,anabun-danceofevernewtheorems,butwhichsimultaneouslydistinguishthemselvesbythelackofanyconnectionwithrealityofanyintuitive,scientificvalue…Thequestion,whethertheFregesystemchangedasindicatedaboveoralsothe‘Mengenlehre’ofZermelowilleverleadtoacontradiction,iscompletelyimmaterialfromthepointofviewofanintellectualanxiety,directedresolutelytowardsreality,andwhichstemsfromanirresistible,intuitivenecessityofbeliefinthe‘truth’ofcertainassumptionsandinthe‘correctness’ofcertaininferences,whichincombinationwiththoseassumptions,leadtocontradiction.Fromthispointofviewtheonlyreal‘solution’tothe‘antinomies’isthemethodofanintuitiveunderminingofthecombinationofinferencesorassumptionswhichcontributetothecontradiction.Anon-intuitivemathematicscontainsnoeffectiveremedyforanymaladyoftheintuition.[Le´sniewski,1992h,177–8].Asdistinctfromapureformalist,ortheCarnapofLogicalSyntax,Le´sniewskiseesdifferencebetweenadeductivetheorythatexpressesintuitivethoughtaboutrealityandamerelyconsistentsystemasbeingofgreatimport.10Thepassageisfollowedbyalongseriesofcom-plaintsabouttheinadequaciesofRussellandWhitehead’sattemptstoexplainthemeaningsoftheexpressionsofthelanguageofPrincipia.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01TheonlypredecessorforwhomLe´sniewskiexpressesanyapprovalisFrege[Le´sniewski,1992h,195note18],forhisattentiontothe“mean-ingfulnessofmathematicalpropositions”.11Ourinterestinthepassageconcernsitscontinuitywithwhatwehaveseen:thepointofadeductivescienceisclearlytoexpressthoughtsacceptedasintuitivelytrue;amereformalismalonecannotsolveanyproblems.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE26—#15\nIntuitionisticFormalism27Commentingonhisturnto“asystematicuseofsymboliclanguageinsteadofacolloquialoneintheyear1920”,Lesniewskimakesclearthathisinterestwassimplyinwardingoffmisunderstanding:ThesymbolismIadopted,basedonformulasconstructedby‘mathe-maticallogicians’,Iusedasatoolwhichistechnicallymuchsimplerthanthecolloquiallanguageandatthesametimelesspronethanthatlanguagetoleadtomisunderstandingsintheformulationofideas.Inattemptingtotranslatethethesesofmy‘generaltheoryofsets’asscrupulouslyaspossiblefromthecolloquiallanguageintomynew‘symboliclanguage’,Ineverthelessconstructedtheproofsofthetheoremsofthattheoryinanintuitivewaywithoutinanywaybas-ingthoseproofsuponsomeclearlycodifiedsystemof‘mathematicallogic’,aprocedureIhaduseduptothen.Thechangetoa‘symbolic’wayofwriting,whichconstitutedafar-reachingrevolutioninmysci-entificlifeinthefieldofsymbolictechnique,wasnotaccompaniedbyanyfar-reachingparalleleventsinthedomainofmy‘logical’views[Le´sniewski,1992h,365–6].LaterLe´sniewskididbecomenearlyobsessedwiththeconstructionofformalrulesthatcapturedthese“intuitivelybinding”inferences,inthe“directives”ofhissystems.WhatwecannoticehereisthatLe´sniewskiwantedtoexpresshisthoughtswiththeminimumpossibleriskofmisun-derstanding.Histhoughts,andhisinferencesamongthem,tookplaceinhismind;whathesoughtwasamediuminwhichtheycouldbeexpressed.Notethattheconceptionofinferencehereisentirelyepis-temic:proofsneedtobe“intuitivelybinding”;thelaterintroductionofexplicitlyformulated“directives”foraddinglinestoaproofsimplymakesforanaddeddegreeofexactnessinexpressingsuchinferences.SincethepointofthesymbolismwastowardoffmistakesonthepartofthereaderastowhatideasLe´sniewskiwastryingtoconveyasmuchaspossible,anoverridinggoalofthesymbolismwastherebytoreducethoseaspectsofthesymbolismthatweren’tspecifiedintermsofthesyntacticformsofexpressionstoaminimum.ThusbegantheobsessionamongLe´sniewskiandhisstudentswithreducingthenumberofaxiomsandprimitivesinadeductivetheory.Thisproject,whichcanlooklikeCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01meticulousnessforitsownsake,waswell-motivatedonIntuitionisticFormalistgrounds:thefewertheprimitives,thefewerthesignswhosemeaningshadtobeexplainedin“colloquial”language;thefewertheaxioms,thefewertheclaimsuponwhichagreementwouldneedtobereachedbeforeonecouldgetdowntothebusinessofderivingintuitivelyvalidclaims.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE27—#16\n28AlfredTarski:PhilosophyofLanguageandLogicOneaspectofLe´sniewski’svisionhereisthataxiomsanddefinitionsmatteredlesstotheintuitiveinterpretationtherebyforceduponhisprimitivesthandidcertaintheorems.Thatis,Le´sniewski,inordertocaptureintuitivemeaning,wantedcertainthesestobetheoremsofhissystemsoastoguaranteeitsexpressionofintuitivecontent,buthewasmoreopenabouttheformulationofaxiomsanddefinitions.Hereishowhecommentsontheformalizationofhis“Ontology”:Iwishedtotakeonestepfurtherand,usingthe‘singular’propositionsofthetype‘Aεb’,tobaseallmydeliberationsonsomeclearlyfor-mulatedaxiomatizationwhichwouldharmonizewithmytheoreticalpracticeinthisdomainatthetime.Inrelationtosuchanaxiomati-zationIrequiredthatno‘constant’termsshouldappearinitwiththeexceptionoftheexpression‘ε’inpropositionsofthetype‘Aεb’andtermswhichappearinthe‘deductivetheory’…Feelingtheneedofadeductivetheoryaxiomatizedinthewaydescribed,Ineverthelesscouldnotfindareadytheoryofthatsortintheworkofeitherthe‘traditionallogicians’orthe‘mathematicallogicians’…Ionedayformulated…anumberoftheseswhichwouldbevalidonthebasisofthetheorymentioned.IcannotrecallwhichthesesImentionedthen—aftersomanyyears—butIcanstatewithcompleteconfidence…thatatthattimeIwouldnothaveacceptedassufficientformytheoreticalpurposes,asdescribedaboveanyaxiom-atizationwhichdidnotguaranteethepossibilityofderivingfromit—whilepermitting,ofcourse,operationswithdefinitionswhichharmonizedwithmywayofusingthecorrespondingexpressionsinmyscientificpractice,—‘symbolic’equivalentsofthesesstatingrespectivelythat(1)someaisbwhenandonlywhen,forsomeX,(Xisa,andXisb)…(6)Aisawhenandonlywhen,(eachAisa,andatmostoneobjectisA).Fromamongthedifferenttheses,whose‘symbolic’equivalentswerevalidinthedeductivetheorywhichIwishedtobuild,Ihaveselectedheretheses(1)–(6),astheyappeartomeaconvenientintuitivetran-sitiontootherdeliberations:thesis(6),oranyotherthesismoreorCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01lesssimilarto(6),hasplayedafar-reachingroleatthecreationoftheaforesaidtheory,whosesoleAxiomIdevisedbyanalysisofthatthe-sis;theses(1)–(5),selectedtoavoidintuitivedifficultiesforthereader,throwsomelight(directlyorindirectly)onthewayinwhichIusedtheexpressionsappearinginthethesis(6)[Le´sniewski,1992h,367–8].10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE28—#17\nIntuitionisticFormalism29Here,quiteclearly,theroleofcertainthesesinthesystemistosecureagreementastothemeaningoftheprimitivetermsofthesystem,andthedesiredaxiomatizationisheldtothestandardthattheybeimplied.Bycontrast,theaxiomitselfissimplyrequiredtoimply(1)–(6)bythedirectives;itneednotbeintuitiveinitsownright.Thatsaid,implicationof(1)–(6)ofcoursedidn’tmakethesystemimmunetomisunderstanding.Le´sniewskicommentsatonepointonthetopicofsecuringagreementabouttheprimitivetermofhis“Ontology”:ItoftenhappenedthatsomeonewithwhomIspokedidnotreal-izethesenseinwhichIwasusingthesign‘ε’inmyontology,andexperiencedinterpretationaldifficultiesinattemptingtorelatethegivensignsemanticallytooneorothersignshandeddownbythetraditionof‘mathematicallogic’and‘thetheoryofsets’.Inthetrou-blesometheoreticalsituationsinwhichIfoundmyselfduetothisstateofaffairs,Isharedthemisfortuneofallwhoareforcedbycir-cumstancestoconveyin‘theirownwords’thesenseofthevariousprimitivetermsofdeductivetheorieswhichtheywereconstructing[Le´sniewski,1992h,374–5].Theremarkisthenfollowedbyadiscussionofvarious“colloquial”formulationsthatmighthelpsomeonetocottonontoLe´sniewski’ssense.NoticeagainthecontrastherewithHilbertandespeciallywithCarnap,whomwehaveseeninLogicalSyntaxtakingthisveryprob-lemtobethereasonthatoneoughttoignoreanythingthatisn’tdeterminedbythestructureofadeductivetheory.ForLe´sniewski,bycontrast,thecolloquialexplanation,thoughproblematic,wasofcentralimportance.Onechangeintheshiftfromtheearlytothelateviewthatislikelyasso-ciatedwiththemovetoconsiderationofformalizedlanguagesisthattheearlyworkisconcernedprimarilywithsub-sententialconventions,whilementionoftheseisabsentinthelaterpassage.OnLe´sniewski’smatureviewconventionsthatdeterminetheintuitivethoughtsexpressedbysentencesareconceivedofasthe“directives”governingasystem,rulesforaddingnewthesestoit.Thedeterminationofsub-sententialmean-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01ingsexpressedisthenanindirectmatteroftheroleofasignasestablishedbythetheoremsinwhichitappears,anditisthisdeterminationthatcomestobethecentralissue,aswiththeexampleof”ε”above.Thisshiftde-emphasizestheearlierdemandthattheaccountbecomposi-tionalandLe´sniewskiinthelaterworkdoesn’tcommentonthatearlier10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE29—#18\n30AlfredTarski:PhilosophyofLanguageandLogicrequirement.Aswewillseein§2.2.1,inTarski’shandstheIntuitionisticFormalistconceptionofintuitivemeaningisnotcompositional.Thepicturethatemerges,then,isarathersimpleone.Le´sniewskihasideasandthoughts.Thereisnoquestionforhimastoexactlywhattheyare,buthewantstoconveythemtoanother.Forthisanadequatesym-bolismisabettertoolthannaturallanguageandsothethoughtsarebestexpressedintheformofaformalizeddeductivetheorygovernedbyrulesfortheformationofexpressionsandfortheconditionsunderwhichonesentencecanbeassertedonthebasisofothers.Sinceonepointoftheexerciseistoruleoutpossibilitiesofmisunderstanding,axiomsandprimitivesaretobekepttoaminimumand,sincetheotherpointoftheexerciseistostate(whatLe´sniewskiisconvincedare)truths,whatmattersisthattheaxiomsseemtrue—recallhereLe´sniewski’sentirelynon-epistemic,psychologisticnotionof“intuition”—andthattheinfer-encesfromthemtotheoremsseemvalid.12Sincerelationsinstitutedinadeductivesciencebyconventionsareconcernedwithsentences,aprimaryissueisthewayinwhichsentencesheldtobetruecon-strainthepossibleinterpretationoftheprimitives.ThisisIntuitionisticFormalism.13OnLe´sniewski’sconceptionthepointofaconvention—and,sincedefinitionsareakindofconventionforLe´sniewski,thepointofadefinition—istodetermine,jointlywithotherconventions,whatintu-itiveconceptatermexpresses,notbydirectlyanalyzingthiscontent,butbyimplyingthings,inconjunctionwithothertheses,andinaccordwiththedirectives,thatconstraintheinterpretationofit.Definitionswere,forLe´sniewski,simplyfurtheraxiomsthatcontributedtocon-straintheconceptexpressedbyaterm;theydifferedfromotheraxiomsinshowingalsohowtoeliminatethetermfromtheoremsofthesys-tem.Le´sniewskicreditsTarski’swork,publishedas[Tarski,1983j],forallowinghimfullytorealizethisconceptionofdefinitionsforPro-tothetic,hissystemofsententiallogic[Le´sniewski,1992e,418–19],[Le´sniewski,1929,11].ThusthereisroominLe´sniewski’sconceptionfordefinitionsthatdoinfact,whenaddedtoatheory,determinethatatermexpressesacertainintuitiveconceptwithoutsomehowthem-selvesbeing,orbeingexpressionsof,ananalysisoftheconcept.WhatCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01mattersissimplythatthatconnotationbeproperlycapturedinthesystemasawhole.WhatthisanticipatesispreciselythemismatchwehavealreadyexaminedbetweenTarski’sdefinitionsandwhatonewouldexpectoftraditionalphilosophicalanalysesofconcepts,aswellastherelationshipbetweenadefinitionoftruth,theT-sentencesandConventionT.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE30—#19\nIntuitionisticFormalism311.3Kotarbinski´SofarwehavefoundaratherclearviewexpressedinLe´sniewski’s1911–14work,andthengeneralstatementsconsistentwithitinlaterwork.SinceTarskionlybeganhisstudiesin1918,weneedtowardoffthesuggestionthatperhapstheviewwasn’treallyaroundforhimtoinherit.However,turningtoKotarbi´nski[Kotarbi´nski,1966],wefindthesameviewaswehavediscernedinLe´sniewski’searlywork.TarskinotesattheoutsetinCTFLthat“inwritingthepresentarticleIhaverepeatedlyconsultedthisbookandinmanypointsadheredtotheterminologytheresuggested”[Tarski,1983a,153].TotheextentthatTarskihasviewsinthephilosophyoflanguagewecanexpectthattheycomefromKotarbi´nski,andsinceKotarbi´nski’sviewsarequiteconsistentwithwhatwefindinLe´sniewski,wecanbeconfidentthattheviewsketchedsofarisinfactTarski’s“IntuitionisticFormalism”.14Kotarbi´nski’sbasicconceptionofmeaningincludeselementsofinten-tionandconvention:ThatsentenceSisa(direct)statementofJohn’sthoughtastocontent,means:Johnthoughtthatp(“p”standsforthewholesentence)andutteredSsothatthelistenermightrecognizethatJohnthoughtthatp.Obviously,insuchcasesthereisnoobstacletoagivenstatement,whichisnotadirectstatementofanyperson’sthoughtastocontent,beingusableassuchastatementinconformitywiththelawsofagivenlanguage,thatis,expressingthatthoughtastocontentindirectly[Kotarbi´nski,1966,4].Thedistinctionbetweendirectandindirectstatementsissimpleenough;asentencestatesanexperiencedirectlywhen“itisspokenorwrittenwiththeintentionthatadefiniteperson,oranypersonwhateverwhoper-ceivesthatphrase,andunderstandsthelanguageinquestion,shouldknowfromthatphrasethatthespeakerexperiencedpreciselythat”[Kotarbi´nski,1966,3–4],while“agivenphrasestatesacertainexperi-enceindirectlywhenever,withoutstatingitdirectly,suchcan,inviewoflinguisticusageoradoptedconventions,beusedinagivenlanguageasadirectstatementofsuchexperience”[Kotarbi´nski,1966,4].15Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Kotarbi´nski’sphrase“thelawsofagivenlanguage”matchesLe´sniewski’stalkof“conventional–normativeschemas”.Kotarbi´nskioffersnoparticularanalysisofconventionor“lawsoflanguage”,nothinglikeanaccountoftheconditionsunderwhichaconventionisinsti-tutedinapopulationintermsofthementalstatesofitsconstituentpeopleinthestylefamiliartousfrom[Lewis,1969],butitisn’tafar10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE31—#20\n32AlfredTarski:PhilosophyofLanguageandLogicstretchtodeterminewhathehastobeassuming.Onecanintendtouseasentencetoexpressaparticularthought;thisisdirectstatement.Itcanalsobethecasethatthelawsorconventionsofagivenlanguagedeter-minethatthataparticularsentenceexpressessomethought,wherethiscomestoitsbeingthecasethatapersonwhohadacertainthoughtandusedthelanguagecorrectlywouldusethatsentencedirectlytoexpressit.Sotheassumptionhastobethat“conventions”and“laws”oflan-guageareagreed-uponrulesthatrelatethoughtstothesentencesthatoughttobeused,inthatlanguage,toexpressthem.16Aconventionthenobtainsinapopulationwhenitsmembersdoagreethatthesen-tencesoughttobeuseddirectlytoexpressthosethoughts.Oncesuchaconventionisinplace,aphraseindirectlyexpressesonalloccasionsthatthoughtwhichitwouldbeuseddirectlytoexpressifitwereusedcorrectlybysomeone.17Considerherethisremarkonthemoregen-eralphenomenonof“expressing”,whichisbasicallystatementlesstheintentiontocommunicate:Theword‘help!’,exclaimedbyJohnatthemomentofdanger,expressesdirectlyonlythedesire,experiencedatagivenmoment,toobtainhelp.Butindirectly,inviewoftheEnglishlanguageusage,itexpressesalsoasimilardesirewhichPeterwouldhaveexpressedevenifithappenedthathewasdrownedwithoutbeingabletocallforhelp.Whatismore,theword‘fy’,shapedbychancefromthetwistsofacreepingplant,expressesintheEnglishlanguageindirectly(and,ofcourse,unintentionally)theexperiencesofdisgustofvariouspersons,evenifthosepersonsdidnotintendtoexpressthemandcouldnotspeakEnglish.[Kotarbi´nski,1966,3].Thelastexamplemakesclearthatthe“rules”aredirectedatwordsandsignsconceivedofasitemswithcertainshapesortheiracousticanalogues.18Animportantelement,though,isthewayinwhichcon-ventionsrelatetheformofanexpressiontoitsmeaning.Thislinkopensupthepossibilityofapurelyformalstudyofmeaning,providedthattheconventionsaregiven.TheideathatconventionsrelatesyntacticformtointuitivemeaningwillbethesourceofTarski’sideasaboutmetamathe-maticsasthestudyofsystemsofmeaningfulsentenceswheretheirformsCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01goproxyfortheirmeanings[Tarski,1983c,62].Thisaccountofdirectandindirectstatements,combinedwiththeroleofconventioninindirectstatement,opensthespaceforpreciselytheproblemthatexercisedLe´sniewski:underwhatconditionsdoesagivensignindirectlyexpressexactlythatwhichonewantsdirectlytoexpresswithit?Theset-upcallsforaprogramofgoodhygienein10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE32—#21\nIntuitionisticFormalism33designingconventionsfortheindirectexpressionofthethoughtsonewantsdirectlytoexpressaboutagiventopic.ItalsomakesclearthatsomethingistakentobeunproblematicbyKotarbi´nski,justasitwasbyLe´sniewski:thoughtsare“experienced”,andpresumablyoneknowswhatone’sexperiencesare;inthisthemembersoftheLvov–WarsawSchoolaregoodBrentanists.19Theprojectinsettingupalanguageistodeviseconventionsthatareclearandagreeduponunderwhichsentencesindirectlyexpressexactlywhatonewantsdirectlytoexpressbythem.Thisdone,oneisn’tlefttothehappenstanceofaparticularcommunica-tivesituationtogetone’sthoughtsacross;ifallgoeswell,fellowpartiestotheconventionswilltakesentencesindirectlytoexpressexactlywhatonedirectlyexpressesbythem.Noticealso,forfuturereference,thattheemergingpictureleavesopenthepossibilitythattheconventionsofalanguagemightfail:poordesigncouldleaveitunsettledwhatasen-tenceindirectlyexpresses,ordetermineanindirectexpressionthatfailstoaccordwithone’sintentionsortheymightassociate,iftakenseriously,everythoughtwitheverysentence.Kotarbi´nski’snotionofmeaningistheobviousauxiliarytohiscon-ceptionofindirectexpression:Ifwewanttoanswerthequestion“Whatdoesagivensentencemean?”wecannotdescribeaperson’sexperienceandsaythatthisis“whatthatsentencemeans”.Wemusttryotherways,andprobablythesimplestconsistsinansweringbythephrase“meansthat”.If,forinstance,someoneaskswhatthesaying“Johnistransportedwithjoy”means,wemayanswerthatthesayingmeansthatJohnistransportedwithjoy[Kotarbi´nski,1966,5].Thisapplies,however,onlytosentences.Kotarbi´nskicontinues:Someonemightask,forinstance,whattheword“goliard”means,whichhehasfoundinatextreferringtothelateMiddleAges.Itwouldbeabsurdtoanswer:“goliard”meansthatetc.,sincetheword“goliard”isnotasentenceandhencedoesnotstateanyone’smentalexperience.Itisonlywithreferencetoasentencethatwecanreason-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01ablysayit“meansthat…”…Howthenshallweanswerreasonablythequestionconcerningtheword“goliard”?Thebestsolutionisper-hapsasfollows:thesentence“Apersonisagoliard”meansthatthepersonisanitinerantminstrelcomposingsatiricalLatinverse.Who-everis,andrightlyso,dissatisfiedwith“aperson”canusetheletterxinsteadandwillarriveattheformula:thestatement“xisagoliard”10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE33—#22\n34AlfredTarski:PhilosophyofLanguageandLogicmeansthatxisanitinerantminstrelcomposingsatiricalLatinverse[Kotarbi´nski,1966,6].Inallthosecaseswhensomeoneaskswhatagivenphrasemeans,theproperanswerisobtainedbyusingthephrase“meansthat”,eitheruseddirectlyafterthewordorphrasetowhichthequestionpertains,orsomehowimpliedinaccordancewiththesyntacticfunctionofthewordorphrase…Ananalyticdefinitionofthephrase“meansthat”asappliedtosentencescanbebuiltwiththeaidofthewords“statesastocontent”(or“isastatementastocontent”)inthefollowingmanner:“ThesentenceSmeansthatq”isthesameas“ThesentenceSstatesastocontent,directlyorindirectly,aperson’sthoughtthatq”[Kotarbi´nski,1966,7].Wecharacterizethemeaningofasub-sententialexpressiononlyindi-rectly,byschematicallyindicatingwhatasentenceincludingitwillmean,whilesentencemeaningis,inturn,directlyanalyzedintermsoftheexpressionofthought.Kotarbi´nskiallowshere“directorindi-rect”,butindirectexpressionismoretothepoint:wecharacterizewhatasub-sententialexpressionmeans,accordingtolinguisticusage,byindi-catingschematicallywhatwouldindirectlybeexpressedbyasentenceincorporatingitwhenusedcorrectly.Thisschematicmodeofspecifyingmeaningbearsanobviousconnec-tiontoTarski’snotionofsatisfaction.Italsomakesclearthatconventionsgoverningsentencesrelatetosub-sententialexpressionsonlyindirectly.ForKotarbi´nskithebasicnotionofmeaningisthatasentencemeansthatpifandonlyiftheconventionsofitslanguagedeterminethatitisusedcorrectlyexactlytoexpressthebeliefthatp.Now,strictlyspeaking,whatwehavehereisatheoryofhow“meansthat”and“means”areused:Kotarbi´nskihastoldushowtoformulateanascriptionofmeaning.Butwhatisitthatisascribed?WegettheanswerinKotarbi´nski’saccountofconnotation:Itistimenowtointroduceanewtechnicalterm,whichwillfacilitateafurtheranalysisoftherelationofdenoting,especiallyinthecaseofgeneralterms.Thisis“connotation”.Nowinsteadofsayingthat,forexample,“xisice”meansthatxissolidwater,wemaysaythattheCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01term“ice”connotesthepropertiesofwaterinessandsolidity…Thepropertiesjointlyconnotedbyagiventermformtogethertheintensionofthatterm,oritssense,oritsconnotation;thismustbecontrastedwiththedesignatawhichjointlyformtheextensionofthatterm,oritsdenotation.Wemaythussaythateveryobjectwhichhasallthepropertiesjointlyconnotedbyagiventermwithrespectto10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE34—#23\nIntuitionisticFormalism35itsgivenmeaningisadesignatumofthattermwithrespecttothatmeaning.Inotherwords,everyobjectwhichhastheintensionofagiventermwithrespecttoitsgivenmeaningbelongstotheextensionofthattermwithrespecttothatmeaning[Kotarbi´nski,1966,11].ThepicturehereisthesameasinLe´sniewski:termshaveintensionsthataresetsofpropertiesandtheirextensioniseverythingthathastheproperties;aswithLe´sniewski,thesourceoftheviewis[Twar-dowski,1977,8–10].Notethatonly“terms”—roughly,subjectsandpredicatesofallorders—haveconnotations,whileallsyntacticallysig-nificantexpressions(e.g.sententialconnectives[Kotarbi´nski,1966,6–7])havemeanings.Thoughtandlanguagearebroughttogetherhereinthedoctrinethatthemeaningofaterm,itsconnotation,isalsotheintensionoftheconceptexpressedbyit:Insemanticterminology,theintensionofaconceptisunderstoodastheconnotationofthecorrespondingterm,thatisthepropertiesjointlyconnotedbytheterm.Morestrictly,theintensionofthecon-ceptofMisthesameasthesetofpropertiesjointlyconnotedbytheterm“M”[Kotarbi´nski,1966,88].ThisdoctrineofintensionistheprecursorofTarski’snotionofthecontentofaconceptinCTFL.Wecanconsiderthepossibilityofamismatchthattheaccountallows.Ontheonehand,terms’meaningsaresupposedtobedeterminedbytheconventionsforsentences.Ontheother,aterm’smeaningissupposedtobeacertainconnotation,thecontentofasub-propositionalmentalstate:graspingaconcept.Thequestionraisedis:howdowemakeconventionsforsentencesexpressingthoughtscorrectlydeterminetheconnotationsthattermsareintendedtohave?ThoughKotarbi´nskidiscussesdenotationatsomelength,theimpor-tantthingforourpurposesisthatdenotationiswithinhisconceptionaderivativefeaturethatisirrelevanttothebasic,psychologisticaccountofthesignificanceoflinguisticexpressionsandtheirroleincommunica-tion.ForKotarbi´nski,followinginthefootstepsofTwardowski,thoughthoughtandlanguagecanbecharacterizedinreferential,semanticCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01terms,thesetermsplaynoroleinthebasicaccountofmeaningandcom-municationinlanguage.ComparehereTwardowskiinacharacteristicpassage:Theword‘sun,’Millmaintains,isthenameofthesunandnotthenameofourpresentationofthesun;yethedoesnotwanttodeny10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE35—#24\n36AlfredTarski:PhilosophyofLanguageandLogicthatthepresentationalone,andnotthething,isrecalledthroughthenameorcommunicatedtoalistener.Thetaskofanameisthustwofold:thenamecommunicatestoalistenerthecontentofapre-sentationand,atthesametime,itnamesanobject.However,wesaidthatwemustdiscern,notjustatwofold,butathreefoldaspectofeverypresentation:theact,thecontent,andtheobject.Andifanamereallyyieldsanaccuratelinguisticpictureofthementalstateofaffairswhichcorrespondstoit,thenitmustalsoshowacorrelatetotheactofpresentation…What,then,isthetaskofnames?Obviously,itistoarouseinthelisteneracertaincontentofapresentation.Someonewhouttersanameintendstoawakeninthelistenerthesamementalcontentwhichappearsinhimself;whensomeonesays:“sun,moon,andstars,”hewantsthosewholistentothink,justashedoes,ofthesun,themoon,andthestars…Inthismanner,anamealreadyful-fillstwotasks.Firstly,itmakesknownthattheuserofthenameispresentedwithsomething;itsignifiestheexistenceofamentalactinthespeaker.Secondly,itawakensinthelisteneracertainmentalcontent.Itisthiscontentwhichisthe“meaning”ofaname.[note]Wedesignateineverycaseasthemeaningofanexpressionthatmentalcontentwhichitistherealtask,thegoal,ofthenametoarouseinthelistener.[Twardowski,1977,8–9].AsforKotarbi´nski,thoughtsareaboutobjectsandnamesrefertothem.But,alsoasforKotarbi´nski,theunderstandingoflinguisticsignifi-canceandmeaningisentirelyintermsofthethoughtsandthought-constituentsassociatedwiththem.20Asignificantlinguisticactisonethatisintendedtoarousethoughtsandideasinanaudience.21OnemarkofthisisthatLe´sniewskiandKotarbi´nskifindnopuzzleaboutthelinguisticsignificanceofemptynames:theyhaveconnotations,andthisiswhatisrequiredforbeingmeaningful.Denotationisoptional.ForKotarbi´nskitherulesoflanguagedeterminewhichsentencesindirectlyexpresswhichthoughts;theyarealsosupposedtodeter-mine,derivatively,thatcertainsub-sententialelementsexpress“parts”ofthoughtsinthesensethatthoughtsaboutagivenobject,prop-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01erty,etc.areexpressedbysentencesthathavecommoncomponents.TheissuethatcomestotheforeforTarskiconcernstheconditionsunderwhichtheconventionsthatdeterminewhichsentencesareassert-iblecorrectlyzeroinontheconnotationsthataresupposedtobeexpressedbyvarioussub-sententialexpressions.ThemainresultofourexaminationofKotarbi´nskiisagrasponthisroleofconnotationin10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE36—#25\nIntuitionisticFormalism37Kotarbi´nski’saccount.Connotationstandsbetweenintuitiveconceptsandtheconventionalrolesofthetermsthatexpressthem.Themainthingthattheformalroleofatermissupposedtocaptureisitscon-notation.WecanthusbeonthelookoutforsomethingsimilarinTarski.AswithLe´sniewski,Kotarbinski’sconceptionofconsequenceinadeductivetheoryisexplicitlyepistemicandderivational:Adeductivesystemisasetofsentencesdividedintotwosubsets,suchthateverysentencewhichbelongstothesecondsubsethasbeenderivedexclusivelyfromsentencesbelongingtothefirstsubset,bymeansoftransformationsinaccordancewithknowledge-formingrulesalone[Kotarbi´nski,1966,242].Kotarbi´nskialsoshareswithLe´sniewskitheviewthattheaxiomsofadeductivesystemneednotthemselvesdisplayanykindofself-evidence[Kotarbi´nski,1966,246ff].ThisisaconceptionofconsequenceandtheroleofadeductivesystemsharedwithLe´sniewski,anditisonethatwewillfindthatTarskiacceptsatfirstaswell.ThuswefindthesameviewinKotarbi´nski’s1929workaswefoundinLe´sniewski’sworksofnearly20yearsearlier.Thebasicnotionofmeaningisthatofthethoughtthataspeakerintendstoconveytoanaudience.Languageisheldtobegovernedbyasystemofconventionalrulesthatdeterminewhichthought-contentsareexpressedwhenexpressionsareusedcorrectly,andthetheoremsofadeductivesciencearesupposedtobe“intuitivelyvalid”,thatis,toseemtruegiventheconventions.Sentencesheldtrueastheoremsarethensupposedtodeterminetheconnotationsexpressedbytheexpressionstheyinvolve,butthequestioncanbeaskedofanygivenproposeddeductivesciencewhetherthesystemofconventionsgoverningitadequatelyanswerstospeakers’“referentialintentions”byrulingoutunintendedassignmentsofconnotationtoitstermsonthepartofitsspeakers.1.4Tarskiincontext1.4.1TheaxiomaticmethodCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01SofarwehavelookedatthephilosophyoflanguageofLe´sniewskiandKotarbi´nski,butwehaveyettopaysufficientattentiontotheirviewsabout“deductivesciences”(axiomatictheories)ingeneral.InordertoseehowIntuitionisticFormalismpansoutasaviewaboutdeductivesciences,wecanconsider,incontrasttotheviewwefindinCarnapinHilbert,alineofreflectiononthe“axiomaticmethod”thatrunsfrom10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE37—#26\n38AlfredTarski:PhilosophyofLanguageandLogicapaperofPadoathatTarskiciteswithenthusiasm,throughKotarbi´nskiandontoremarksofTarskithatpersistaslateas1937’s“Surlaméthodedeductive”andtheinclusionofthelatter,withadditions,inTarski’slogictextbook.Thissortofviewaddstotheformalistonetheideathatthepointorfunctionofadeductivetheory,thoughitcanbeunderstoodonitsownintheformalistmanner,istoexpresscertainintuitivelyvalidthoughtsofitscreator.ThislineofthoughtthusjoinstheaccountoflanguagewehavefoundinLe´sniewskiandKotarbi´nskitotheformalstudyofdeductivetheoriesand,inthatway,allowsustoseefurtherwhat“IntuionisticFormalism”meanttoLe´sniewskiandTarski.Padoawrites:Thenecessarypointofdepartureforanydeductivetheorywhatsoeverisasystemofundefinedsymbolsandasystemofunprovedproposi-tions.Besides,itisobviousthatadeductivetheoryhasnopracticalsignificanceifitsundefinedsymbolsanditsunprovedpropositionsdonotrepresent(orcannotrepresent)ideasandfacts,respectively.Thusthepsychologicaloriginofadeductivetheoryisempirical;nev-ertheless,itslogicalpointofdeparturecanbeconsideredamatterofconvention.Indeed,duringtheperiodofelaborationofanydeductivetheorywechoosetheideastoberepresentedbytheundefinedsymbolsandthefactstobestatedbytheunprovedpropositions;but,whenwebegintoformulatethetheory,wecanimaginethattheundefinedsymbolsarecompletelydevoidofmeaningandthattheunprovedpropositions(insteadofstatingfacts,thatis,relationsbetweentheideasrepresentedbytheundefinedsymbols)aresimplyconditionsimposedupontheundefinedsymbols.Thenthesystemofideasthatwehaveinitiallychosenissimplyoneinterpretationofthesystemofundefinedsymbols;butfromthededuc-tivepointofviewthisinterpretationcanbeignoredbythereader,whoisfreetoreplaceitinhismindbyanotherinterpretationthatsatisfiestheconditionsstatedbytheunprovedpropositions[Padoa,1967,120–121].Inthisquotewefindaclearseparationoftheideas“chosen”toberepre-sentedbyundefinedsymbolsandthefacts“chosen”tobestatedbytheCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01axiomsfromthebasicformalistideawehavealreadyfoundinHilbert.WhatPadoa’sconceptionaddsistheideathatoncethetheoryiscon-structedoneneednotknowanythingoftheoriginalideasandthoughtsitsexpressionsandsentenceswere“chosen”toexpress,theideathatthetheoryisintendedtoallowacertainfavoredinterpretation.22Thisisits“empirical”originandthesourceofits“practical”significance.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE38—#27\nIntuitionisticFormalism39Kotarbi´nskiechoesthesentiment(notwithcitationofPadoa,thoughsomelaterworkofPadoaiscitedbyKotarbi´nski)towardtheendofasectiononprovingtheindependenceandconsistencyofaxiomsthroughimposingvaryinginterpretationsonprimitiveterms,writing:Itcanbeseen,however,bothfromthegeneralcharacteristicofthemethodsofprovingindependenceandtheconsistenceofaxioms,andfromtheexamplesanalyzedabove,thatinthisprocedureweacceptastrueorfalsecertainsentencestowhichwereducegiveninscriptionsbycorrespondinginterpretationofundefinedconstants.Forinstance,wehaveacceptedastruethesentence:“ifXisapointandYisapoint,thensomestraightlinecontainsXandY”…Toroundoffinformationonformalisminthebuildingofadeduc-tivesystem,wemayaddthatsuchformalismrecommendsitselfbyacertaineconomyofwork.Byapplyingaformalprocedureinprovingtheoremsofagivendisciplinewemaytherebyalsoprovetheoremsofsomeotherdiscipline,iftheobjectsinvestigatedbythelattersocor-respondtotheobjectsinvestigatedbytheformerthatsentencesofthesameform,accordingtotheinterpretationoftheconstantstheyinclude,statetheoremseitheroftheformerorofthelatterdiscipline[Kotarbi´nski,1966,256].ThiscontrastswithHilbertinawaybroadlyintunewithPadoa:unlikeHilbert,theprimitivetermsofanaxiomsystemareassociatedwithideasnotdeterminedbytheaxioms,anditissimplyanintriguingsortofgeneralitythatitcanturnoutthatthesameaxiomsystemcanreceivedifferentintuitiveinterpretations.Thesystembearsanintendedinter-pretationnotsettledbyitsstructure,thoughgeneralitycanbeachievedbyvaryingtheinterpretationimposeduponit.Tarski’sownremarksonformalismhaveasimilarflair.Considerfirstthisone,fromSurLaMethodeDéductive[Tarski,1986f],whichwaslaterincorporatedintoTarski’slogictextbook:23Asisknown,everymathematicalscienceisfoundeduponasuitablychosensystemofprimitivetermsandaxioms.Ourknowledgeoftheobjectsdesignatedbytheprimitivetermscouldbequiteextensive,Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01andtheacceptedaxiomsneednotexhaustit.Yetthisknowledgeisourprivateaffair,exercisingnottheleastinfluenceupontheconstruc-tionofthescience.Thus,inderivingatheoremfromtheaxioms,wemakenouseofthisknowledge—weconductourselvesasthoughwedidnotunderstandthesenseoftheconceptswithwhichweoccupyourselvesintheseconsiderations,asthoughweknewnothingbeyond10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE39—#28\n40AlfredTarski:PhilosophyofLanguageandLogicthatwhichwasexplicitlystatedintheaxioms;weabstractfromthesenseoftheprimitivetermsweadmitandweattendtonothingbuttheformoftheaxiomsinwhichthesetermsappear[Tarski,1986f,331](mytranslation).Interestingly,intheEnglishversionofthepassageinthetextbookof1941,Tarskihedgesbysayingthatourgraspontheprimitivetermsis“sotospeak”ourprivateaffair,whileintheGermanversionthedec-larationisunhedged,asintheFrench,buttheGermanassertsthattheaxioms“mustnot”exhaustourknowledge(mußkeinesfallsdurchdieangenommenenAxiomeerschöpftwerden).PadoaanticipatessomeaspectsofIntuitionisticFormalism.Theinsis-tencethattheintendedinterpretationofatheoryissomehowimportant,butalsostrictlyirrelevanttothetheoryitselfisoneimportantpointhere.WehavealreadyseenexactlythisemphasisontheseparabilityofadeductivetheoryfromitsintendedinterpretationintheopeningpagesofODS[Tarski,1983f,112].PerhapsevenmoreimportantisPadoa’sentirelypsychologistictakeon“interpretation”,whichhereaselsewhereinthearticleisclearlysomething“inthemind”.24ThisissomethingtowhichLe´sniewskiandKotarbi´nskiwereentirelysympathetic,andTarski’stalkofourknowledgeofthedenotationsoftheprimitivetermsbeing“ourprivateaffair”showsthathewas,too.25Toreturnagaintoourcon-trastfromtheintroduction,IntuitionisticFormalismisaviewwithinexpressiveratherthanrepresentationalsemantics.26Itissometimesargued(e.g.[Hodges,2009,482])thatIntuitionis-ticFormalismisactuallyincompatiblewithrecognizingthatanaxiomsystemcanbesatisfiedbymanymodels.Thisisn’tthecase.Infactthedoctrinetookthisfullyintoaccount.RecallthepsychologisticaspectofIntuitionisticFormalismthatisalreadyatworkinPadoaandcontinuesintoLe´sniewski.NoticethatLe´sniewskitalksof“plac-ing”or“putting”determinatesensesonthethesesofhissystem,andofthesetheses,takenwiththesemeanings,having“intuitivevalid-ity”forhim.ThepictureisoneofLe´sniewskihavingputhimself(andperhapsothers)inapositiontoassociatecertainthoughtswithcer-tainformulaeofthesystem,thesysteminturnbeingdesignedinCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01suchawaythatinferenceaccordingtoitsrulesalwaysleadsfromsen-tencesthatexpressintuitivelyvalidthoughtstoothersthatdosoaswellviaintuitivelyvalidinferences.Thereneedbenoideaherethatthesystemcouldn’tbearotherinterpretationsaslongastheinfer-entialrelationsamongtheotherthoughtswerethesame.Wewillseemoreaboutthispsychologismandhowitaffectsthedoctrine,10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE40—#29\nIntuitionisticFormalism41especiallywhenwelookatTarski’sunderstandingofPadoabelow(§2.2.3).1.4.2MonismvstoleranceLe´sniewski,intendedthethoughtsexpressedinhisformalsystemstocapturethetruenatureofthings;see[Le´sniewski,1992h,177],dis-cussedabove,on“non-intuitive”,merelynon-contradictorysystemsascontrastedwithan“intuitive”mathematicsthatis“directedresolutelytowardsreality”,aswellashisvariousdisparagingremarksaboutformal-ism.Le´sniewski’ssystems,andthetheoryofsemanticalcategoriesonwhichtheyrested(see§4.2.2),weresupposedtobethecorrectsystemoflogicandfoundationsofmathematics.BythetimeofLogicalSyntaxtherethuscomestobeamajordifferencebetweenCarnapandLe´sniewskionwhatCarnapcallsthe“principleoftolerance”.In1929theCarnapoftheAbrissworksentirelywithintheSTT:27“Logistic”justissimpli-fiedtypetheorydirectedtowardwardingoffparadoxbyprohibitionof“level-mixing”[Carnap,1929,20],butitisalsoa“sharptool”fortheanalysisofassertionsandconceptsineverydomain[Carnap,1929,iii].By1934Carnaphadmovedtoaforthrightlypluralisticattitudetowardlogic:Thefollowingexplanationsareconcernedwiththesymbol‘=’con-sideredasthesymbolofidentityinthenarrowersense(thatistosay,asusedbetweenZorbetweenobject-designations)andnotthesym-bolofequivalence(thatistosay,asusedbetweenS).ThesymbolofidentityoccursinLanguagesIandII(andalsointhelanguagesofFrege,Behmann,Hilbert)asanundefinedsymbol.FollowingLeibniz,Russelldefines‘x=y’inthefollowingway:“xandyagreeinalltheir(elementary)properties.”Wittgensteinrejectsthesymbolaltogetherandsuggestsanewmethodfortheuseofvariablesbywhichitmaybeavoided.Philosophicaldiscussionsconcerningthejustificationofthesevar-iousmethodsseemtoustobewrong.Thewholethingisonlyaquestionconcerningtheestablishmentofaconventionwhosetech-nicalefficiencycanbediscussed.NofundamentalreasonsexistwhyCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01thesecondorthirdofthesemethodsshouldnotbeusedinsteadofthefirstinLanguagesIandII[Carnap,2002,49].Ourattitudetowardrequirementsofthiskindisgivenageneralfor-mulationinthePrincipleofTolerance:Itisnotourbusinesstosetupprohibitions,buttoarriveatconventions[Carnap,2002,51].10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE41—#30\n42AlfredTarski:PhilosophyofLanguageandLogicInlogic,therearenomorals.Everyoneisatlibertytobuilduphisownlogic,i.e.hisownformoflanguage,ashewishes.Allthatisrequiredofhimisthat,ifhewishestodiscussit,hemuststatehismethodsclearly,andgivesyntacticalrulesinsteadofphilosophicalarguments[Carnap,2002,52].AsRickettsnotes,withthisCarnaplosestheconnectiontotheexpressionofthoughtsthatwassocentraltoFrege’sprojectinLogic:Inrepudiatingtheidealofaunique,science-embracinglogic,CarnaprejectsFrege’sassumptionofacommonstoreoflogicallyinterre-latedthoughtsexpressedbythesentencesofcolloquiallanguageandperspicuouslyexpressiblebysentencescouchedintheframeworkofBegriffsschrift.Carnapcannotthenrelyonanyoverarchingnotionofcontenttogivehissyntacticinvestigationstheirapplicationtoactualorhypotheticallanguagesforscience[Ricketts,2004,191].WecannowappreciatethedepthofCarnap’srejectionofFrege’scon-ceptionofathoughtasthatforwhichthequestionoftrutharises.Linguisticbehavioris,sotospeak,initselflogicallyamorphous.Webringlogictoitbycoordinatingacalculuswithit.Thescientificphilosopher,thelogicianofscience,describesvariouscalculi.Shethencanfreelypickanyofthesecalculi,andenvisionhergrouptospeakalanguagecoordinatedwithit.Inthiswaysheappliestohergroup’shypotheticalutterances,thesyntacticexplicationsofepis-temicnotionsthatthesyntacticdescriptionofthecalculusmakesavailable.Applicationoftermslike“true,”“false,”“consequence,”and“consistent”inthelogicofsciencebecometolerablypreciseonlyviasuchcoordination…Thisishowtoleranceinlogicispossible[Ricketts,2004,193].Le´sniewski,andwithhimKotarbi´nski,wouldnotapprove:forthemthewholepointofdesignedconventionsoflanguagewastofaciltatetheclearexpressionofthoughtsthatwereindependentofthesystemandassumedtobeclearlygrasped.RecallinthiscontextthatFregeistheonlypredecessorforwhomLe´sniewskiexpressesmuchrespectin[Le´sniewski,Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-011992h].WewillseeTarski’sviewchangeonthisissuebetweenthebodyofCTFLandthePostscript.WhatisofsignificanceforusatthisjunctureissimplythestronglyepistemicconceptionoflogicalconsequencethatisattheheartofIntuitionisticFormalismasLe´sniewskiconceivesofit.Adeductivetheory,forLe´sniewski,isatoolfortheexpressionofthoughtandreasoning.Itisn’tanobjectofstudyinitsownright,anditmust10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE42—#31\nIntuitionisticFormalism43therebybecraftedtorespectthelogicalrelationshipsamongthoughtsintuitivelyjudged.1.4.3FivedoctrinesWehavenowseenfivemainaspectsofIntuitionisticFormalisminLe´sniewskiandintherelatedphilosophyoflanguageofKotarbi´nski:1.Theunderstandingoflanguageasamediumfortheexpressionofthoughtviaaspeaker’sintentions.2.Theconceptionofconventionsoflanguageasdeterminingwhichthoughtsareexpressedbytheuseofagivensentencewhenalanguageisusedcorrectly.3.Theconceptionofaxiomsandtheoremsassentencesthatareassert-ibleinthesenseofseemingtrue,giventheconventions,tothosewhoarepartytothem.4.Theconceptionoftheintuitivemeaningorcontentofatermintermsofthetraditionallogicalnotionofconnotation.5.Thegoal,intheoryconstruction,ofconstrainingtheinterpretationsoftheprimitivesasmuchaspossiblethroughtheimplicationoftheo-remsthatconstraintheassignmentofintuitivemeaningtothetermsofatheory.AlloftheseaspectsoftheviewremaininTarski’sworkoftheperiodinwhichweareinterested,savethatinhisconceptionofthefourthhemakesnoreferencetodoctrinesaboutconnotationfromtraditionallogic.28Hekeeps,however,theideathatsomethinglikeconceptualanal-ysistraditionallyconceivedispossibleandimportant,andhisdefinitionsareintendedtoanswer,thoughnotinthewayusuallyassumed,tosuchanalyses.LetusherebrieflymarshalsomeprimafacieevidencethattheseviewsaretobefoundinTarski’sworksoftheperiod.(1)LanguageasExpressiveThatlanguageisamediumofexpressionviaspeakers’intentionsis,aswesaw,animportantdoctrineinTwardowskiwhichisworkedoutmorethoroughlyinKotarbi´nski’saccountofdirectstatement.NowTarskiwritesthat“normallyexpressionsareregardedastheproductofhumanactivity”andlinksthistotheissueofnominal-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01ism[Tarski,1983a,174,note2].The“humanactivity”herecanonlybeexpressingthoughtsasunderstoodbyKotarbi´nski.Thereferencetoactionsandproducts,aswellastonominalism,areclearreferencestoTwardowskiandKotarbi´nskiandTarski’ssheepishnessaboutneverthe-lesssettingthepointasideforthesakeofdoinglogicindicatesthathestillacceptstheseviewsasessentiallyright.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE43—#32\n44AlfredTarski:PhilosophyofLanguageandLogicAnotherindicationofTarski’ssharingthisorientationcomesfromapointobliteratedbyWoodger’stranslation.Considerthispassage:Nowformalizedlanguageshavehithertobeenconstructedexclusivelyforthepurposeofstudyingdeductivesciences[Tarski,1983a,166].AsHodges[Hodges,2008,98]notes,Woodger’s“studyingdeductivesciences”rendersTarski’s“uprawiacsformalizowanenaukidedukcyjne´”[Tarski,1933,16],and“uprawiac´”meansnotstudybutratherpractice.Blausteinhasthisrightwith“formalisiertededuktiveWissenschaftenzubetreiben”[Tarski,1986a,72].TheideaisessentiallyLe´sniewski’s,asHodgesdiscusses:Tarskirefersseveraltimesto‘practising’or‘performing’adeductivetheory;sometimesheadds‘onthebasisofitslanguage’…Thereisreallyonlyoneactivityhecanhaveinmind.Thisistheactivityofsittinginfrontofapieceofpaperandwritingasequenceofformulasinthelanguageofthetheory,whereeachformulaiseitheranaxiomofthetheory,oradefinition,orsomethingderivedfrompreviousrowsbyrulesofthetheory.Therearenotmanypublishedexamplesofthisactivity…ForvirtuallypureexamplesoneshouldlookatpapersofLe´sniewski…[whosepaper]onAbeliangroupsiselevenpageslong,andnearlyeightpagesconsistofaformalderivationusingnothingoutsideoftheformallanguageandsymbolstoindextheapplicationsofrules[Hodges,2008,97].Theworkingmathematicianprovesthingsinformallyatfirst.Butproofsneedtobecheckedandcommunicatedtootherpeople.Forthesepurposesmathematicianshaveagreedoncertainwaysoffor-malizingtheirintuitivethoughts,namelyasdeductivetheories.Theyhaveagreed,forexample,thatweshouldtakeasprimitivetermsofourdeductivetheoriesexpressionsthat‘seemtoustobeimmediatelyunderstandable’,andweshouldtakeasaxioms‘statementswhosetruthappearstousevident’…Theidealmathematiciansetsupdeduc-tivetheoriesthatmeettheserequirementsandthen‘practises’thesedeductivetheories.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01IbelievethatthepreviousparagraphsummarizeswhatLe´sniewskimeantwhenhecalledhimselfanIntuitionisticFormalist,andwhatTarskimeantwhenhesaidin1930…thatthiswashisposition,too.[Hodges,2008,103].Thisfurtherexplainswhytheexpressionsofthedeductivetheorieswestudyindefiningtruthhavetobemeaningful:thetheoriesareactually10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE44—#33\nIntuitionisticFormalism45tobeusedtoexpressintuitivelytruethoughts.WhatisinterestinghereisthatalthoughTarski,asametamathematician,isinterestedinstudy-ingdeductivesciences,asaphilosopherhisviewisthatthepointofadeductivescienceisthatitistobeusedclearlytoexpressthoughts,justasLe´sniewskimaintained.Indeed,aswewillseeintheremainder,inthelate1920sthroughtheearly1930sTarski’saiminstudyingformal-izedlanguageswastostudytheconditionsunderwhichthegoalssetfortheirusesoconceivedweremet.Itistothisendthatheconceivedtheprojectofseeingtowhatextentthemetamathematicalnotionsneededtostudyandevaluateadeductivesciencecouldthemselvesbeincorpo-ratedwithinadeductivesciencethatcouldbeusedrigorouslytoexpressmetatheoreticthought.Thus,sinceTarskiholdsthatdeductivetheoriesaretobe“practised”hehastobethinkingoftheminthefirstinstanceasvehiclesfortheexpressionofthought.Commentatorsoftennotefamiliarcontrastsbetweenlogicconceivedofasamediumfortheexpressionofthought,andlogic,orlogicalsystems,asobjectsofstudyandapplythemtoTarski.Sundholm[Sund-holm,2003]stressesthedistinctionbetweenwhathecalls“languageswithmeaning”and“languageswithoutuse”,wheretheformerarelan-guagesdesigned,inthetraditionofFrege,tobeactuallyusedtoexpressmathematicsandthelatterare,inthetraditionofThomaeandHilbert,tobeobjectsofmetamathematicalstudy.SundholmcharacterizesTarski’sprogressionduringtheperiodfromtheformulationofCTFLin1929tothePostscriptof1935asashiftfromaLe´sniewskianadherencetothefirsttraditiontoamoremathematicalattachmenttothesecond.OnethingIwilldohereistostressthat,totheextentthatthetwoperspectivesadequatelycharacterizetheendpointsofTarski’sdevelopmentduringtheperiod,therewasanintermediatephase,andalloftheimportantworkoccursduringit:namely,aphaseinwhichTarskitriedtoblendbothtraditionsbydesigningdeductivetheoriessituatedwithinthefirsttraditionpreciselytobeusedforthesecondsortofactivity.Thesamepointthusapplies,mutatismutandistotherelatedcharacterizationsoftwoperspectivesonlanguageof[Hintikka,1996](linguauniversalisvscalculusratiocinator)and[vanHeijenoort,1967](logicaslanguagevslogicascalculus).Ialsodatetheabandonmentoftheearlierperspec-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01tive,intheformonIntuitionisticFormalism,laterthanSundholmdoes,sinceIfinditnotinthe1935PostscripttoCTFLbutinthedefinitionofconsequenceintermsofmodels,sincethepostscriptisstillcommit-tedtoaderivationalconceptionofconsequence,centraltoIntuitionisticFormalism,whichisoverturnedonlyin“OntheConceptofFollowingLogically”(CFL).10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE45—#34\n46AlfredTarski:PhilosophyofLanguageandLogic(2)ConventionOnesignificantpieceofevidencehereistheoft-notedfactthatTarskicallsConventionTa“Convention”(Konvention,umowa).ThisaccordsperfectlywellwithLe´sniewski’sviewthatadefinitionisakindofconventionfortheuseofaterm.Nowtheappealtoconventioncanbeexpectedtobesomewhatindirect,sinceinLe´sniewski’slaterworktheappealtoconventionistakenupinformationrulesanddirectivesofhissystems:rulesfortheformationofsentences,andrulesfortheassertionofonesentenceonthebasisofothers.Inlightoftheearlierwork,however,wecanseethatsuchrulesareconceivedofasplayingtheroleoflayingdownprinciplesfortheuseofsignsthatsuitthemtoexpressvariousintuitivecontents.ThroughoutthearticleswithwhichweareconcernedTarskiofcoursethinksofdeductivesciencesaspreciselygovernedbysuchrules[Tarski,1983f,113–116],[Tarski,1983a,166].SinceTarskiisconcernedwiththeexpressionofintuitivemeaningindeductivesciencesasLe´sniewskiwas,hisconceptionofsuchruleswouldhavebeenalongLe´sniewski’slinesaswell,andsinceLe´sniewski’sconceptionoflanguageandmeaningwasalliedwithKotarbi´nski’s,Tarski’sconceptionoftherelationbetweentherulesgoverningsentencehoodandtheoremhoodinasystemwouldhavebeeninaccordwithKotarbi´nski’saccountoftheroleofconventionindeterminingwhatis“indirectlystated”byasentence.Onefurtherpieceofapparentevidencerequiresmoredelicatehan-dling.TarskiusescertainphrasesthatcomeoutinWoodger’stransla-tionsoundingagooddealliketheEnglishtranslationofKotarbi´nski’sElementy.Forexample,inCTFLwefindthisfamiliarpassage:Wetakethescheme(2)andreplacethesymbol‘x’initbythenameofthegivensentence,and’p’byitstranslationintothemetalan-

guage.Allsentencesobtainedinthisway,e.g.‘(ι+ι)isa121,22,1truesentenceifandonlyifforanyclassesaandbwehavea⊆borb⊆a’,naturallybelongtothemetalanguageandexplaininapreciseway,inaccordancewithlinguisticusage,themeaningofphrasesoftheform‘xisatruesentence’whichoccurinthem[Tarski,1983a,187].ThisseemstocomparefavorablytopassageslikethisinKotarbi´nski:Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Agivenphrasestatesacertainexperienceindirectlywhenever,with-outstatingitdirectly,suchcan,inviewoflinguisticusageoradoptedconventions(naobyczajtegojke˛zykalubustalonecodoniegoumowy),beusedinagivenlanguageasadirectstatementofsuchanexperience[Kotarbi´nski,1966,4][Kotarbi´nski,1961,14].10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE46—#35\nIntuitionisticFormalism47UnfortunatelythisisoneofthepointsatwhichTarskiisactuallywork-ingbehindthescenestoburythepervasiveappealtointuitionfoundintheoriginalPolishofCTFL.ThePolishphrasethatWoodgerhasas“inaccordancewithlinguisticusage”,followingBlaustein’s“mitdemSprachgebrauchübereinstimmender”[Tarski,1986a,45],is“zgodnyzintu-icjka˛”[Tarski,1933,39]:consistentwithintuition.(Thesameswitchcanbefoundonthepreviouspage.)Thisisofcoursehelpfultousinargu-ingthatTarskiwasconcernedwithintuitivemeaningintheoriginalwork,butitrendersproblematicthecurrentattempttogatherevidencethatTarskiconceivedoflinguisticexpressionsassignificantbecausetheyaregovernedbyconventionalrulesthatdeterminewhatintuitivecon-tentstheyexpress.SinceBlausteindoctoredthetranslationunderTarski’sdirectionIthinkwecanbefairlyconfidentthattheGermanwordingthatWoodgerfollowsisTarski’s.ButsinceTarskiwashappywithcog-natesof“intuitive”inFrenchandPolish,butnotinGermanandEnglish(cf.§4.1.1),themotivationforthechangeintheGermantranslationwaspresumablythedesirenottoraisehacklesamongGermanreaders.(ForanotherexampleofTarski’sstrategic“translations”fromPolishtoGerman,see§4.3.3.)Sothequestioniswhether“mitdemSprachgebrauchübereinstimmender”wasconceivedofbyTarskiasappealingtotheconventionalistideasexpressedinKotarbi´nski’s“naobyczajtegojke˛zykalubustalonecodoniegoumowy”,orwasratheraphraseheknewfromsomeGerman-languageauthorthathethoughtwouldgodownbetterinGermanyandAustriathan“mitdenIntuitionübereinstimminder”.CertainlytheagreementwithKotarbi´nski’sviewwouldhavebeenmanifesttoTarskiandtheappealtolinguisticusagewouldalsolikelyhavestruckhimasagoodsubstitutefortheappealtointuition,backedupbyKotarbi´nski’sviews,sinceinwell-designedconventionslinguisticusagedoescorrespondtoandexpressourintuitionsaboutwhatexpressionsmean.SoIaminclinedtothinkthatthisissomeevidencethatTarskiacceptedKotarbi´nski’snotionofcon-vention,butmattersinthatrespectaren’tentirelyclear.WhatiscleareristhatthewordingisTarski’s,sinceBlausteinisn’totherwisesofreewithhistranslations,sowecangrantthatTarskiwashappyexpressinghimselfbyappealtotheideathatwhathewastryingtocaptureinadeductiveCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01sciencewastheordinaryusageofphrases,andsurelyhereincorrectusagewouldn’thavebeentothepoint,sothereissomenormativenotionclosetotheappealtoconventionpackedintothephrase.Inanycase,intheendtheappealtoconventionhereisn’tterriblycontroversialasascribedtoTarski:ifonethinksofanaxiomatictheoryasintendedtohavethe-oremsthatconveycertainthoughts,onehastothinkofitasgoverned10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE47—#36\n48AlfredTarski:PhilosophyofLanguageandLogicbystipulationsastohowexpressionsaretobeused,andthesearejustrulesfortheuseofthesentencesuponwhichusersofthesystemagree.(3)AssertabilityItiscrucialtokeepinmindthatTarskitakesthemeaningoftheprimitivesofadeductivesciencetobesettledbywhichsentencesaretreatedastrue,notwhichsentencesactuallyaretrue[Ray,2003].Tarski’sconceptionofthemeaningsofprimitivetermsispartofageneraltheoryofdeductivetheories—setsofsentencesclosedundersomeassumedrelationofconsequence[Tarski,1983d,40]—thegener-alityofwhichwouldbepointlesslycompromisedifthesentencesinquestionhadtobetrue.Whenevertheclaimthatprimitivetermshavetheirmeaningssettledbyaxiomsisunderdiscussion,Tarskitalksaboutsentencesasbeing“heldtrue”orofwhat“canbeasserted”,e.g.[Tarski,1983a,167],[Tarski,1983k,299]and[Tarski,1983k,299,note1].29Theclaimthatprimitivetermshavetheirmeaningsestablishedbysentencesthatareheldtrueratherthanbysentencesthataretrueiscrucialtounder-standingTarski’sconceptionofaninconsistentlanguage,aswewillseelaterin§5.2.2.(4)ContentandConnotationTarskispeaksoftenofdefining“thecon-tent”ofaconcept,ratherthanjustdefiningaconcept[Tarski,1983a,187],[Tarski,1983b,401][Tarski,1983h,409].At[Tarski,1983f,112]wehavethe“content”oftheterm“definablesetofrealnumbers”,butweknowfromKotarbi´nskithatthisisalsothecontentoftheconceptofadefinablesetofrealnumbers.Thislocutioncanlooktothecontem-poraryreaderlikeaverbalhiccup,butitisinfactanacknowledgementofdoctrinesaboutthecontentofmentalactshailingfromTwardowskiandacceptedbyKotarbi´nskiandtheearlyLe´sniewski.Onthisview,a“concept”isamentalparticular,whileits“content”iswhatintheperiodmighthavebeencalledits“intentionalobject”.Tarski’s“contentsofconcepts”arethereforefairlyclosetowhatthecontemporaryreaderwouldmeanby“concept”itself.Tarskiappearstohavefoundthelocu-tionuncomfortable,becausehealsosometimestreats“contentoftheconcept”and“theconcept”assynonymous—adefiniteno-nobyTwar-dowski’sstandards,butonethatmatchesbettertheuseof“concept”outsideofPoland.Nevertheless,theappearanceofthelocution,andthetreatmentofthecontentofaconceptaswhatadefinitionissupposedtoCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01capture,indicatethatthebasicideasatworkaretheoneswehavefoundinLe´sniewskiandKotarbi´nski.(5)CapturingContentsPassagesonthewayinwhichaxiomsandtheo-remsexpresspropertiesofconceptsordeterminemeaningscanbefoundat[Tarski,1983g,31],[Tarski,1983c,63]and[Tarski,1983f,112].OurmaintaskinthenexttwochapterswillbetoestablishthatTarski’sproject10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE48—#37\nIntuitionisticFormalism49wasthecaptureofthecontentsofmetamathematicalconceptswithindeductivemetatheories,butthecitedpassagesmakeatleasttheinitialcasethatTarskiisthinkingofthingsalongLe´sniewski’slines.HavinginhandagoodideaofwhatIntuitionisticFormalismreallywasforTarski,wecannowreadafamiliarpassageinanunfamiliarbutmuchmoreilluminatingway:Itremainsperhapstoaddthatwearenotinterestedherein‘formal’languagesandsciencesinonespecialsenseoftheword‘formal’,namelysciencesofthesignsandexpressionstowhichnomeaning(inhaltlichSinn)isattached.Forsuchsciencestheproblemheredis-cussedhasnorelevance,itisnotevenmeaningful.Weshallalwaysascribequiteconcreteand,forus,intelligiblemeanings(Bedeutungen)tothesignswhichoccurinthelanguagesweshallconsider[Tarski,1983a,166–7].Itisgenerallythoughtaboutthispassagethatitexpressescommitmenttotheideathattheconceptoftruthwillbestudiedinlanguagesthatare“interpreted”inthemodel-theoreticsense.Wecannowsee,however,thatthepassageisactuallyanexpressionofIntuitionisticFormalism.The“concrete,intelligiblemeanings”atissuearen’tmodel-theoreticinter-pretants,butintuitiveconcepts.30Substantiatingthisclaimwillhavetowaituponafullerunderstandingofthemannerinwhich,in1931,Tarskisimplycouldn’thavebeenthinkingofmeaningintermsofhissemantics.311.4.4Tarski’sprojectAsnotedabove,whenitcomestotherelationbetweentheintuitivecon-ceptintendedtobeexpressedbyatermandtheformalroleinstitutedforthattermbythestructureofanaxiomatictheory,thepossibilityofvariouskindsofmismatchandfailurethreatens:theconventionsgov-erningtheuseofasentenceinvolvingatermintendedtohaveacertainconnotationmightfailtoconstraintheassociationofthatconnotationwiththetermdowntotheintendedone.Aprojectthusbeckons:investi-gatingtheconditionsunderwhichtheusageoftermsisadequatetotheCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01connotationsgraspedintheconceptswithwhichtheyareassociated.Asimpleexamplewillillustratethepoint;wewillmeetTarski’sownapplicationssoonenough.Considerthreepredicates,F,GandH,intendedtoconnotethepropertiesofbeingscarlet,beingred,andbeingblue,respectively,andsupposeforthesakeofexamplethatitissomesortofconceptualtruththatanythingscarletisred.If,accordingtothe10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE49—#38\n50AlfredTarski:PhilosophyofLanguageandLogicconventionsofalanguage,(∀x)(Fx→Gx)isn’tassertablethentheconno-tationofFhasn’tbeenfullycaptured,sinceanyonewiththeconceptofbeingscarletwillacceptthatanythingscarletisred,whileif(∃x)(Fx&Hx)isassertable,thentheconnotationofFhasn’tbeencapturedatall,sinceanyonewhopossessestheconceptofbeingscarletwillacceptthatnoth-ingscarletisblue.Onlyalanguageinwhich(∀x)(Fx→Gx)isassertablebut(∃x)(Fx&Hx)isnotisalanguagethatdoestheLe´sniewskianjobofrepresentingtheintuitiveconnectionsamongthoughtsexpressedwhenF,GandHareintendedtoexpresstheconceptsofbeingscarlet,beingred,andbeingblue.Sowehavetwoquestionsaboutanygiventerminadeductivethe-ory:aretheconventionsgoverningtheassertablesentencesinwhichitappearssufficienttoexpressitsintendedconnotation,andaretheconventionsgoverningthemconsistentwithitsexpressingitsintendedconnotation?Therequirementthatthesequestionsbeansweredintheaffirmativeis,inembryo,the“materialadequacy”requirementofConventionT.32Theproject,then,istocraftadeductivetheorythatforces,thoughitstheorems,itsprimitivetermstoexpressthecontentsofcertaincon-cepts.ThisiswhatremainsinTarski’sworkoftheappealtoconnotationinLe´sniewskiandKotarbi´nski.TheusualsortofconceptualanalysisiswhatTarskireferstowhenheremarksthatthereaderisfamiliarwiththeconceptoftruthandcanfinditdiscussedinworksonthe“theoryofknowledge”at[Tarski,1983a,153]anditiswhathereferstowhenhesaysthatheisconcernedwiththe“content”of“definablesetofrealnumbers”andwillremindthereaderofitat[Tarski,1983f,112].ThenthepointthatarisesfromLe´sniewski’streatmentisthattheroleofasignintendedtoexpressaconceptwithinadeductivetheory,determinedbyconventions,mustcaptureitsintuitivecontent.Inwhatwayisthecontentofaconcepttobecapturedinadeductivetheory?Thesimplestwayisforastatementoftheanalysistobeamongthetheoremsofthetheory.This,aswesaw(§1.2.2)wasLe´sniewski’swayofsettlingtheinterpretationof“ε”inhisOntology[Le´sniewski,1992h,367–8].WewillfindanexampleofthisinthenextchapterinTarski’searlytreatmentoftheconsequencerelation.ButtherelationshipneedCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01notinvolveanexplicitstatementoftheanalysisappearingamongthetheorems:whatisrequiredissimplythatacceptingtheconventionsandacceptingtheaxiomsastrueconstrainsarationalthinkertoassociatetherightcontentwiththetermsofthetheory.LaterwewillseethatTarski’streatmentofthecontentoftheconceptoftruthexemplifiesthisinex-plicitformofcapture:theT-sentencesbeingtheoremsofthemetatheory,10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE50—#39\nIntuitionisticFormalism51giventheinterpretationsoftheotherexpressionsoftheobject-andmet-alanguageswillforce“istrue”toexpresstruthconceivedaccordingtothe“semanticaldefinition”[Tarski,1983a,155]eventhoughtheT-sentencesaren’tthemselvespartoforimpliedbythisanalysisoftruth.Anidealdeductivetheory,fromanIntuitionisticFormalistperspective,wouldbeonemaximallyimmunefrommisunderstanding,atheorysuchthattwothinkerswhocontemplatedit,acceptedtheconventionsgov-erningtheconditionsunderwhichsentencesareassertable,andaccepteditstheoremsastruewouldhavetoassociatethesameconceptswithitsterms.Inmostcases,forobviousreasons,thisisunattainable:thethe-oryhasprimitiveterms,andtheinterpretationsofthesecanbevaried;indeed,wehaveseenthatitcanbeofsomevalue,inKotarbi´nski’sphrase,foran“economyofwork”.Butatleastthefollowingisagoal:theaxiomsandprimitivesshouldbekepttoaminimumsothattheworkofsecuringagreementamonginterlocutorsonintuitivemeaningismadeassimpleaspossible,andasmuchaspossiblethatisofinterestinthedomaininquestionshouldbesettledcompletelyoncetheaxiomsareaccepted.Thelimiting,idealcase,onTarski’sview,however,isatheorythatdoesuniquelydeterminetheconceptsexpressedbyitsprimitiveterms(§2.3).Tarski,havingacceptedIntuitionisticFormalism,setouttostudytheextenttoandtheconditionsunderwhichthiscouldbeachieved.ButTarskialsofocusedonaparticularapplication:“metamathematics”,con-ceivedofasthedisciplineinwhichdeductivetheorieswhoseintuitivesubjectmatterwasotherdeductivetheoriesareused:Thedeductivedisciplinesconstitutethesubject-matterofthemethod-ologyofthedeductivesciences,whichtoday,followingHilbert,isusuallycalledmetamathematics,inmuchthesamesenseinwhichspatialenti-tiesconstitutethesubject-matterofgeometryandanimalsthatofzoology…Thepresentstudies…aimistomakeprecisethemeaningofaseriesofimportantmetamathematicalconceptswhicharecommontothespe-cialmetadisciplines,andtoestablishthefundamentalpropertiesoftheseconcepts[Tarski,1983c,60]Tarskiwantsto“makeprecisethemeaning”ofconceptswhichareCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01deployedinmetatheoreticthought,andtodosowithindeductivetheories.TheneedtobringmetamathematicsuptothestandardsofLe´sniewskianIntuitionisticFormalismmeantthatTarskihadtobegintoformalizethingsthatpreviouslywentunformalized,andhewasproudofthisachievement:“asfarasIknowthemetatheoryhasneverbeengivenintheformofanaxiomatizedsystem”[Tarski,1983a,173note3].10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE51—#40\n52AlfredTarski:PhilosophyofLanguageandLogicInthe“HistoricalNotes”Tarskiclaimsindependentdiscoveryof“theaxiomaticconstructionofthemetasystem…andinconnexionwiththis…thediscussionsonpp.184f.ontheinterpretationofthemeta-systeminarithmetic…whichwasdevelopedfarmorecompletelyandindependentlybyGödel”[Tarski,1983a,277–8].WeturnnexttoTarski’sfirstcontributiontothisproject.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-0110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP01”—2011/11/29—14:24—PAGE52—#41\n2TarskiasIntuitionisticFormalistAcceptingthebasicsofIntuitionisticFormalism,Tarskisoughttoextendit.Inparticular,inspiredperhapsbyLe´sniewski’sstepstowardformal-izedsyntax[Le´sniewski,1929],[Le´sniewski,1992e],Tarskibegantolookattheissuesthatarosewhenone’sdiscourseaboutadeductivetheorywasitselfexpressedinadeductivetheory.Thisisthesortofprojectthatwouldhaveattractedanambitiousyoungcontributor:investigatewhetherthegoingviewamonghisteachersabout“metamathematics”couldhandlemetamathematicalconceptsthemselves.Hisfirstpublishedcontributionstothisprojectwereseveralpapersonaxiomsfortheconse-quencerelation.WewilllookatthoseworksinordertoseethebeginningofTarski’sproject.Afterthatwewilllookathismostextensivecontribu-tiontothegeneralproject,“SomeMethodologicalInvestigationsontheDefinabilityofConcepts”,whichcollectsworkfrom1926and1932.2.1Theearlymetamathematicalworks2.1.1AxiomatizingconsequenceTheaimofthepaperscollectedasessaysIIItoVofLogic,Semantics,Metamathematicsistocaptureaxiomaticallythecentralconceptoflogic:consequence,onesentence’sfollowingfromsomeothers.1Thepaperswillbeofinteresttousbothbecausetheyprovideanearlyexam-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01pleoftheIntuitionisticFormalistprojectandbecausethetreatmentoftheconsequencerelationisoneofthemostimportantaspectsofTarski’sdevelopmentinthe1920sand1930s.TarskidevelopsextensionalsemanticsasacontributiontotheIntuitionisticFormalistanalysisofmetamathematicalconcepts,butwellintothisproject,aswewillsee—aslateas1934–5,infact—Tarskiremainsweddedtoaconceptionof5310.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE53—#1\n54AlfredTarski:PhilosophyofLanguageandLogicconsequenceonwhichforonesentencetofollowfromothersissimplyforittobederivedfromthemthroughintuitivelyvalidrulesofinference.Tarskistateshisgoalsinthefirstofthethreepapersasfollows:Ourobjectinthiscommunicationistodefinethemeaning(denSinn…zupräzisieren),andtoestablishtheelementaryproperties(elementarenEigenschaftenfestzustellen),ofsomeimportantconceptsbelongingtothemethodologyofdeductivesciences,which,followingHilbert,itiscustomarytocallmetamathematics…Formalizeddeductivedisciplinesformthefieldofresearchinmeta-mathematicsroughlyinthesamesenseinwhichspatialentitiesformthefieldofresearchingeometry.Thesedisciplinesareregarded,fromthestandpointofmetamathematics,assetsofsentences.Thosesen-tenceswhich(followingasuggestionofS.Le´sniewski)arealsocalledmeaningfulsentences(sinnvolleAussagen),arethemselvesregardedascertaininscriptionsofawell-definedform…FromthesentencesofanysetXcertainothersentencescanbeobtainedbymeansofcertainoperationscalledrulesofinference.Thesesentencesarecalledconse-quencesofthesetX.Thesetofallconsequencesisdenotedbythesymbol‘Cn(X)’.Anexactdefinitionofthetwoconcepts,ofsentenceandofcon-sequence,canbegivenonlyinthosebranchesofmetamathematicsinwhichthefieldofinvestigationisaconcreteformalizeddiscipline.Onaccountofthegeneralityofthepresentconsiderations,however,theseconceptswillhereberegardedasprimitiveandwillbecharacter-ized(charakteriziert)bymeansofaseriesofaxioms.Inthecustomarynotationofgeneralsettheorytheseaxiomscanbeformulatedinthefollowingway:¯¯Axiom1.S
ℵ0.Axiom2.IfX⊆S,thenX⊆Cn(X)⊆S.Axiom3.IfX⊆S,thenCn(Cn(X))=Cn(X).
Axiom4.IfX⊆S,thenCn(X)=¯¯Cn(Y).Y⊆XandY
ℵ0Axiom5.Thereexistsasentencex∈SsuchthatCn({x})=S.[Tarski,1983g,30–31][Tarski,1986i,311]ThepassageconveysthebasicIntuitionisticFormalistideas.TheappealCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01toLe´sniewskiindistinguishingaclosedwell-formedformulaasa“mean-ingful”sentenceisoneobviouspoint.Next,thereisaconceptofconsequencethatcanbecapturedwithinavarietyofdeductivetheo-riesbutwhichisindependentofthem—itcanbedefinedwithinonetheoryandaxiomatizedwithinanother.Note,relatedly,that“defining”therebyhastomeansomethingotherthan“analyzing”inasensewe10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE54—#2\nTarskiasIntuitionisticFormalist55wouldrecognize(cf.§1.1.1),sinceaparticulardefinitionofconsequencewithinthemetatheoryforasingledeductivetheoryhardlycountsasageneralanalysisoftheconceptofconsequence.Furthermore,thebasicwayaconceptis“characterized”withinadeductivetheoryistohavethetermthatexpressesitconstrainedbyasetofsentencesheldtrue.Inlayingdownhisaxioms,Tarskiislayingdownconventionsfortheuseofhistermsthatdeterminethat“Cn(X)”hastoexpressthecontentoftheconceptofconsequence.Theexerciseis,then,simultaneouslyphilosophicalandformal.Thephilosophicalaspectofit,thoughreducedtoaminimuminthecaseathand,isarticulatinganddefendingananalysisoftheconceptofcon-sequence.Tarskitakestheanalysisoftheconceptofconsequencetoinvolvetheideathatconsequenceistransitive,thatasetofsentencesisamongitsownconsequences,andsoforth.ThesearenotcontroversialclaimswhenitcomestoconsequenceandsoTarskidoesnottarryoverthem,butthebasicintuitionisiticformaliststructureisinplaceinthesearticles:weexpressaconceptwithinadeductivetheorybylayingdownaxiomsthataresufficienttocaptureitsassociatedconception,inthiscasedirectlybyimplyingastatementofit.2Oneimportantpointabouttheanalysisofconsequencehereneedstobetakennoteof.ItcouldappearfromTarski’sideathatthereisageneralnotionofconsequencewhichcanonlybeworkedoutindetailinaparticulardeductivesystemthathisconceptionofconsequenceis,atthetimeofwritingbefore1928,actuallyindependentoftherulesofproofandhencesomehownon-epistemic:hehas,itmightappear,aconceptionofonethingfollowingfromothersthatisn’ttiedtoparticular“transformationrules”.Thisappearanceis,however,misleading.WhatTarskiassumesinthispassageisthatwehaveisageneralconceptionofonething’sfollowingfromothersbyrulesthatweleaveunspecified.Itisstillthecaseherethattofollowfromistofollowbyrulesofproof.Thisisimportantforustosee,sincewhenwebegintolookatTarski’sdevelopmentofextensionalsemanticswewillhavetonotethathedoesnotseethepossibilityofunderstandingconsequenceitselfintermsofituntilverylateand,aswewilldiscussin§7.3.2,thoughpeoplewerestudyingmodelsoftheoriesinthe1920sandearlier,nobody,includingCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01HilbertandGödel,wasthinkingoflogicalconsequenceitselfotherthanderivationally.2.1.2RelativizationtoadeductivescienceWenowcometoasignificantquestionaboutwhathappenstoIntu-itionisticFormalisminTarski’shands.AsLe´sniewskiunderstoodit,the10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE55—#3\n56AlfredTarski:PhilosophyofLanguageandLogicgoalintheconstructionofadeductivetheorywastocraftanaxiomaticstructurethatconstrainedthecontentsoftheconceptsexpressedbytheprimitivetermsofthetheoryastightlyaspossible—ideallydowntouniqueness,thoughthatgoalisnotachievableformostconceptsandtheories.Protothetic,Ontology,andMereologyaresupposedtobesotightlyconstructedthattheyzeroinonparticularconcepts;Ontology,forinstance,issupposedtocomeascloseaspossibletosinglingouttheoneandonlyconceptofonething’sbearingtheε-relationtoanother.Tarski,bycontrast,isconcernedwithconceptsthathaveanessentialrelativizationtoalanguage.Aparticulardeductivetheoryconcerningtheconsequencerelation,whetheritbeaxiomaticorwhetheritintroduce“Cn(X)”bydefinition,isrestrictedtoaparticularobjectlanguage:Strictlyspeakingmetamathematicsisnottoberegardedasasingletheory.Forthepurposeofinvestigatingeachdeductivedisciplineaspecialmetadisciplineshouldbeconstructed.Thepresentstudies,however,areofamoregeneralcharacter:theiraimistomakeprecisethemeaningofaseriesofimportantmetamathematicalconceptswhicharecommontothespecialmetadisciplines,andtoestablishthefunda-mentalpropertiesoftheseconcepts.Oneresultofthisapproachisthatsomeconceptswhichcanbedefinedonthebasisofspecialmetadis-ciplineswillhereberegardedasprimitiveconceptsandcharacterizedbyaseriesofaxioms[Tarski,1983c,60].InadditiontostatingthebasicIntuitionisticFormalistideas,thispas-sageinvitesaquestionthatisveryfamiliarfromtheliteratureonTarski’struth-definitions:howcanTarskiclaimtodefine“the”conceptofcon-sequenceinametatheorythatconcernsonlyasingleobjectlanguage?Therelatedquestionabouttheconceptoftruthandhisparticulartruth-definitionshasbeenasourceofconsternationfordecades:howcanTarskiclaimtodefine“the”conceptoftruthinatheorythatconcernsonlyasingleobjectlanguage?Weshouldbegivenpausehere,however,bythefactthatevenTarski’streatmentofthelowlyconceptofasentenceraisesexactlythesamequestions.Intheearlyarticlesontheconsequencerelationhesimplytakesthenotionofasentence(closedformula)forgranted,butwhenhedefinesbeingasententialfunctionandthenbeingCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01asentenceatDefinition10and12ofCTFL[Tarski,1983a,177–8]hisdefinitionraiseseveryquestionraisedbythemuchmorethoroughlydiscussedDefinition23.3Onewaytoputtheobviousproblemisthis.Supposewegrantthat“S”isgivensomeaxiomatizationordefinitioninthemetatheorythatsuitsittoapplytoallandonlythesentencesoftheobjectlanguageL.Whyhold10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE56—#4\nTarskiasIntuitionisticFormalist57onthisbasisthat“S”expresses“the”conceptofasentence?Afterall,forallthatthetheorysettles,itcouldeitherexpresstheconceptofbeingasentenceingeneral,ortheconceptofeitherbeingasentenceofLorasevencharacterstringofsomeotherlanguage(orbeingeitherasentenceofLoraturnip,or...).HowcanTarskiholdthatconstrainingatermbysometheoremsinatheorythatappliestoonlyonelanguagesettlestheissuethatoneconceptistherebyexpressedasopposedtoothersthatagreewithitasappliedtothatlanguage?Metamathematics,asTarskiconstruesit,isconcernedwith“formal-izeddeductivedisciplines”[Tarski,1983g,30][Tarski,1983c,60]or,whatcomestothesamethingforhim,“formalizedlanguages”[Tarski,1983a,165].“These”,hewrites,“canberoughlycharacterizedasarti-ficiallyconstructedlanguagesinwhichthesenseofeveryexpressionisuniquelydeterminedbyitsform”[Tarski,1983a,165–6].Thelackofthiscentralfeatureislikewisewhatmakesunformalizeddisciplinesorlanguagesunsuitedforstudy[Tarski,1983c,60].Anassumptionofmetamathematics,then,isthatexpressionsdifferinformiftheydifferinmeaning.NeverthelessTarskifollowsLe´sniewskiintakingtheexpres-sionsofthelanguagehestudiestohaveintuitivemeaning[Tarski,1983c,62]:“theexpressionswhichwecallsentencesstillremainsentencesafterthesignswhichoccurinthemhavebeentranslatedintocolloquiallan-guage”[Tarski,1983a,167].Thepointiseasytomiss,butattheoutsetof[Tarski,1983g]and[Tarski,1983c],andnotfarinto[Tarski,1983a]Tarskihasalreadyestablishedtohissatisfactionthatalthoughmeaningfullan-guageistheobjectofstudy,expressionsandsentenceswillbetreatedasmeaningfulonlyinsofarastheirsyntacticformscangoproxyfortheirmeanings.Nowonenotionforwhichtheformofanexpressioncannotgoproxyisthenotionofanexpression’sbeinginalanguage[Tarski,1983a,264].Thoughexpressionsgettheirmeaningsfromtheconventionsoflanguage—inthecaseofformalizedlanguages,formationrulesandrulesofproof,rulesofdefinitionandstipulationsofaxiomaticstatus[Tarski,1983c,63].[Tarski,1983f,112–13]andespecially[Tarski,1983a,166]—thenotionofanexpression’sbeinginalanguagesimplyisn’tsuitableformetamathematicaltreatment,despiteTarski’sacknowledgementofCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01thedependenceofmeaningonrulesoflanguageinpassagessuchas[Tarski,1983a,153].HenceTarskialwaysdependsonthereadertokeepinmindthatsentencehoodisalwaysunderstoodrelativetothepartic-ularobjectlanguageforwhichagivenmetatheoryissetup,withitsconventionsassumedtobeinplace.Inthisrespect,therelativizationtotheobjectlanguage,whenitcomestoassociatinganintuitivemeaning10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE57—#5\n58AlfredTarski:PhilosophyofLanguageandLogicwiththerelevanttermsofthemetalanguage,isleftinthemindofthereaderinthesamewayasistheinterpretationoftheprimitivesofthemetatheorytotheextentthatthetheorydoesn’tsettleit.Hereaselse-where(e.g.,aswithPadoa’smethod,aswewillseein§2.2.3)Tarski’sgoalistogetasmuchaspossibleexplicitlyonthepage,butnoteverythingcansuitablybeputthere.(This,bytheway,explainsthepuzzlingfactthatTarskirequiresthateachmetatheoryconcernonlyasingleobjectlanguage[Tarski,1983c,60]:ifametatheorywasaboutmorethanoneobjectlanguage,itwouldhaveexplicitlytoinvolvethenon-formalizablenotionofbeinginalanguage.)Wewilldowell,then,tosettlethesemattersnow,focusingonthesimpleconceptofasentence(or,incurrentterms,aclosedwell-formedformula)andthenturningtoconsequence.ThereasonthatTarskidoesn’ttrytogiveanabsolutelygeneraldefinitionofasentenceintermsoftheformofexpressionsisperfectlyclear:tobeaclosedwell-formedformulaisjusttobedeclaredsuchbythetheformationrulesofagivenlanguage.Somethingisasentenceonlyrelativetoalanguage,andsinceTarski,followingKotarbi´nskiandLe´sniewski,thinksofalanguageintermsofconventions,onlyrelativetoasetofconventions.EquallycleariswhyTarskidoesn’tgiveadefinitionofasentenceofLforvariableL:herethereisagainnothingtothedefinitionotherthanthattobeasentenceistobedeclaredonebytheformationrulesofsomelanguageorother.Lessclear,however,iswhyTarskisuppressesreferencetotherelativiza-tionofsentencehoodtoaparticularlanguage,andevenlessclearthanthatiswhyheiswillingtotalkaboutaxiomatizingordefiningoneandthesameconceptofasentencerelativetoanyparticularlanguageor,gen-erally,tosomelanguageorother,asheclearlyisintheabovepassage.Sincetheextensionof“sentence”differsbylanguage,onemightwellthinkthattheconceptitexpressesdoes,too.Theparallelwiththedis-cussionoftruthhereisclear:Tarskidoesn’tdefinetruthinLforvariableL,andwhenhedoesdefinetruthrelativetoaparticularobjectlanguagehesuppressesthisrelativization,yetgliblyretainstalkof“the”contentoftheconceptoftruth.Whatwehavehere,then,isaquestionaboutrelativizedconcepts.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Totakeareasonablyaptcomparison,beinganillegalactispresumablyrelative:thereisnosuchthingasdoingsomethingillegalsimpliciter,thereisonlydoingsomethingillegalrelativetosomesetoflawsorother.Wethushavetwooptions:wecanholdthatthereisasingleconceptofanact’sbeingillegalandthattherelativizationtoasetoflawsrequiredtodetermineanextensionforitis“external”totheconcept,orwecanhold10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE58—#6\nTarskiasIntuitionisticFormalist59thattherereallyisn’tageneralconceptofbeingillegalandthatthereis,rather,onlyasetofconceptsofbeingillegalaccordingtothelawsofNewYorkState,beingillegalrelativetothelawsoftheStateofCalifornia,andsoon.Tarski’spracticeclearlyindicatesthatheassumesthefirstsortofviewwhenitcomestolinguisticconceptsandconventionsoflanguage.Unfortunatelythereisnothingonthetopicwhereonemightexpecttofinditin[Kotarbi´nski,1966].Wole´nskisaysthatsuchaviewwas“cus-tomaryintheWarsawSchool”[Wole´nski,1989,205],andinalettertoNeurathTarskiwritesthat“onemustnotforgetthatsemanticexpres-sionslikedesignate,etc.,havearelativecharacter,thattheymustalwaysberelativizedtoaparticularlanguage”[Tarski,1992,16–17](mytrans-lation)andattributesthisviewtotheWarsawschoolgenerally.ThoughIhavetoconfesstolackingclearenoughintuitionsabouttheindividu-ationof“concepts”tosettletheissuemyself,thischoicedoesseemtocorrespondtoagooddealofordinarythoughtandtalkaboutillegality:thoughwerecognizethatthelawsofNewYorkandCaliforniadifferinsomerespects,wetreatthelawsassettlingtheapplicationconditionsofasinglenotionoflegality.Now[Kotarbi´nski,1966,22]doescontainadiscussionofacloselyrelatedtopic:indexicalityor“occasionality”.AnaspectofthisdiscussionbearsmentionnowbecauseitinvolvesanunclaritythatseemsalsotoinfectTarski’susage.Itbearsmentioningthatthediscussionoccursinachapterentitled“OnDefectivenessinaLanguage”;thatalanguagethatexpressesindexicalor“occasional”conceptsis“defective”endsupbeingoneofthereasonsthatTarskileavesrelativizationtoalanguageinthemindofthereader.Hereisthepassage:Amongsituationsinwhichwereadilyfallvictimstoambiguity,itseemsadvisabletomentionthoseinvolvingwhatiscalledoccasional-ity,thatissituationsinwhichthemeaningofawordvariesaccordingtotheoccasion.Thewords“I”,“here”,and“now”arestrikingillus-trationsofoccasionality…Thewords“I”and“my”,themeaningsofwhich,toputitfreely,haveacertainpersonalshade,varyrela-tivelyastotheirmeaningaccordinglytobywhomtheyareused;thewords“now”and“present”themeaningsofwhichhaveatempo-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01ralshade,varyrelativelyastotheirmeaningaccordingtowhentheyareused…Forinstance,“I”,usedbyPeterhassuchameaningthatitdenotesPeter,andusedbyJohnhassuchameaningthatitdenotesJohn…Thismaygiverisetomisunderstandings,asinthecaseofthedecep-tivesign-boardinashop“Tomorrowoncredit,todayforcash”.On10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE59—#7\n60AlfredTarski:PhilosophyofLanguageandLogicreadingthat,ascustomermightbeinclinedtocomeonthenextdaysoastobeabletobuygoodsoncredit,butthentheownerwouldpointtothesignboardandrefusetoselloncredit.Thetextonthesignboardpreservesitsform,butchangesitsmeaningeveryday.Hencethetrickresortedtobytheshopownerisnothonest:hemakesconsecutivestatementswhichcontradictoneanother,indoingwhichheavailshimselfofthefallacythattwosentenceswhichhavethesameformalwaysmeanthesame.Butthisallegedprincipledoesnotholdwhenitcomestosentencesthatincludeoccasionalexpressions[Kotarbi´nski,1966,22–3].Firstwecannotethatthis,combinedwithTarski’sviewsaboutmeta-mathematicsaccordingtowhichin“formalizedlanguages…thesenseofeveryexpressionisuniquelydeterminedbyitsform”[Tarski,1983a,165–6],itfollowsthattherecanbeno“occasional”expressionsinformal-izedlanguages,andhencetakingmetalinguistictermslike“sentence”or“consequence”or“truth”tobeindexicalforTarskiinhismetalanguagesisnogood.4Moreimportantforourpurposes,though,isKotarbi´nski’swaveringbetweenamoreandalesssensibleviewofindexicalityinthepassage.ThelesssensibleviewisthatwhenJohnandBillbothsay“Iamhere”,theyuttersentencesthatmerelyshareaform,buthavecompletelydistinctmeaningsaswellascompletelydistinctdenotations.ThisleavesouttheobvioussenseinwhichJohnandBillboth“saythesamething”withthesentence.Indeed,themoststraightforwardapplicationofKotarbi´nski’sownaccountofindirectstatementappliedtoindexicalswouldsuggestthattheconventionsofEnglishassociateasinglethought-contentwithalltokeningsoftheexpressiontype“I”:generalconventionsoflanguagecouldnotrelateparticulartokeningsof“I”totheirspeakers.Themoresensibleviewcropsupinturnsofphrasesuchas“themeaningsofwhichhaveacertainpersonal(temporal)shade”;onthisview,“Iamhere”hasaconventionalmeaningofwhichBillandJohnavailthemselvestosaysomethingthatinvolvesdifferentdenotations.5Thepassagestrikesmeasimplyconfused;Kotarbi´nski,ifhehadthoughtitthrough,couldnothaveacceptedthefirstview,buttheCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01firstviewiscertainlymoreclearlypresentinthepassage.Unfortunately,thiswafflingintroducesasimilarambiguityintoTarski’sconception.At[Tarski,1983a,263]Tarskiwritesthat“theconceptoftruthdepends,asregardsbothextensionandcontent,onthelanguage”andthat“assoonasthediscussionconcernsmorethanonelanguage,theexpressionceasestobeunambiguous”.Thismakesitsoundasthoughalinguisticterm,10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE60—#8\nTarskiasIntuitionisticFormalist61appliedtodifferentformalizedlanguages,changesnotonlyitsextensionbutalsoitscontent.ButifthisisreallyTarski’sview,thenhereallycan’tclaimtoshowhowtointroduceanexpressionfor“the”conceptoftruthintoawholefamilyofmetatheories,sincetheintroducedexpressionsreallywouldhavenothingincommon.However,somethinghasgonewrongat[Tarski,1983a,263],sincethepassagesaysthattheviewthatrel-ativizationtoalanguagemakesforadifferencebothinextensionandincontentwas“alreadyemphasizedintheIntroduction”,buttherelevantpassageoftheintroduction,[Tarski,1983a,153],saysnothingaboutcon-tent,butonlythatextensionvariesbylanguage.Tarskinevercompletelysettledhisusageoftheterminologyhere,butwhatisclearisthatwhenitcomestolinguisticconceptsandtheirrelativizationtoalanguage,the“contentoftheconcept”thatTarskitriestocapturehastobesomethingthatisinvariantacrossapplicationstodifferentlanguages,something,aswiththethetreatmentofconsequencediscussedabove,isintroducedintosometheoriesbydefinitionandintoothersaxiomatically.6Ourreadingwillthusbethatsomeconceptsareessentiallyrelativeinthesensethattheyhaveextensionsonlyrelativetosomefurtherparame-terandlinguisticconceptsareamongsuchconcepts.Asnotedabove,wehaveanexplicitstatementtothiseffectinaroughlycontemporaneouslettertoNeurath.Fregemightdisapprove,butthisiswheretheevidencepoints.ThoughTarskitalksinafewplacesaboutthe“content”or“mean-ing”ofametalinguistictermbeingrelativetothelanguagetowhichitisapplied,thistalkcannotsupporthisinsistencethattheimplicitlyrela-tivizedaxiomatizationsanddefinitionshegivesalwayscaptureoneandthesameconcept.Tarski’saiminboththeaxiomatizationof[Tarski,1983g]andDefini-tion12ofCTFListointroduceatermthatexpressestheoneandonlyconceptofbeingaclosedwell-formedformula,aconceptoverextensionsforwhichthefirstgeneralizes,andwhichisgrantedanextensioninthesecondcasebytheassumedrelativizationtothelanguageofthecalculusofclassesasTarskiformulatesit.ForTarskithen,theconceptappliesonlyrelativetoalanguage.Since,asnoted,beinginalanguageisn’tsome-thingforwhichtheformofanexpressioncangoproxy,TarskicountsonthereadertokeeptherelativizationinmindwhileleavingitunstatedCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01inthemetalanguage,sincethemetatheoryissupposedtoamounttoamathematicaltreatmentoftheobjectlanguage.Inthisrespecttherel-ativizationtoalanguageisofthesamestatusastheinterpretationoftheprimitivetermsofthemetatheory—totheextentthatthetheoremsdon’tsettleit,itissomethingexternaltotheformalizeddisciplinebutrequiredtobeknowntothereaderinordertounderstandthethoughtsconveyedbytheoremsofthesystem.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE61—#9\n62AlfredTarski:PhilosophyofLanguageandLogicThus,onceweaccountforTarski’sassumedviewsabouttheconceptsinquestion,thereisactuallynodepartureherefromLe´sniewskiatall.Tarski’saimindefiningoraxiomatizinganexpressionthatexpressestheconceptofasentenceistocapturetheoneandonlyconceptofasentence.Itisjustthatthisconcept,unliketheconceptexpressedbyLe´sniewski’s“ε”,getsanextensiononlyrelativetoalanguage,andthisrelativizationmustbekeptinthemindoftheonewho“practises”thedeductivemetatheoryfortheobjectlanguage.Whensorelativized,theintendedconceptisexpressedaslongascertaincharacteristicclaimsarestatedbytheoremsofthemetatheory.Themetatheorymightaswellthenbeexplicitlyrelativizedtotheobjectlanguage,butsincethiswouldbringtheproblematicnotionofanexpressionbeing“inalanguage”,andthustheconcomitantviewsaboutconventions,ontothepageforexplicittreatment,Tarskiforcestherelativizationtoalanguageintothemindofthereader.WecanthussaythesamefortheconceptofconsequenceasTarskiconstruesitintheworksunderconsideration:thereisasingleconceptofonesentence’sfollowingfromotherswhichdeterminesanextensionrelativetoasetofrulesofproof;theaimofbothageneralaxiomatictheoryofthisrelationandanydefinitionrelativizedtosomeparticu-lardeductivescienceistointroduceanexpressionproperlyconstrainedbytheaxiomsofthemetatheorytoexpressthisconcept,withtherela-tivizationtotheobjectlanguage,orsomeobjectlanguageorother,leftinthemindofthereader.Theaimis,aselsewhereinmetamathematicsconstruedintheIntuitionisticFormalistway,tointroduceanexpressionthatexpressesanintuitiveconceptbutisitselfdefinedentirelybytermsfrom“themorphologyoflanguage”.2.2ExplicitdefinitionIturnnowto“SomeMethodologicalInvestigationsontheDefinabilityofConcepts”(DC),thepaperTarskipublishedintheproceedingsofthePragueVorkonferenzfortheUnityofScienceCongresstobeheldinParisin1935.Thoughthepaperwasdeliveredin1934andpublishedin1934(Polish)and1935(German),itcontainsmaterialthatdatesfrom1926Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01(§1)and1932(§2)[Tarski,1983k,297].SinceitwasdeliveredwellafterTarskihaddevelopedhissemantictechniquesthefactthathewaswillingtopresentandpublishitatthoselatedatesshowsthataslateasthe1934VorkonferenzTarskiwasstillacommittedIntuitionisticFormalist,despitethefactthatsemanticshasa(now)obviousapplicationtothetopicsdiscussed.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE62—#10\nTarskiasIntuitionisticFormalist63OnmyreadingTarskiconceivedoftheworkasastudyofthecondi-tionsunderwhichatheoryachievestheIntuitionisticFormalist’sgoalofsettingoutatheorythatcapturesintuitivemeaningaseffectivelyaspossible.Thepaperiswidelyneglectedinpartbecauseittreatssomestan-dardtopicsinawaythatinvolvessignificantidiosyncrasyandbecause,asHodgesnotes,“Themathematicalinterestisalmosttrivial;certainlyaningeniousandambitiousmathematicianlikeTarskiwouldneverhavepublisheditasacontributiontomathematics”[Hodges,2008,108].However,ifweviewthepaperasacontributiontotheIntuitionisticFor-malistprojectofexpressingconceptsbycapturingtheiranalyseswithinadeductivetheory,theotherwisestrangemodificationsTarskimakestoPadoa’smethodandtothenotionofcategoricitymakegoodsense.ItwillalsobeofusetoustostudythepaperbecauseitcontainsTarski’smostextendeddiscussionofexplicitdefinition.2.2.1DefiningdefinitionTarskidefines“definition”andexplicitlytiesitssignificancetothetheassertibilityofcertainsentencesonthebasisofadeductivetheoryinthispassage:Everysentenceoftheform:(x):x=a.≡.φ(x;b,b,...),where‘φ(x;b,b,...)’standsforanysententialfunctionwhichcon-tains‘x’astheonlyrealvariable,andinwhichnoextra-logicalconstantsotherthan‘b’,‘b’,…ofthesetBoccur,willbecalledapossibledefinitionorsimplyadefinitionoftheterm‘a’bymeansofthetermsofthesetB.Weshallsaythattheterm‘a’isdefinablebymeansofthetermsofthesetBonthebasisofthesetXofsentences,if‘a’andalltermsofBoccurinthesentencesofthesetXandifatthesametimeatleastonepossibledefinitionoftheterm‘a’bymeansofthetermsofBisderivablefromthesentencesofX[Tarski,1983k,299].Itisnotdifficulttoseewhytheconceptofdefinability,aswellasallderivedconcepts,mustberelatedtoasetofsentences:thereisnosenseindiscussingwhetheratermcanbedefinedbymeansofCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01othertermsbeforethemeaningofthosetermshasbeenestablished,andonthebasisofadeductivetheorywecanestablishthemeaningofatermwhichhasnotpreviouslybeendefinedonlybydescribingthesentencesinwhichthetermoccursandwhichweacceptastrue[Tarski,1983k,299,note1].10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE63—#11\n64AlfredTarski:PhilosophyofLanguageandLogic(Thecurious“possibledefinition”canbebefoundat[Kotarbi´nski,1966,198].)Thenoteexpressesthepaper’scommitmenttoIntuitionisticFor-malism.TheIntuitionisticFormalistisinterestedinassociatingintuitiveconceptswiththeexpressionsofatheory.Nowwhichintuitiveconceptscanbeassociatedwiththetermsofatheorydependsonwhichsentencesofthelanguageofthetheoryareacceptedastruewithinthetheory.Thesignificanceofanexplicitdefinitionthusliesinthedemonstrationthattheaxioms,plustheformationandtransformationrules,fullyindivid-uatetheintuitiveinterpretationofatermrelativetotheinterpretationsofotherprimitivenon-logicalterms.NowitisinthismomentthatanimportantfeatureofIntuitionisticFormalismcomestolight.Theentirearticleisconcernedwith“thedefin-abilityofconcepts”.Thismeans,Itakeit,thatadefinitiondeterminesthecontentofthetheconceptexpressedbyitsdefiniendum,andtheonlypossibledeterminationcouldbeidentity:atermdefinedbyanother(simpleorcomplex)sharesitsintuitivemeaningwithit.Whatshouldstrikeushereisthatitfollowsthattheintuitivemeaningsoftermsaren’tdeterminedcompositionally:acomplexsymbol’smeaningisn’tentirelydeterminedbythemeaningsofitspartsandhowtheyaresyntacticallyputtogether,itdependsalsoontheintuitivemeaningsofothersym-bolsforwhichpossibledefinitionsequatingthemanditareprovableinthetheory.WeseeherehowverydifferenttheIntuitionisticFormalistaccountofmeaningisfromthereferentialsemanticstowhichTarski’sownsemanticworkgaverise.Furthermore,asmuchasLe´sniewskihim-selfdemandedacompositionalaccountinhisearlywork[Le´sniewski,1992b],inTarski’shandsIntuitionisticFormalismisincompatiblewithitbecausetheintuitivecontentassignedtoanexpressiondependsonthesetoftheoremsoftheoveralldeductivetheory,andnotmerelyontheintuitivecontentsofitsparts—indeed,allexpressionsgettheirintuitivecontentassigned“topdown”bylookingatthesetoftheoremsinwhichtheyfigure.Whattheprovabilityofadefinitionshowsisthatdefiniensanddefinien-dumsharetheirmeaningtotheextentthateitheroftheirmeaningsisdeterminedbythetheory.Totheextentthatthetheoryfailstodeter-minethis,theintuitivecontentexpressedbybothsymbolsisleftopen.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01ItisbecauseofthissortofpossibilitythatTarskiintroducestheconceptofatheory’sbeingcompletewithrespecttoitsspecifictermsin§2ofthearticle,forinatheorythatmeetsthisconditiontheintuitivemeaningsofalltermsareuniquelyindividuated.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE64—#12\nTarskiasIntuitionisticFormalist652.2.2TwoconceptionsofdefinitionAnimportantthingtorecognizeaboutTarski’sconceptionofexplicitdefinitionisthathestandswithinaminoritytraditionin20thcen-turythoughtaboutwhatanexplicitdefinitionis.Callthetwoviewsofexplicitdefinitionatissuetheabbreviativeconceptionofdefinitionandthetheory-relativeconceptionofdefinition.Adefinitionstatessomesortofequivalencebetweenadefiniendumdanditsdefiniensd.Ontheabbreviativeconception,thisequivalenceisamatterofstipulatedequiv-alenceanddefinitionistherebyreallyametalinguisticaffair.Onthetheory-relativeconception,ontheotherhand,adefinitionissimplyanequivalencethatfollowsfromatheory.(See[Belnap,1993]foraclearrecenttreatmentofthelatterconception.)Unliketheabbreviativecon-ception,thetheory-relativeconceptionholdsthatthereisnodifferencebetweenthe‘↔’ofordinarymutualimplicationofthesesinatheoryandthe‘=df’ofdefinition.Thatthisdistinctionissooftenmadeinformalandinformalpresentationisasymptomofhowentrenchedtheabbre-viativeconceptionis.Tarski,bycontrast,gothisstartinpublicationwithaninnovationthatmadeitpossibletogetridofthisdistinctionandthusbaseasystemofthesententialcalculusonasingleconnec-tive[Tarski,1983j],somethingLe´sniewskihadwantedbutwasunabletosupplyhimself[Le´sniewski,1992h,418ff].Thecontrastbetweenthesetwoconceptionswasatopicfordiscus-sionamongTarskiandhisteachers(cf.[Wole´nski,1989,100,107],[Hodges,2008,101–2]).7BothKotarbi´nskiandLe´sniewskidiscussthetopicandTarskibrieflynotesitat[Tarski,1983e,166].8Kotarbi´nskisideswiththestipulativeconception.EarlyintheElementyhedistin-guishes“axiomaticpseudo-definitions”inwhichbothdefiniendumanddefiniensareused—thatisdefinitionstakenasthetheory-relativecon-ceptiontakesthem,asordinarybiconditionals—andmaintainsthataxiomaticpseudo-definitionsarenotdefinitions“inthestrictsenseoftheterm”[Kotarbi´nski,1966,27].Inalatersectionondefinitionsin“TheDeductiveMethod”,hewrites:Definitionsincludesuchformulasas“meansthesameas”or“isequiv-alentto”,andalsonamesofthesymbolsbeingdefined…defintionsCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01cannotthemselvesbelongtothesententialcalculus;thesameholdsforotherdefinitionsofmanydeductivesystems…Foreveryproperdefinition,wecanchoose…acorrespondingpseudo-definitionasbelongstothesystem,whereastheproperdefinitionitselfremains10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE65—#13\n66AlfredTarski:PhilosophyofLanguageandLogicoutsidethesystemasanotindispensablecommentconcerningthesamenessofmeaningofsometwoinscriptions…theyprovideinfor-mationthatagivensymbolmaybeusedtoreplaceanothergivensymbolintheprocessofinference…Inpurelyformaltransformationsaformulationofthekind“AisagraphicalsubstituteofB”wouldsuf-fice.Suchdefinitionsmightbecalledgraphical.Butusuallywehavetodowithsemanticdefinitionsformulatedbymeansofsuchphrasesas“meansthesameas”…Suchdefinitions,apartfromprovidinginfor-mationaboutpermissiblesubstitutions,alsoprovidetheinformationthathewhoformulatesthemassociatesthesamemeaningwiththedefiniendumaswiththedefiniens.Thisisimportantincommunication,ifthelistenerorreaderistounderstandthesentencesofthesysteminthesamesenseasdoestheirauthororexpounder[Kotarbi´nski,1966,244–5].9(NoteherethatthepsychologisitcconceptionofmeaningthatinformsIntuitionisticFormalismisclearlyinevidenceinthepenultimatesen-tence.)ThoughKotarbi´nskiherestressesthecommunicativeutilityofabbreviative,metatheoreticdefinitions,headmitstheirdispensabilityandnotesthedispensabilityofthenotionofmeaningwhenitcomesto“formal”transformations.WhatTarskisawmoreclearlyhere,inlinewithIntuitionisticFormalismasheunderstoodit,wasthat“axiomatic”definitionssimplyrecordinfactwhattheothersortmerelypurporttodoinword:expresstheintersubstitutabilityoftwoexpressions.ThoughLe´sniewski’spublishedworkcontainsnoexplicitdiscussionofthetwosortsofdefinition,Le´sniewskiwasawareofthedifferenceatleasttotheextentthathispositionchangedbetweenhisearlyandlatework.EarlyonhemaintainsthatInadequaciesofverbalrepresentation…may,however,hinderresearchwhen,e.g.,inadequatesymbolsofcertaincontentsaremistakenlyregardedasadequate.Insuchcaseswholeseriesoffun-damentallyfalsetheoriesmaygrowoutofinadequaterepresentation;suchtheoriescanbeexemplifiedby,say,thefrequentopinionthatdefinitionsareanalyticpropositionsabouttheobjectsrepresentedbythesubject…Thisopinion…[is]basedonthefactthatinsteadoffor-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01mulatingapropositionabouttheexpressionwearegoingtodefine,weformulateapropositionabouttheobjectofwhichtheexpressioninquestioncanonlybeasymbol[Le´sniewski,1992b].Later,however,hecamearoundtoaviewsharedwithTarskithatKotarbi´nski’s“axiomaticpseudo-definitions”aretheonlyrealdefini-tions:10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE66—#14\nTarskiasIntuitionisticFormalist67Indefiningthefunctionsofthetheoryofdeductionintermsofothersuchfunctions,bothSchefferandNicoduseaspecialequal-signfordefinitionswhichtheydonotdefineintermsoftheprimitivefunctionsofthesystem…ThiscircumstancemakesitdifficulttosaywhetherNicod’stheoryofdeductionisinfactconstructedoutofthesingleprimitiveterm‘|’.In1921Irealizedthatasystemofthetheoryofdeductioncontain-ingdefinitionswouldactuallybeconstructedfromasingletermonlyifthedefinitionswerewrittendownwithjustthatprimitivetermandwithoutrecoursetoaspecialequal-signfordefinitions[Le´sniewski,1992e,418].TarskisideswithLe´sniewskiinhislaterperiodhere,holdingthatadefinitionissimplyanequivalenceprovablefromatheory.AnotherrestrictiononTarski’sconceptionmustbenoted:thatanexpressioniseverywheresubstitutableforanotherpreservingtheorem-hoodifandonlyifanexplicitdefinitionisprovablefromthesametheorydependsonspecificfeaturesoftheassumedlogic.Foratoyexample,supposemytheoryconsistsofthetwoaxiomsFaandFbandthelogicisthenulllogicinwhichnothingfollowsfromanything.Hereaandbareeverywheresubstitutablepreservingtheoremhood,butobviouslynoexplicitdefinitionisprovablefromthetheory.Tarski’sconceptionassumestheanalogueofBeth’sdefinabilitytheoremforSTT[Vaught,1986,864],anditwouldmakethesamedemandsofanyothersystemtowhichitwasapplied.2.2.3Padoa’smethodTheexplicitaimof§1of“SomeMethodologicalInvestigationsontheDefinabilityofConcepts”issomehowtosupporttheapplicaibilityofPadoa’smethodforestablishingthatnoneoftheprimitivetermsofadeductivetheoryaremutuallydefinable:ManyyearsagoA.Padoasketchedamethodwhichenablesustoestab-lish,inparticularcases,theundefinabilityofatermbymeansofotherterms.Inorder,bythismethod,toshowthataterm‘a’cannotbedefinedbymeansofthetermsofasetBonthebasisofasetXofsen-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01tences,itsufficestogivetwointerpretations(“Interpretationen”)ofallextra-logicalconstantswhichoccurinthesentencesofX,suchthat(1)inbothinterpretations(“Interpretationen”)allsentencesofthesetXaresatisfied(“erfüllt”)and(2)inbothinterpretations(“Interpretatio-nen”)alltermsofthesetBaregiventhesamesense(“Sinn”),but(3)thesense(“Sinn”)oftheterm‘a’undergoesachange.Weshallhere10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE67—#15\n68AlfredTarski:PhilosophyofLanguageandLogicpresentsomeresultswhichprovideatheoreticaljustificationforthemethodofPadoaand,apartfromthis,appeartothrowaninterestinglightontheproblemofdefinability[Tarski,1983k,299–300][Tarski,1986b,640].ThequotationmarksareintheGermantextbutareomittedbyWoodger.(Le´sniewskialsoscare-quotes“interpretation”whilediscussing“thewellknownmethodof‘interpretation’”at[Le´sniewski,1992h,261].)Woodger’somissionofthemissignificant:whatTarskiproposestodoin§1istotellushowtoremovethescarequotes;thatis,heintendstoreplacePadoa’sappealto“interpretation”withsomethingsupposedlymorerigorous.GiventhepositionofthepaperinLogic,Semantics,Metamathematics—afterCTFLandODS—anditspublicationin1934,onemightexpectthatTarskiwouldgetridof“sense”,“interpretation”and“satisfaction”infavor,intheobviousway,ofhissemantics.Butthisisnotwhathedoes(cf.[Hodges,2008,125])andtheappearancein1934ofmaterialnotupdatedsince1926showsjustwhatthestateofTarski’sthinkingabouthissemanticsanditsrelationtomeaningwasin1934.10WewillreturntowhatTarskicouldhavedonein1934,andwhatthefactthathedidn’tdoitshows,below.HerethepointistodescribewhatTarskidoesdo.ThoughitistemptingtothinkofPadoaassomesortofproto-model-theorist,asalreadynotedthereisnothingsemanticabout[Padoa,1967].11ThepassageinwhichPadoapresentshis“method”forprovingthatnoprimitivetermofadeductivetheorycanbedefinedintermsofothersreadsthus:Letusassumethatafteraninterpretationofthesystemofundefinedsymbolsthatverifiesthesystemofunprovedpropositionshasbeendetermined,allthesepropositionsarestillverifiedifwesuitablychangethemeaningoftheundefinedsymbolxonly.Then,sincethemeaningofxisnotindividualizedoncewehavechosenaninterpreta-tionoftheotherundefinedsymbols,wecanassertthatitisimpossibletodeducearelationoftheformx=a,whereaisasequenceofotherundefinedsymbols,fromtheunprovenpropositions.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Conversely,inordertobeabletoassertthatitisimpossibletodeduce,fromtheunprovedpropositions,arelationoftheformjustmentionedwemustshowthatthemeaningofxisnotindividualizedoncewehavechosenaninterpretationofthesystemoftheotherundefinedsymbols;andthiswedobyestablishinganinterpretationofthesystemofundefinedsymbolsthatverifiesthesystemofunproved10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE68—#16\nTarskiasIntuitionisticFormalist69propositionsandthatstilldoessoifwesuitablychangethemeaningofxonly.[Padoa,1967,122].ThismightsoundsemanticuntilwetakenoteofPadoa’sconceptionofinterpretation,meaningandrelatednotions,which,aswehaveseen(§1.4.1)[Padoa,1967,120–21]iscompletelypsychologistic:Padoa’scon-ceptionofinterpretationexactlymatchestheIntuitionisticFormalist’sideaofassociatinganintuitivemeaningwithasymbol.12Viewedinthislight,theinterestofPadoa’smethodisthatitcon-cernstheextenttowhichtheformalroleofasymbolwithinatheoryisconstrainedbythetheorydowntothepointwhere,giventheinterpre-tationsoftheothersymbols,arationalthinkerwhoacceptsthetheorycanassociateonlyoneparticularintuitiveconceptwithasymbol.ThisiswhatcapturesTarski’sattentioninPadoa’smethod.Tarski’sinterestisinunderstandingtheextenttowhichatheoryadequatelyachievestheIntuitionisticFormalistidealofbearingauniqueintuitiveinterpretationasdeterminatelyaspossible.Now[Tarski,1983k]isafaithfulrenderingofPadoa’sapproachunder-stoodasPadoaunderstoodit.WhatTarskidoesistoreplacePadoa’stalkaboutwhetherornotthe“meaning”ofasymbolis“individualized”byamentalchoiceofmeaningsforothersymbolsbyadeductivecondi-tion:that,foratermaandasetofotherprimitivetermsB,itbethecasethatifweintroduceanothertermaandrestatementsofallaxiomsinvolvingabytheirsubstitutionvariantswitha,a=abeprovablefromtheextendedtheory.Whenthisconditionobtains,onecannotassociatetwodifferentmeaningswithaandawhiletakingthetheoremsoftheextendedtheorytobetrue(assumingthatthetheoryisextensional).ThemoregeneralstatementsoftheconceptioncomeasTheorems2and3,inwhichdefinabilityandindefinabilityareconceivedofasinvolvinggen-eralizationsontheprovabilityor(syntactic)consistencyoftherelevantidentitiesortheirnegations[Tarski,1983k,303–4].WhatinPadoa’spresentationgoesoninone’smindwhilecontem-platingthedeductivetheory,inTarski’srefinementgoesondeductivelywithinthesystemitself;itisforthisreasonthatTarskiholdsthathehasprovided“thepropertheoreticalfoundationforthemethodofPadoa”Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01[Tarski,1983k,305].Theideaofassociatingmeaningswithsymbolsisstillpresentinthearticle,sincetheinterestintheconditionsstatedinTheorems2and3concernsthepossibilityofassociatingdifferentmeaningswithsymbolsofthetheory,butPadoa’sideathatwewillfind,whenwecontemplatethesystem,thatweeithercanorcannot“varythemeaning”weassociatewithasymbol“inthemind”whilestilltakingthe10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE69—#17\n70AlfredTarski:PhilosophyofLanguageandLogicprimitivesentencesofthetheorytobe“verified”hasbeenreplacedbyconditionsonderivabilityinvolvingidentitystatementsformulatedwiththesymbols.13Thisisexactlyinlinewith“metamathematics”asTarskiconstruesit.Ifdefinabilityandeliminabilityofprimitivesistobestudiedmetamathematically,theissueshavetobecastinaformwherewhattheyconcernistheformofexpressions.This,then,demandsthatPadoa’smentalvariationofmeaningsbyreplacedbytheintroductionoftwosymbolsdifferinginform,eachofwhichbearsoneofthesemeanings.2.3Categoricityandcompletenessofterms2.3.1ProvablemonotransformabilityWeturnnowtothesecondsectionof“DefinabilityofConcepts”,rep-resentingworkthatwascompletebyJune,1932.Onemightexpect,giventhesection’sfocusoncategoricityandthelatedate,thatitwouldbegroupedwithTarski’ssemanticworks,butthereisnothingsemanticaboutit,either:itis,rather,afurthercontributiontotoIntuitionis-ticFormalism.Indeed,IthinkitranksasTarski’sdeepestattempttocontributetothebasicIntuitionisticFormalistprojectasheunderstoodit,forinthesectionTarskiattemptstoestablishsomegeneralresultsabouttheconditionsunderwhichadeductivetheory:(a)completelydeterminestheconceptsexpressedbyitsnon-logicalvocabulary,and(b)expressesalloftheintuitiveconceptsofthedomainfromwhichtheseconceptsaredrawn—e.g.,theconditionsunderwhichatheoryexpresses,forinstance,allgeometricalconcepts.Thesecondpointofcourseinvolvesaviewaboutwhatmakesaconceptageometricalconcept,etc.andwewillexaminewhatTarskidoeswiththis.FulldeterminationoftheconceptsexpressedbythesentencesheldtrueinatheorywastheIntuitionisticFormalistideal,andinthissectionTarskigivescriteriafortheconditionsunderwhichithasbeenrealized.Tarskifirstintroducesthenotionofasetofsentencesbeing“essentiallyricher”thananother“withrespecttospecificterms”:Theproblemofthecompletenessofconcepts,towhichwenowturn,isalsocloselyrelatedtocertainproblemsconcerningsystemsofsen-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01tences,andindeedtotheproblemsofcompletenessandcategoricity,althoughtheanalogydoesnotextendsofarasinthepreviouscase.Inordertomaketheproblemofcompletenessprecise,wefirstintro-duceanauxiliaryconcept.LetXandYbeanytwosetsofsentences.WeshallsaythatthesetYisessentiallyricherthanthesetXwithrespecttospecificterms,if(1)everysentenceofthesetXalsobelongstothe10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE70—#18\nTarskiasIntuitionisticFormalist71setY(andthereforeeveryspecifictermofXalsooccursinY)andif(2)inthesentencesofYthereoccurspecifictermswhichareabsentfromthesentencesofXandcannotbedefined,evenonthebasisofasetY,exclusivelybymeansofthosetermswhichoccurinX.[Tarski,1983k,308].14AsTarskinotesnext,foranysetofsentencesthereistriviallyanessen-tiallyricherset:justaddalogicaltruthwithatermthatdoesn’toccurintheoriginalset.Inordertodeviseaconditionofsomeinteresthere,Tarskimakesuseofthenotionofcategoricity,butwithpeculiartwists.Heintroducesitassomethingfamiliar:“asiswellknown,asetofsen-tencesiscalledcategoricalifanytwointerpretations(“Interpretationen”)(realizations(“Realisierungen”))ofthissetareisomorphic”[Tarski,1983k,309].(OnceagainthequotationmarksareintheoriginalbuthavebeenstrippedbyWoodger.)Tarski’sfootnotetoVeblenencouragesthethoughtthatgenuinesemanticsisnowatissuein§2butitturnsoutimmediatelythatTarskiisinterestedentirelyinadeductiveconceptionofcategoricity:ConsidernowanyfinitesetYofsentences;‘a’,‘b’,‘c’,…areallspe-cifictermswhichoccurinthesentencesofY,and‘ψ(a,b,c,...)istheconjunctionofallthesesentences.ThesetYiscalledcategorical(orprovablycategorical)iftheformula(V)(x,x,y,y,z,z,...):ψ(x,y,z,...).ψ(x,y,z,...).x,y,z,...⊃.(∃R).(Rx,y,z,...)islogicallyprovable[Tarski,1983k,310].HeretheR-formulagetsthefollowinginterpretation:Letussaythattheformulax,y,z,...Rx,y,z,...istohavethesamemeaningastheconjunction…inwords:‘Risaone-onemappingoftheclassofallindividualsontoitself,bywhichx,y,z,…aremappedontox,y,z,…,respectively[Tarski,1983k,310].Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Ratherthanthinkingofcategoricityintermsofitsbeingatheoremofametatheoryforthedeductivetheoryatissuethatanytworealizationsofitsaxiomsareisomorphicwhetherthiscanbeprovedinthetheoryornot,categoricityishereconstruedasitsbeingprovableinthebackgroundlogicofthetheory(simplifiedtypetheory,asusual)thatforanytwoorderedn-tuplesofthings(ofanytype;asTarskinotesthenotationhere10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE71—#19\n72AlfredTarski:PhilosophyofLanguageandLogicisreallyschematicoverthetype-theoretichierarchy[Tarski,1983k,310note1])iftheypairwisehavethefeaturesattributedtothembytheaxioms,thenthereisa1–1mappingofthesetofallindividualsontoitselfthatmapsthemontooneanother.15CategoricityisimportanttoTarskibecauseheholdsthat:Onvariousgrounds,whichwillnotbeenteredintofurther,greatimportanceisascribedtocategoricity.Anon-categoricalsetofsen-tences(especiallyifitisusedasanaxiomsystemofadeductivetheory)doesnotgivetheimpressionofaclosedandorganicunityanddoesnotseemtodeterminepreciselythemeaningoftheconceptscon-tainedinit.Weshallthereforesubjecttheoriginaldefinitionoftheconceptofcompletenesstothefollowingmodification:asetXofsentencesissaidtobecompletewithrespecttoitsspecifictermsifitisimpossibletoconstructacategoricalsetYofsentenceswhichisessen-tiallyricherthanXwithrespecttospecificterms.Inordertoestablishtheincompletenessofasetofsentencesitisfromnowonwardsrequi-sitetoconstructasetofsentenceswhichisnotonlyessentiallyricherbutalsocategorical.Trivialconstructionsinwhichthemeaningofthenewlyintroducedspecifictermsisquiteindeterminatearethusexcludedfromthebeginning[Tarski,1983k,311][Tarski,1986b,647].Acategoricalaxiomsystem,then,“determinesprecisely”themeaningsofthetermsitcontains.(ItakeitthatTarski’s“concepts”inthesecondsentenceistobeignoredinfavorof“terms”fromthelast,sinceitisonlythelatterthatareactually“contained”inanaxiomsystem.)ThisiswhatanIntuitionisticFormalistwants:thatadeductivetheoryconstraintheconceptsexpressedbyitstermsdowntouniqueness.Completenessisthenanadditionalmatter,correspondingto(b)above:adeductivethe-oryiscompletewithrespecttoitsspecifictermsifonecannotextendittocontainanewtermthemeaningofwhichisalsocompletelydetermined.Therearetwomodificationstothefamiliarconceptionofcategoricityhere:provabilityintheassumedlogicandtherequirementofa1–1map-pingfromthesetofallindividualsontoitself.Contrastthefirstwiththestandardmodel-theoreticconceptionofcategoricity.Ifamodelofasetofsentencesisasetofassignmentsofvaluestovariables,denotationstoCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01namesandsoon,andadomainforthequantifiersthatsatisfiesallofthesentences,thenasetofsentencesiscategoricaljustincaseeverymodelisisomorphic.Withthatnotionofamodelonboard,anyonewithevenapassingacquaintancewiththenotionofcategoricitywoulddefineitinnootherway.ComparehereVeblen’sremarksonthecategoricityofhisaxiomsforgeometry:10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE72—#20\nTarskiasIntuitionisticFormalist73Inasmuchasthetermspointandorderareundefined,onehasaright,inthinkingofthepropositions,toapplythetermsinconnectionwithanyclassofobjectsofwhichtheaxiomsarevalidpropositions.Itispartofourpurposehowevertoshowthatthereisessentiallyonlyoneclassofwhichthetwelveaxiomsarevalid.Inmoreexactlanguage,anytwoclassesKandKofobjectsthatsatisfythetwelveaxiomsarecapableofaone-to-onecorrespondencesuchthatifanythreeelementsA,B,CofKareintheorderABC,thecorrespondingelementsofKarealsointheorderABC.Consequentlyanypropositionwhichcanbemadeintermsofpointsandordereitherisincontradictionwithouraxiomsorisequallytrueofallclassesthatverifyouraxioms[Veblen,1904,346].TheunderstandinginVeblenisfullysemantic:thereisnothingherethatofitselfwouldrestrictapplicationofthenotionofcategoricitytofinitesetsofsentences,asTarski’sconceptiondoes,andthereisalsonothingaboutitbeingalogicaltruthofthelogicofthelanguageinwhichtheaxiomsarestatedthatanytwosetsofthingsofwhichtheyholdaresuchthatthereisa1–1mappingofVontoitselfthat1–1mapsthemontooneanother.Tarski’sfocus,bycontrast,isonconstruingcategoricityintermsofwhatcanbederivedfromwhat.Noticeonthisscorethattheparenthetical“(orprovablycategorical)”oftheEnglishdoesnotappearintheGermanofthearticle,butisaddedtothe1956Englishtranslation[Tarski,1986b,650];by1956Tarskiwanteditknownthathewasawarethattheearlierconceptionhadbeenbenttohis1932purposes.Tarskialsonotesattheendofthetranslationthattheremarksoncategoricitydonotreflecthisviewsatthetimeofthetranslation.Whatdoesthisaddupto?Tarskiistryingtomakecategoricityamatterofwhatisprovableinthelanguageofatheory,ratherthansomethingthatisprovableaboutatheory.Thisisentirelyinlinewithmetamath-ematicsasheconstruesitintheearly1930s:justasPadoa’smethodreceivesits“theoreticaljustification”bybeingrecastasadeductivecon-ditionconcerningtwosymbolsratherthananintuitiveconditionaboutthevariabilityofthemeaningonementallyattachestoasinglesymbol,sotoothenotionofcategoricityisn’tconstruedintermsofmodeltheoryCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01orintermsofmentallyassociatedconcepts,butisratherbeingconceivedofasadeductivecondition.Thesenseinwhichanon-categoricalsetofaxiomsdoesn’tfullydeterminetheconceptsexpressedbythespecifictermsitemploysisexplicatedintermsoftheextenttowhichtheintu-itiveconceptsthatcouldbeexpressedbytermsareconstrainedbytheaxiomsbeingheldtrue.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE73—#21\n74AlfredTarski:PhilosophyofLanguageandLogicNowTarski’spresentationdistinguishescategoricityfromwhathecalls“monotransformability”,whereamonotransformablesetofsentencesissuchthatonecanprove:(VI)(x,x,y,y,z,z,...,R,R):ψ(x,y,z,...).ψ(x,y,z,...).x,y,z,...x,y,z,...⊃.(Rx,y,z,...).(Rx,y,z,...).⊃R=R[Tarski,1983k,313]Thatis,amonotransformablesetofsentencesisonethatiscategori-calandwhere,inaddition,thereisonlyone1–1mappingofthesetofindividualsontoitselfcompatiblewiththetruthofthetheory.Mono-transformabilityinheritstherequirementof“provability”inTarski’spresentation,soweareagaintalkingaboutadeductiveconditionhere.TarskiholdsthatmonotransformabilityissufficientforthecompletenessofatheorywithrespecttoitsspecifictermsinTheoremIV[Tarski,1983k,314].Theconverseisleftopenand,asVaughtnotes[Vaught,1986,874],itmaystillbeopen,thoughaswewillseethisissurelyduetothelackofanylogicalhandleon“competenesswithrespecttospecificterms”otherthanmonotransformability.Thismakesclearagainthatthemean-inginthearticlecannotbeequatedwithextension,forifmeaningwereextensionthencompletenesswithrespecttospecifictermsandmono-transformabilitywouldbetransparentlyoneandthesameproperty.ThemeaningatissuewhenTarskiholdsthatprovablemonotransform-abilityissufficientforcompletedeterminationofthemeaningofathe-ory’stermsisintuitivemeaning.ForTarski’spurposes,however,thedistinctionbetweencategoricalandnon-categoricalsentencesmatterslittle;inbothcasesonecouldexchangetheideasassociatedwiththespecifictermsofthetheorywhilestilltakingthetheorytobetrue.Monotransformabilityistheimportantproperty,sinceamonotrans-formabletheoryconstrains(thecontentsof)theconceptsassociatedwithitsspecifictermsdowntouniqueness.ConstruingprovablecategoricityasIntuitionisticFormalismdemands,consideroneself,inthepositionoftheLe´sniewskianmathematiciantry-ingtoassociateintuitiveconceptswiththetermsofatheory.FollowingTarski’sexamples,consideratheoryforone-dimensionaldescriptiveormetricgeometry;thefirsthas“xisbetweenyandz”asaprimitive;theCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01secondadds“thesegmentxyiscongruenttothesegmentzw”.Nowcallingtomindone’sideaoftherealline,withpoints0and1distin-guished,oneintuitivelydistinguishes,e.g.,thesegmentwithendpoints0and1,andthesegmentwithendpoints1and2.Butonealsoseesthatanydefinedtermofone-dimensionalmetricgeometrywithwhichonemighttrytoassociateone’sideaofthesegmentwithendpoints0and110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE74—#22\nTarskiasIntuitionisticFormalist75couldjustaswellhaveone’sideaofthesegmentwithendpoints1and2associatedwithit:nothinginthetheoryitselfdemandsoneinterpreta-tionasopposedtotheother,eventhoughthetheoryiscategorical.Therearenon-identityautomorphisms,andhencedifferentsetsofintuitiveconceptscanbeassociatedwiththetermsoftheory:onecanassociatetheconceptoftheintervalfrom0to1with“01”oronecanassociatetheconceptoftheintervalfrom1to2withit.Ifatheoryisprovablycategor-ical(ornotcategoricalatall)butprovablynotmonotransformable,thenoneknowsthatthereismorethanonewaytoassociateintuitivemean-ingswithitsspecifictermscompatiblewithtakingittobetrue.Ifthetheoryismonotransformable,then,sincetheidentityautomorphismisavailableandonecanprove(VI),oneknowsthattheconceptsassociatedwiththetermsofthetheoryareconstraineddowntouniqueness.Nowsincethetheoryisextensional,itwon’tdetermineintuitivemeaningspasttheirbeingindividuatedextensionally.Thismightseemadifficultyonthegroundsthatthereareobviouslyexpressions(e.g.“Hesperus”and“Phosphorus”)thatintuitivelydifferinmeaningbutareextensionallyequivalent.ButTarskiwouldrejecttheexample:anytwotermswhoseequivalence(inthecaseofnames,anidentitystatement)isprovableinthetheoryhavetobeattributedthesameintuitivemeaning,sinceTarski’sconceptionofdefinitionsdoesn’tdistinguishthemfromothersortsofequivalences,anddefinitionsdefine“concepts”.16Tarski’swordingcertainlyindicatesthatheholdsthatmonotransformabilityissufficientfordeterminingintuitivemeaningscompletely;witnessagainhiscommentthat“Anon-categoricalsetofsentences…doesnotgivetheimpressionofaclosedandorganicunity,anddoesnotseemtodeterminepreciselythemeaningoftheconceptscontainedinit”[Tarski,1983k,311].Thiswouldseemtoindicatethatacategorical,andcertainlyamonotransformablesetofsentencesdoesdeterminetheintuitivemean-ingsofitstermscompletely.Butsincethetheoryisextensional,thismeansthatintuitivemeaningsareindividuatedatbestextensionally.17Unfortunately,atthispointin§2ofDCitbecomesclearthatTarski,searchingformathematicalexpressionsofhisviewsaboutconcepts,haslandedhimselfinapositionthatsimplyisn’tconsistentwiththeviewsthatanimatetheotherpapers:conceptsaccordingtoIntuitionisticCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Formalismaren’textensionallyindividuated.Thisisanobviousconse-quenceofLe´sniewskiandKotarbi´nski’sviewsaboutconnotationanddenotation.InparticularthemajorgoalofCTFL,toexpresstheconcep-tionoftruthexpressedinthe“semanticaldefinition”of§1,wouldbelostifconceptsandtheircontentswereindividuatedextensionally,since,aswewillsee(§4.3.1),Tarskirecognizesthatdefinitionsthatclearlyaren’t10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE75—#23\n76AlfredTarski:PhilosophyofLanguageandLogicintuitivelyadequatecanstillbeextensionallyadequate.ForthisreasonIwillset§2ofDCasidefortheremainderofthisworkassomethingofanexperimentthatwentwrong.Theproblem,onecansee,waswaitingforTarskiallalong:asalogicianhewasstridentlyextensionalist,whileasaphilosopherheacceptedanaccountofconceptsonwhichtheyweren’tindividuatedextensionally.Theproblemsof§2ofDCshowthistensionfinallycomingtolightwhenTarskisetshismindtorigorouslyworkingoutcertainthesesaboutconceptsinadeductivetheory.Thetension,itseems,wasneverreallyresolved;itceasedtobeimportantwhenTarskileftIntuitionisticFormalismbehindin1935.2.3.2AbsolutemonotransformabilityThatsaid,§2isstillinterestinginitself,andsowewillcontinueourexaminationofit.Aprovablymonotransformabletheoryuniquelyindi-viduatestheintuitiveconceptsexpressedbyitsspecifictermsandassuchperfectlymeetstheIntuitionisticFormalistidealofcapturingthecontentsofourconceptswithinthedeductivestructureofanaxiomatictheory.Weturnnowtotheothermodificationoftheusualnotionofcategoricityinthearticle.Tarskicommentsonitinanote:Weusetheword‘categorical’inadifferent,somewhatstrongersensethaniscustomary:usuallyitisrequiredoftherelationRwhichoccursin(V)onlythatitmapsx,y,z,…ontox,y,z,…respectively,butnotthatitmapstheclassofallindividualsontoitself.Thesetsofsentenceswhicharecategoricalinthecustomarysensecanbecalledintrinsicallycategorical,thoseinthenewsenseabsolutelycategorical.Theaxiomsystemsofvariousdeductivetheoriesareforthemostpartintrinsicallybutnotabsolutelycategorical.Itis,however,easytomakethemabsolutelycategorical.Itsuffices,forexample,toaddasinglesentencetotheaxiomsystemofgeometrywhichassertsthateveryindividualisaapoint(ormoregenerallyonewhichdeterminesthenumberofindividualswhicharenotpoints)[Tarski,1983k,310note2].18Intrinsiccategoricityisn’tsufficientforabsolutecategoricitybecauseifthesetofallindividualsandthesetofindividualstowhichthetheoryCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01iscommittedareinfinitethenwecandescribetwoisomorphicmodelsthatleavetheindividualstotheexistenceofwhichthetheoryisn’tcom-mittedoutof1–1correlation.Totakeasimpleexample,ifmytheoryis“thereareinfinitelymanyindividuals"andthesetofallindividualsisthenaturalnumbers,thentheevennaturalsandthenaturals=3arebothmodelsofmytheory,buttheoddnaturalsand3arenotin1–110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE76—#24\nTarskiasIntuitionisticFormalist77correspondence.Hencefactsaboutinfinitesetsdeterminethattherearetheoriesallmodelsofwhichareisomorphicwhicharenotsuchthatforanytwomodelsthereisa1–1correlationofthesetofallindividualsthatalso1–1correlatesthosetwomodels:onecannot1–1correlatetheevennaturalsandthenaturals=3whilealso1–1correlatingthesetofallnaturalswithitself.19WhydoesTarskiimposethisadditionalrequirement?Andwhydoesheimposeitwhenatheorycanbetriviallybutextraneouslymodifedtosat-isfyit?Thisiswherethesecondaspectof“completenesswithrespecttospecifictermscomein”.AsTarskiunderstandsacompletetheoryinthissense,itiscompletebothinthatitindividuatestheintuitivemeaningsofitsspecifictermsuniquely,andinthatitdoesn’tallowanyadditionaltermsnotdefinableintermsoftheoldonestobeadded.Hereistheexplanation:Ifwewishtousethewords‘categorical’and‘monotransformable’intheintrinsicsense…thenTh.4intheaboveformwouldbefalse.Inorderthatitshouldremainvalidthedefinitionofcompletenessmustbeweakened,absolutecompletenessmustbereplacedbyintrinsiccompletenessorbycompletenesswithrespecttointrinsicconcepts.Forthispurposeitwouldbenecessarytogivefirstanexplanationofwhatismeantby‘intrinsicconcept’.Allthiswouldcomplicatetheexpositionconsiderably[Tarski,1983k,314].Theproblemisthatifthetheoriesinquestiondidn’texhaustthesetofallindividuals,amonotransformabletheorywouldbeextendablebya(provably)categoricaltheorythatcontainedatermnotdefinablebytheoriginaltheory’stermsintheextendedtheory.AsTarskinotes,thisneedforanotherwiseextraneousrecastingofthetheoriesatissuecouldbeavoidedif“intrinsicconcept”couldbedefined:intrinsicallymonomor-phictheorieswouldcompletelydeterminetheir“instrinsic”conceptsandtheexistenceofcategoricalextensionswouldbenoissue.ButTarskishiesawayfromcommittinghimselftosuchanaccountofbeing“intrin-sic”.Ontheotherhand,ifatheoryisabsolutelymonotransformablethennocategoricalextensioncanintroduceatermthathasadetermi-natemeaningbutforwhichapossibledefinitionintermsoftheoldCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01vocabularyisn’tprovable:sinceanabsolutelymonotransformablethe-orydeterminesthewholeofVuptotheidentityautomorphism,thereisnoobjectleftoverintheSTThierarchyoverVforanynewtermtohaveasitsextension.Thisbeingso,bytheresultof§1(BethdefinabilityforSTT),apossibledefinitionwillbeprovablethatreducesthenewtheorytoadefinitionalextensionoftheoldone.2010.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE77—#25\n78AlfredTarski:PhilosophyofLanguageandLogicTarskirefersthereadertotheendof1935’s“OntheLimitationsoftheMeansofExpressionofDeductiveTheories”(whichwelearnat[Tarski,1983i,384n.1]datestothesameperiodas§2of[Tarski,1983k]and“belongstothesamecircleofideas”asit)wherehewrites,oftheapplicationofTheorem1ofthearticle(whichstatesthatthatwhichislogicallydefinableisinvariantunder1–1permutationsoftheclassofallindividuals)toparticulardeductivetheories:Inconclusionwenotethefollowing:whenapplyingTh.1tospecialdeductivetheoriesandtogeneralmetamathematicswemustrestrictourselvestothoseaxiomsystemsfromwhichitfollowsthattherearenoindividualsoutsidethedomainofdiscourseofthetheorydis-cussed.WecouldremovetherestrictionbymeansofanappropriategeneralizationofTh.1.Toobtainthisgeneralizationweshouldintro-duceanewnotion,infactthenotionofasententialfunctionintrinsicunderagivenvariable(orconstant)‘a’or,inotherwords,thenotionofapropertyintrinsicforagivenclassa.Roughlyspeaking,apropertyisintrinsicforaclassaifitinvolvesexclusivelyelementsofa,sub-setsofa,relationsbetweenelementsofa,etc.(andnot,forexample,individualsoutsideofa)[Tarski,1983i,392].Theexplanationof“intrinsic”inthelastsentenceofthequoteisa1956additiontotheEnglishtranslationthatdoesnotappearinthe1935Germanversionofthepaper.Thatruns,rather:Inconclusionwenotethefollowing:whenapplyingTh.1tospecialdeductivetheoriesandtogeneralmathematicswemustrestrictour-selvestothoseaxiomsystemsfromwhichitfollowsthattherearenoindividualsoutsideofthe“domain”ofthetheoryconsidered.Wecouldremovetherestrictionbymeansofanappropriategeneraliza-tionofTh.1andbyconsideringonlysententialfunctionsthatare—inacertainsensethatwewillnotherefurthermakeprecise—“inner”inrelationtotheprimitivetermsofthecontemplatedtheory[Tarski,1986h,212](mytranslationofthealternatematerial).Tarskiappearstobeattemptingtocoverhistracksinthe1956Englishver-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01sionbysupplyinganon-mysterioussemanticunderstandingof“intrin-sic”.Whilethe1956Englishversionoffersaratherclearmodel-theoreticunderstandingofan“intrinsic”conceptrelativetoasub-domainofthesetofallindividuals(compareheretheremarksontruthinanindividualdomainat[Tarski,1983a,199ff])throughastraightforwardrelativiza-tionthatmakesnoattempttocapturewhysomerelativizationsmight10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE78—#26\nTarskiasIntuitionisticFormalist79be“intrinsic”relativetoatheorywhileotherswouldnot,the1935Ger-manversionholdsoutthehopeofcapturingsomesenseof“intrinsic”thatisofphilosophicalinterest.ThispassagealsodoesawaywithanylingeringdoubtthattheconceptionofmeaninginDCispsychologisticratherthansemantic:the1956modificationissoobvious—and,indeed,iscloselyrelatedtothediscussiontruthinadomainat[Tarski,1983a,199,239]—thatifTarskihadintendeditin1932–5hewouldsimplyhavestatedit.Sincehedoesn’t,theremustbesomereason,andmysugges-tionisthatthereasonisthatheisstillthinkingof“interpretation”inIntuitionisticFormalistterms.Presumably,giventhediscussioninthearticleofarithmetic,geometryandmechanics,“intrinsic”conceptsof,say,geometrywouldbeallandonlythegeometricalconcepts.ButdoingthingsthiswaywouldcommitTarskitogiveanaccountofwhatmakesaconceptageometricalconcept,anditisthisphilosophicalheavyweatherthatheshiesawayfrom,preferringinsteadtoformulatethingsintermsofabsolutecategoricity.Thusthemodificationof“categorical”toitsabsolutesenseisafudgethatallowsTarskitoraiseanissuehedoesn’ttakehimselftobeabletosettle.TheeffectofthemoveistotaketheselectionofthesetofintuitiveconceptswearetryingtocaptureoffthetableandtoputitintothemindofthemathematicianintheassumedsuitableselectionofVfortheapplicationofSTT.So,forinstance,spottingourselvestheclaimthatourconceptsofvariouspropertiesofthereallinehangtogetherassomesortofphilosophicallyimportantwhole,makingVthesetofpointsonthereallinethenallowsustostateitasatheoremthatmetricgeometryplustheintroductionoftermsfor0and1resultsinatheorythat:(a)uniquelydeterminestheconceptexpressedbyeverytermofthearithmeticoftherealnumbers,and(b)exhauststhesupplyofconceptsofthepropertiesoftherealline.Tarskiwouldliketofreehimselfoftheneedforthefudgefactor,butlackingawaytodosohehastopushtheissuebackintotheassumptionsheldinmindbythemathematician.So§2ofDCisanattempttostatesomemathematicalconclusionsaboutwhathastobeprovableinatheoryforittomeettheIntuition-isticFormalistgoalofbuildingatheorythatallowsustoexpressallofourthoughtsaboutadomaininsuchawaythatthestructureoftheCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01theoryuniquelysettleswhichconceptsareexpressedbywhichterms,andwhichthoughtsbywhichsentences.Onecanseefromnoteslike[Tarski,1983k,319],Tarski’sinterpolationof“orprovablycategorical”inthe1956translation,aswellasfromtherelated1956modificationtothetreatmentof“instrinsic”in[Tarski,1986h]thatTarskiwaslatersome-whatchagrinedaboutthelengthstowhichhehadgoneinmodifying10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE79—#27\n80AlfredTarski:PhilosophyofLanguageandLogicthemathematicstosuitphilosophicalpurposesinthearticle.AsIhavenoted,substantialaspectsofthesectionareinconsistentwiththerestofTarski’sviewsandforthisreasonIwillleavethesectionoutoftheaccountwheninterpretingtheotherworks.However§2doestellusagooddealaboutTarski’sinterestsaslateas1934:hewasstillafullycommittedIntuitionisticFormalist.Hisinter-estsindefinabilityandcategorictyhadastrongIntuitionisticFormalistbent;bothweresignificantnotmodel-theoreticallybutfortheirroleinformallyconstrainingtheexpressionofintuitiveconceptswithinadeductivetheory.HisfailuretousehissemanticstoworkoutPadoa’smethodortotreatcategoricityanddefinabilitymodel-theoreticallyshows,likewise,thateveninto1934Tarskiwasstillsquarelywithintheearlierparadigm,thinkingofdeductiverelationshipswithinatheoryintermsofprimitivelyvalidtransformationrules,andthinkingofthemeaningofasignasthecontentofamentalstateinawaythatwouldhavebeenfamiliartotheTwardowskiof40yearsearlier[Twardowski,1977,8–10].212.4TheoryandconceptTarski’streatmentofdefinabilityin[Tarski,1983k]isfocusedonthewayinwhichthestructureofadeductivetheoryconstrainstheconceptsitstermscanbetakentoexpress.Theprovabilityofanexplicitdefini-tionfromatheory,undertheconditionsTarskiassumes,isnecessaryandsufficientfordefiniensanddefiniendumtorequirethesameintuitiveinterpretationonpainofirrationalityintheinterpreter,andforthisinterpretationinturntobesettledbytheinterpretationoftheprimitivetermsasfaraspossible.Ifthetheoryisinturncompletewithrespecttoitsspecificterms,thendefinabilityamountstocompletecaptureoftheintuitivemeaningofanexpression.Tarski’streatmentofdefinitionconcernsconstraintsontheassociationofintuitivemeaningwiththeexpressionsofadeductivetheory.Nothinginit,however,saysanythingaboutwhichintuitiveconceptsareassoci-atedwithprimitiveexpressions.ForthisTarskineedsadifferentsortofaccount:foragivenintuitiveconcept,anaccountoftheconditionsCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01underwhichasymbolthatfiguresincertaintheoremsofadeductivetheoryexpressesit.Herenogeneralaccountofthestructureofthethe-oryispossible;whatisneededisatreatmentintermsoftheanalysesoftheseconceptsthemselves.WediscussedthisinthepreviouschapterwithrespecttoKotarbi´nski’saccountofconnotation.Theconnotationofatermisthesetofpropertieshadbyanythinginitsextension.To10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE80—#28\nTarskiasIntuitionisticFormalist81capturetheconnotationisforcertainsentencestobeassertibleofany-thingtowhichthetermisapplied.If,forinstance,atermFistoexpresstheconceptofbeingscarletinatheorywhereGexpressestheconceptofbeingred,then(∀x)(Fx→Gx)mustbeatheorem,sincepartofthecon-notationgraspedbysomeonewiththeconceptofathing’sbeingscarletisthatscarletthingsarered.ThesearchforsuchanaccountofconnotationsuitabletothecasesinwhichTarskiwasinterested—metamathematicalconceptsthemselves—anditsdiscoveryinthecaseof“istrue”andrelatedterms,isthemainphilosophicalachievementofTarski’sIntuitionisticFormalistconcep-tionofmetamathematics,thoughwehavealreadyseenatleastanassumedtreatmentoftheanalogoustopicinhisearlyarticlesontheconsequencerelation.ThoughinthosearticlesTarskihasnotyetfacedtheissuesquarely,thetreatmentassumesasimpleconceptualanalysisof“isaconsequenceof”onwhich,e.g.,consequenceisreflexiveandtran-sitive.Beginninginthenextchapterwewilllookattheissuesastheyarisewithrespecttosemanticconcepts,andinthesubsequentchapterwewilllookathowTarskiarrivedatanIntuitinisticFormalisttreatmentoftheconceptoftruththatsatisfiedhim.Tarski’sdefinitionsthemselvesneednotstateconceptualanalyses.Indeed,onTarski’sviewdefinitionsneednothaveanythingtodowiththeanalysisofaconcept,andtheyneedbeneitherintuitivelyanalyticofitnorsomehowintuitivelytrue.Whatmattersinadefinitionisn’titsownrelationshiptotheconceptsupposedtobeexpressedbyitsdefinien-dum,butwhetherthedeductivetheoryrelativetowhichitisadefinitioncomes,whenitisadded,tohavetheoremsthateitherstatetheconcep-tionoftheconceptorotherwiseconstrainthedefinedterm(ideally)toexpressjustthatconcept.Definitionsthereforeneednothaveanyparticularepistemicorsemanticfeatures—theyneednoteliminatethedefinedconceptinfavorofothersthatareknownwithmorecertainlyorareotherwisemorebasic.22ForTarski,adefinitionofaconceptisalwaysasentencethat,addedtoatheory,forcesthedefinedtermtoexpressthatconcept.23ConsiderhereTarski’sdefinitionofconsequenceforthelanguageofthecalculusofclassesin[Tarski,1983a].Tarskifirstdefines“consequenceCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01tothenthdegree”intheobviousway:xisaconsequenceofthesetofsentencesxtothenthdegreeiffn=0andx∈X,orn>0andxis,e.g.thedisjunctionoftwosentencesthatareconsequencestothen−1thdegree,etc.(Definition15[Tarski,1983a,181]).Hethendefinesconsequenceinthelikewiseobviousway:xisaconsequenceofXifitisaconsequencetothenthdegreeforsomen[Tarski,1983a,182].Theaxiomsoftheearly10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE81—#29\n82AlfredTarski:PhilosophyofLanguageandLogicarticlesarethentrivialconsequencesofthedefinition.Hereitisn’tthedefinitionthatisdirectlyrelatedtotheconceptualanalysis,butcertaintheorems;cf.Le´sniewski’sremarksdiscussedabove(§1.2.2).Boththeaxiomatictheoryofconsequencein[Tarski,1983c]andthemetatheoryforthelanguageofthecalculusofclassessupplementedbyTarski’sdefinitionsinCTFLsuit“∈Cn(X)”toexpresstheconceptofbeingaconsequenceofthesetofsentencesXinvirtueofthetheoriesasawholehavingtherelevanttheorems.Thedefinitionneednotitselfstatetheanalysis,anditneednotimplytheanalysiswithouthelpfromtherestofthetheory.WenowhavesomeinsightintoTarski’sfrequentuseofenumerativedefinitions.Itisoftenwonderedhowa“definition”thatmerelylistscasescouldpossiblybesuitedfortheanalyticaltasksthatTarskisetshimself([Field,1972],[Davidson,1990],[Mou,2001]andmanyothers).Theansweristhatthedefinitiononitsownisn’tintendedtocarryouttheanalyticaltask;thatjobisdonebythetheoryasawholeinimplyingtherighttheoremsinvolvingthedefiniendumoncethedefinitionisadded.WhenTarskiislookingto“catchholdofthemeaningofanoldnotion”[Tarski,1986e,665],itisalwaysthedeductivetheoryasawholethatdoesthis,notaparticulardefinition.Anotherexamplefrom[Tarski,1983a]makesthepointwell.Tarski’spresentationofDefinition14containsinmicrocosmalloftheelementsofTarski’sviewabouthowanintuitiveconceptiscapturedinadeductivetheorybyaddingadefinition.Hefirstwrites:IntheformulationofthedefinitionoftheconceptofconsequenceIshalluse,amongothers,thefollowingexpression:‘uisanexpressionobtainedfromthesententialfunctionwbysubstitutingthevariablevkforthevariablevl’.Theintuitivemeaning(inhaltlicheSinn,tres´cintuicyjna´)ofthisexpressionisclearandsimple,butinspiteofthisthedefinitionhasasomewhatcomplicatedform.[Tarski,1983a,180].Whatfollowsisafifteen-linedefinitionthatcoverssixcases.Clearly,asimpleconceptneednothaveasimpledefinition.Sowhatsuitsittocapturetheconcept?Notanyfeatureofthedefinitionitself,butratheritsconsequences:Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Forexamle,itfollowsfromthisdefinitionthattheexpresionsι1,1,3(ι3,1+ι1,3)andι1,3+2ι2,3areobtainedfromthefunctionsι2,2,3(ι3,2+ι2,3)andι2,3+2ι2,3respectivelybysubstitutingv1forv2.Buttheexpression1ι1,3cannotbeobtainedinthiswayfromthefunction2ι2,3northeexpression1ι1,1fromthefunction2ι2,1[Tarski,1983a,180].10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE82—#30\nTarskiasIntuitionisticFormalist83Thepointhereisthattheseconsequencesofthedefinitioninvolvepairsofexpressionsthatintuitivelydoordonotfallundertheconceptofanexpressionthatresultsfromanotherbysubstitutionofavariable.WehavehereexactlytheprocedurewewilllaterseetreatedatgreaterlengthwithConventionTandatruthdefinition:Tarskiannouncestheintentiontocaptureaconcept,introducesarathercomplicateddefini-tionthatdoesn’tlookanythinglikeatraditionalconceptualanalysis,andthenjudgesthedefinitionasworthyongroundsofthetheoryaug-mentedbythedefinitionhavingconsequencesthatareintuitivelytruegiventheintendedinterpretationoftheexpressionintroducedandtheassumedinterpretationoftheprimitivetermsofthetheory.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-0110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP02”—2011/11/28—11:39—PAGE83—#31\n3SemanticsTarski’sviewsaboutmeaninginthelate1920sandearly1930swereentirelyinlinewithIntuitionisticFormalismasheconceivedofitfol-lowingLe´sniewskiandthecloselyrelatedphilosophyoflanguageofKotarbi´nski.Thefunctionoflanguageistoconveythecontentsofthoughtsandthemeaningofasymbolisacontentofaconcept(inthecaseof“terms”,expressionsinthesimpletype-theoretichierarchy)oritscontributiontodeterminingthemeaningofsuchsymbols(inthecaseofother“technicalsigns”[Tarski,1983a,167note1]).Theprimitivetermsofadeductivetheorygettheirmeaningsfromthesetofsentencesheldtrue,thetheorems.Tarski’sprojectwastocraftdeductivetheoriesthatexpressedimportantmetamathematicalconceptsviatheconstraintsplacedontheirprimitivetermsbytheirtheorems.ThereisnohintanywhereinTarski’searlywritingsthatdenotation,satisfactionandtruthplayanyroleintheaccountofthesignifi-canceoflanguage,communication,oranythingelseofthesort;alongLe´sniewski’slinesaxiomsandtheorems,whentakentoexpressthoughts,seemtobetrue,butforLe´sniewskiandTarskithiscomestonothingmorethanexpressingbeliefsthatonehas,giventheconventionsofthelanguage.Ultimately,asnotedinvariousplacesabove,thisempha-sisonnon-referentialmeaning-theoreticnotionsinthebasicaccountofunderstandingandcommunicationispartofTwardowski’slegacy(e.g.[Twardowski,1977,8–9]).Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01NeverthelessoneofTarski’smostcelebratedcontributionswastothedevelopmentofthebasicsetoftoolsforthinkingaboutmeaningintermsoflanguage–worldrelations,theconceptsandtechniquesofref-erential,truth-conditionalsemantics.IntheremainderofthisworkwewilltellthestoryofTarski’sintroductionofthesetoolsandthedevelopmentofhisthoughtaboutthem.Tarskiinitiallydidn’tthinkof8410.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE84—#1\nSemantics85semanticsasheintroduceditasacontributiontothestudyofmeaningatall.Tarskidevelopedsemanticsnotasacontributiontothebasictheoryofmeaning,butasabitofdetailworkinhisprojectofgivingIntuitionisticFormalisttreatmentsofimportantmetamathematicalcon-cepts.Inthisrespect,theearlydevelopmentofsemanticsissimplyabitofworkthatextendshisworkontheconsequenceconstruedderivationallyaround1930.Unlikewithhisearlyworkonconsequence,however,Tarski’smotiva-tionforprovidingtreatmentsofsemanticconceptswasinparttosecuretheirlegitimacy.Le´sniewskiandCarnapwereopenlyhostiletosemanticconcepts,whiletheirroleinKotarbi´nski’saccountisperipheralatmost.Theirunderstandingofvariables,sententialfunctionsandquantificationwassubstitutionalwhenitwasclear.Ontheotherhand,someofTarski’sownlogicalandmathematicalworkhadfocusedinthelate1920sonwhatwecannowrecognizeasearlycontributionstomodeltheorybytheAmericanPostulateTheorists,andLöwenheimandSkolem,HilbertandAckermann,andothers.Tarski’sinterestingivingdefinitionsofseman-ticconceptswasdirectedtowardshowingthelegitimacyoftheseearlymodeltheoreticstudiesinthefaceoftheskepticismofLe´sniewskiandCarnapbyshowinghowtermsexpressingsemanticconceptscouldbeintroducedintorigorousdeductivetheoriesbydefinitionsandinaccordwiththeconstraintsimposedbyIntuitionisticFormalism.3.1PhilosophicalresistanceThoughtaboutsemanticsbeforeTarskishowsastrongbifurcation:ontheonehand,thereweredevelopmentsinmathematicallogicthatinvolvedthetreatmentofsemanticconceptsoftruthandsatisfactionasunprob-lematic,whileontheotherhandinphilosophicalcirclestherewasagreatdealofskepticismaboutsemanticnotions.Wewilldiscussthelat-terhereandtheformerinthenextsection.TheblanketskepticismaboutsemanticnotionsthatonecanfindintheViennaCircle,forinstance,derivedfromtheputatively“unscientific”characterofthenotionoftruththatcamefromitsfrequentoccurrenceinmetaphysicaldisputes(see[Soames,1984,415]andthereferencesthere)andfromtheideaCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01thattheconceptionoftruthascorrespondencecommittedonetosomesortofincoherentconceptionoftheconfrontationofstatementswithfacts(theprimarymotivationforeschewingsemanticconceptsascribed,especiallytoNeurath,byHempel[Hempel,1935,50–51]).InthissectionwewilllookinmoredetailattwotopicsdirectlyconnectedwithTarski’sinterests:theinterpretationofquantificationandsemanticparadox.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE85—#2\n86AlfredTarski:PhilosophyofLanguageandLogic3.1.1ThequantifierItisimportanttonotehowdeeplymiredinasubstitutionalconceptionofvariablesandquantificationsomeofthefigurestowhomTarskipaidthemostattentionwere.1OneplacewecanseethisisinCarnap’sAbrissderLogistik,aworkTarskirepeatedlycites:Ifanumberofconstituentsignsareremovedfromacomplexsign,therearisenewcomplexsignsbytheinsertionofdifferentsignsinthethepositionstherebymadefree.Tomakethefreepositionsthem-selvesrecognizablewecouldsimplyleavethemempty,butwewillasaruleplaceothersignsinthefreeplacesthathaveonlythispur-pose;thesethemselveshavenomeaning(beteutenalsoselbstnichts),butmerelyholdthepositionsopenfortheinsertionofothersigns.Wesayofthecompleteexpressionwiththeseemptyplacesthatitsig-nifies(bezeichne)a“function”;avariablethatmakesanopenpositionrecognizableareiscalledan“argument”(Argument)ofthefunction.Ifconstantsaresetin(or“substituted”)forthevariables,theseconstantsarecalled“values”ofthearguments,andthemeaning(Bedeutung)oftheresultingcomplexsigniscalledthe“value”(Wert)ofthefunctionforthegivenargumentvalues(Argumentwert)”[Carnap,1929,3](mytranslation).Hereafunctionisanopensentence,theargumentofafunctionisavariable,andthevalueofavariableisaconstant.Somehopeisgivenbytheapparentdistinctioninthelastsentencebetweenthevalueofafunction(=aclosedsentence)anditsBedeutung,butthiscanhardlybetakenseriously,sinceBedeutungis,forCarnap,an“apparent”relation(Scheinrelation)[Carnap,1929,21].Soattentionnaturallyturnstotheapparentlyrelational“bezeichen”inthepassage.This,however,provideslittlecomfort:ifafunctionissomethingtowhichanopensentencebearssomerelation,thenitdoesn’tmakeanysensetocallthevariables“arguments”ofthefunction,rathertheargumentsshouldbethatwhichisbezeichnetbythevariables,butvariablesthemselves,wehavealreadybeentold,havenomeaningsatall.Obviouslywhatisneededhereistheideathatthevariablesinanopensentencerangeoversomeobjects.WithoutasemanticconceptionCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01ofvariablesandquantification,Carnapislefttoflounder—though,beingbasicallyaclearthinker,thelaterconstructionofclasses,cardinalnum-bersandsoonisunaffectedbyconfusionatthefoundationallevel.Asafurtherexample,however,ofthefoundationalunclarity,considerCarnap’sdefinitionsoftherangeofvaluesofafunctionandofasetabstract:10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE86—#3\nSemantics87“admissibleargumentvalues”arethosewhich,whensubstitutedintoacomplexexpression,giveitameaning;forthesethefunctionhasavalue…Therangeofadmissiblevalues(BereichderzulässigenWerte)ofavariableiscalledits“rangeofvalues”(Wertbereich)[Carnap,1929,3].zˆ(ϕz)willmean:‘everyvalueofzthatsatisfies(befriedigen)ϕ’;suchanexpressionsignifiesa“class”,inthiscase“theclassdeterminedbythesententialfunctionϕ”[Carnap,1929,16].Takingthisseriouslyalongwiththeearlierdefinitionofanargumentvalueasaconstantthatcanbesubstitutedforavariabletoproduceatrue,meaningfulsentence,therangeofvaluesofasententialfunctionisasetofnames,andtheclassdeterminedbyasententialfunctionis,again,asetofnames.Classes,therefore,wouldseemtobesetsofnames.OnemightsuspectthatthisisnotwhatCarnapreallywantstosaybutitiswhathesays.Indeed,andinhindsightastonishingly,whenpushcametoshoveCarnapactuallytookthisseriously.ThedefinitionofanalyticityforLanguageIIofLogicalSyntaxgoestosuchlengthtomakenamesdotheworkofobjectsintheaccountofquantificationthatCarnapwouldratherintroducenondenumerablesetsofnamestoprovideinterpretationsforhigherordervariablesthansimplytodothingsobjectually(§6.2).Wewillreturntothisinchapter6.2Carnapisstuckwiththis,ofcourse,becauseheholdsthatBedeutungisaScheinrelation“whichdoesnotappearatallinapurifiedlanguage”[Carnap,1929,21]and,correlatively,that“‘true’and‘false’areunde-finableconcepts”[Carnap,1929,3].TheearlypagesoftheAbrisscryoutforTarski’streatment:everythingwouldgomuchmoresmoothlyforCarnapifhehadsomewayoftalkingabouttherelationsbetweenexpres-sionsandobjects.However,withouttakingsemanticrelationsseriously,onecanhardlytakevariablesandquantificationasanythingotherthansubstitutional;theonlynotionofgeneralityaccessibletothesemanticskepticissubstitutionalgenerality.3Kotarbi´nskilikewiseunderstandsthevariablesinasententialfunctionsubstitutionally:Sententialfunctionsdifferastothekindofvariables:therearetermvariables,sententialvariables,conjunctionvariables,etc.…avariableCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01representsallconstantsoftheappropriatecategory;henceatermvari-ablerepresentsallterms;asententialvariable,allsentences,etc.Sinceithasbecometheusagetocalltheconstantsrepresentedbyavari-ablethevaluesofthatvariable(whichhasitssourceinthelanguageofmathematics,wherenumericalvariablesareused,forexample,inthecaseofthesententialfunction“x=2y”,wherethe“namesof10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE87—#4\n88AlfredTarski:PhilosophyofLanguageandLogicnumbers”formtherangeofxandy),itcanalsobesaidthatavariablerepresentsallitsvalues[Kotarbi´nski,1966,19].Theworkdoescontainaratherextensivediscussionofdenotation,buttreatedonlyasanauxiliaryfeatureofmeaningfultermsandentirelywithinthetermsoftraditionallogic.ThepassageabovemakesclearthatKotarbi´nskigraspednopossibilityofaroleforreferentialseman-ticnotionsintheinterpretationofthebasicexpressiveresourcesoflogicitself.Le´sniewski’sviewsonvariablesandquantificationareamatterofdis-pute.Hishostilitytosemantics(seebelow)makesclearthathecouldn’thavehadanythingtodowithaproperlyobjectualreadingofthequan-tifiersnorwiththecorrespondingconceptionofwhatavariablerangesover,butinterpreters(e.g.[Küng,1977])rejectQuine’ssuggestion[vanOrmanQuine,1973,99]thatLe´sniewski’sunderstandingissubstitu-tional.Someclueisgiven(asuggestionadoptedby[Küng,1977])byhistypicalformulationsintheearlyworkoftheform“forsomemean-ingoftheword‘a’”(e.g.[Le´sniewski,1992g,125]),butwhenitcomesthelaterworksayingsomethingresponsiblywouldtakeustoofarintotheLe´sniewskiliteratureforourpurposes,soIwillleavethematterhere.4Tarskihimselfmakesstatementsaboutvariablesandquantification,especiallyearlyon,thataresubstitutionalorplaywiththeideathatquantificationsomehowrangesover“meanings”.Forinstance,intheearlyLe´sniewskian“OnthePrimitiveTermofLogistic”areferenceto“quantifiers”isgiventhisnote:
InthesenseofPierce…whogivesthisnametothesymbols‘’(uni-versalquantifier)and‘’(particularorexistentialquantifier)repre-sentingabbreviationsoftheexpressions:‘foreverysignificationoftheterms…’and‘forsomesignificationoftheterms…’[Tarski,1983j,1].Thismakesfairlyclearthatatleastin1923TarskiadheredtoLe´sniewski’sviewofquantification.Indeedasubstitutionalunderstandingofsenten-tialvariablespersistsinTarski’swritingallthewaythroughtheEnglishversionofthetextbook,e.g.[Tarski,1995,38].ThereisalsoapeculiarCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01remarkatonepointin“TheConceptofTruth”itself:“Intheintuitiveinterpretationofthelanguage,whichIalwayshaveinmindhere,thevariablesrepresentnamesofclassesofindividuals”[Tarski,1983a,169].Intriguingly,thoughquantificationintheobjectlanguagegetsanobjec-tualtreatmentinthemetalanguage,the“intuitive”readingoftheobjectlanguageisglossedsubstitutionally.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE88—#5\nSemantics893.1.2ParadoxTarskiattributeshisconcernwithsemanticparadoxtoLe´sniewski[Tarski,1983a,154]andtonobodyelse.Theloneattributionmaywellbeappo-site,asstraightforwardstatementsoftheviewthatsemanticconceptsarenottobeincludedinserioustheorybecauseofthepossibilityofpara-doxareinfactdifficulttocomeacrossintheliteraturebeforeTarski.Theliarparadox,ofcourse,hadbeenknownsinceantiquity,butinthecrit-icalperiodfollowingthediscoveryofRussell’sparadox5thefocuswasontheset-theoreticparadoxes.WiththeadventoftheRichardparadox[Richard,1967]Peanoinitiatedatraditionofdistinguishingthesemanticfromthesettheoreticparadoxesandwritingtheformeroffas“linguistic”or“epistemological”(e.g.[Fraenkel,1928,210]),6butIhavebeenunabletofindaclearstatementoftheviewthatsemanticsistobeeliminatedbecauseoftheparadoxesinanyoftheplacesIwouldhaveexpectedtofindit.Forinstance,althoughTarskimakesuseofŁukasiewicz’sformula-tionoftheliarparadox[Tarski,1983a,157–8],Łukasiewiczhimselfdoesnotseemtohavetakenittocastdoubtontheconceptoftruth,prefer-ringinsteadaversionofthefamiliarstrategyoffindingsomereasonthatsentencesthatpurportedlyrefertoorquantifyoverthemselvesarenotwell-formed(remarktranslatedat[Wole´nski,1994,89]).ThereisnootherclueinTarski’sworkastowhowerethe“special-istsinthestudyoflanguage”hehadinmindamongwhomsemanticconcepts“foralongtime…havehadanevilreputation”[Tarski,1983a,252].Thesameissueariseswithrespectto“definable”in[Tarski,1983f]:“mathematicians”aresaidtohaveanattitudeof“distrustandreserve”withrespecttothenotionofdefinability,butaswewillseeactualmathe-maticianssuchasSkolemandtheAmericanpostulatetheoristsdisplayednosuchreticence.SincetheopinionwasquitecommoninWarsawthata“mathematician”wasanyonewhostudiedaxiomatizedtheories[Kotarbi´nski,1966,318ff]thereferencecouldagainbetoLe´sniewskiandTarskihimself(cf.[Hodges,2008,128]).In1935Tarskiagaincred-itsLe´sniewskiwithappreciatingtheproblem,butthecontextmakesitrelativelyclearthatwhatLe´sniewskiappreciatedwasthatthesemanticparadoxesshowthat“semanticalconceptssimplyhavenoplaceinthelanguagetowhichtheyrelate”[Tarski,1983b,402],whiletheskepti-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01calattitudelongtakentowardtheseconceptsisattributedtonooneinparticular.TarskiattributeshisremarksonparadoxinnaturallanguagetoLe´sniewski[Tarski,1983a,155]inlecturesanddiscussionsfrom1919onward.As[Betti,2004]argues,someoftheapproachTarskilatertakesupisforeshadowedinLe´sniewski’s1913paper“TheCritiqueof10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE89—#6\n90AlfredTarski:PhilosophyofLanguageandLogictheLogicalPrincipleoftheExcludedMiddle”,inasmuchasLesniewskiatleastgivestheT-sentencesaroleinthe‘paradoxofEpimenides’[Le´sniewski,1992c,77]andholdsthatnotall“‘naturalintuitions’oflanguage”[Le´sniewski,1992c,82]canbepreservedina“solution”totheparadox.However,theearlyLe´sniewskiprovidesasolutionbasedonabanonself-reference,appliedinacontextwheresentencetokensaretakentobetruthbearers[Le´sniewski,1992c,80].The“solution”there-forestandsinthemedievaltraditionofOckhamandBuridan,anditsclosestcontemporaryrelativeis[Simmons,1993],asBettinotes[Betti,2004,274].TheattitudethatLe´sniewskitheretakestowardthepreser-vationofintuitionsinthesolutionoftheparadoxesalsoseemsdirectlycontrarytotheoneexpressedat[Le´sniewski,1992h,177–8],soitisleftunclearexactlywhatsortofremarksTarskiheardintheyearsafter1919,thoughitseemsfairlyclearfromthetextthatin1913Le´sniewskihadn’tyetarrivedattheviewattributedtohiminESSthat“semanticalconceptshavenoplaceinthelanguagetowhichtheyareapplied”.AsidefromLe´sniewski,theonlyfiguremakingsomesortofpointtotherelevanteffectisCarnap,whointheAbrissappearstorejecttheinclusionofsemantictermsinserioustheory,ifnotongroundsoftheirgivingrisetoparadox,thenatleasttakingtheparadoxestobeasymptomofwhatiswrongwiththem.InadiscussionofthebranchedtheoryoftypesfromPMhewrites:Thebranchedtheoryoftypeswassetforthbecauseitwasbelievedthatcertainantinomiesofaspecialsortcouldnototherwisebeavoided.(Herebelong,forinstance,theantimonyofthesmallestnumberthatcannotbesignified(bezeichnet)withonehundredcharacters,the“heterological”antinomy,andsoon.)Acloserexaminationappearstoshowthattheseantinomiesareof,notalogical,butalinguis-ticsort,thatis,theyareonlyconsequencesofthedeficiencyofeverydaylanguage(dieWortsprache).(Namely,theyallinvolvetheapparentrelation“means”(“Bedeutens”),whichdoesnotoccuratallinapurified(gereinigten)language.)Theproblemoftheseantinomiesisnotyetsolved;ithangstogetherwiththeproblemofthethesisofextensionality(9e)[Carnap,1929,21](mytranslation).7Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Thehistoricalrecord,then,seemstoindicatethatskepticismaboutsemanticsbasedontheparadoxescametoTarskiprimarilyfromLe´sniewskiwithperhapssomehelpfromCarnap;Tarskihimselfseemstobeonthemarkwhenhesayslaterthat“asfarasIknow”Le´sniewskiwasthefirsttobecomefullyawareofthefactthatbecauseoftheparadoxessemanticconceptsdon’tbelonginthelanguagetowhichtheyapply10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE90—#7\nSemantics91[Tarski,1983b,402].Asidefromthesepoints,thesituationseemsrathertohavebeenthatsemanticconceptswereruledoutonthegroundsofablanketconsiderationtotheeffectthattheyobviouslyexpressed,asCar-napputit,“Scheinrelationen”;inthiscontexttheparadoxesweremoreasymptomthanasourceoftheproblem.Le´sniewski’sinfluenceisimportant,then,becauseonlytheworryaboutconsistencymotivatesTarski’sprojectofdefinition.AllofTarski’sstrictlyIntuitionisticFormalistgoalscouldbemetbyaxiomatizations.ItwastheproblemposedbytheparadoxesthatledTarskitothedefinitionsforwhichheisnowremembered.3.2MathematicalacceptanceThesemanticperspectiveonmeaningcouldbetracedinTarski’sinflu-encesbackthroughthealgebraiclogicians,inparticularPierce,whomTarskisinglesoutformentioninadiscussionofthedevelopmentoftheappreciationoftheroleofvariablesinmathematicsinthetext-book[Tarski,1995,14].Moreproximately,however,Tarski’sinfluencesinincorporatingthebasicnotionofthesatisfactionofasententialfunctionintometamathematicsarethegroupknownasthe“AmericanPostu-lateTheorists”,andLöwenheimandSkolem.8Asiswellknown[Vaught,1986,869–70][FefermanandFeferman,2004,73][Hodges,2008,120–4]in1927–1929Tarskiconductedaseminar(actually,theexercisesessionsattachedtoaseminarconductedbyLukasiewicz)inWarsawthatfocusedonSkolem’smethodofquantifiereliminationandonrelatedworkbytheAmericanPostulateTheoristC.H.Langford.ThepointforushereistheissuesthatexercisedLöwenheimandSkolem,aswellasthepostulatethe-orists,wereonesthat,againstCarnap’sscruples,demandedthatsemanticnotionsbesomehowtheoreticallytractable,ifnottoanyone’sknowl-edgedefinableintermsofsomethingmoreuncontroversiallyacceptablebythestandardsofthetime[Vaught,1974,161].Löwenheimformulatesbothsomeofhisbasicdefinitionsandthetheoremthatbearshisnameintermsofthesatisfactionofexpressionsinasensethoroughlyfamil-iartous,thoughundefinedandunremarked-upon[Löwenheim,1967,233–5].Skolem,likewise,simplystatestheresultsin[Skolem,1967]inCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01termsof“domains”,ofwhichaxioms“hold”,orinwhichpropositionsare“satisfied”.AswithLöwenheim,thenotionsareusedasthoughtheywereunremarkable.Langfordspendstheearlypagesof[Langford,1926]explainingissuesofscopeandmultiplequantification,aswellasthereductionoffirst-orderformulastonormalform.Theconcernofthearticleiswithhow10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE91—#8\n92AlfredTarski:PhilosophyofLanguageandLogicthetruth-valuesof“first-order”functionsare“determined”bysetsofpostulates(inthecasesatissue,forlinearorder);theargumentsgobywayofestablishingequivalencesbetweenfirst-orderformulasofvariousformsandtheirnormalforms.LikeLöwenheimandSkolem,LangfordshowsnoCarnapianorLe´sniewskianscrupleabouttheappearanceofunreducedsemanticprimitivesinthediscussion.Langfordspeaksof“consequence”,butinawaythatwecanrecognizeassemanticinaccordwithTarski’slaterdefinition,explicitlydistinguishingitfrom“strictconsequence”,whichisn’tremarkeduponfurther:Thegeneralprocedureoftheproofswhichfollowmaybeoutlined:Everymultiplyquantifiedfunctioninnvariablesisshowntobeequiv-alent,inviewof(1)–(10)[Langford’spostulatesfordenseorder],tosomefunctionintwovariables.Thisequivalenceis,ofcourse,mate-rialandnotstrictsinceitdependson(1)–(10).Thefunctionintwovariablestowhichafunctioninnvariablesisshowntobemateriallyequivalentisthereforeinnosenseanotherformofthesamefunction.Thefunctionshavethesametruth-valuesolelyinviewof(1)–(10)[Langford,1926,28].Notethesensitivitytothesemanticaspectoftheissue:it’snotthatafunctionisequivalenttoanotherinsomesenseofderivabilityor“sayingthesamething”,butsimplythatifthepostulates(1)–(10)hold,itcanbeshownthattheyhavethesametruthvalueonallassignmentstothevariables.LangfordherestandsinthetraditionoftheAmericanPostulateThe-orists,whoseapproachwasthoroughlysemanticormodel-theoreticinthemodernsensefromtheturnofthecenturyon.Keypostulatetheo-ristssuchasVeblen(e.g.[Veblen,1904,344])inturncitethePeanistsandPasch.9Scanlanwritesoftheirpapersthatthey:containnumerousreferencestotheEuropeanresearchliteratureoftheperiodandareclearlymeanttobecontributionstolinesofresearchthatwereongoinginEurope.Theyneverthelessrepresent(1)clearformulations,bothbyexplicitstatementandbyexample,ofapostu-lationalpointofviewthatistheauthor’sown,and(2)advances,inCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01somerespects,overwhatwascurrentinEurope[Scanlan,1991,982].Onecanseethisjustaftertheturnofthe20thCentury.[Huntington,1902]readsinsomewaysasquiteold-fashioned—e.g.atoneplacewegettheexplanationthatanidentity“x=yindicatesthatthetwosymbolsxandyareusedtorepresentthesameobject”[Huntington,1902,266],thearticleisfilledwithtalkof“rulesofcombination”fortheobjectsthat10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE92—#9\nSemantics93belongto“assemblages”,andthenotionofa“class”isgivenaglossintermsofacondition—butthesemanticperspectiveisfullyinevidence;e.g.wehave:TheoremII:—AnytwoassemblagesMandM
whichsatisfythepostulates1–6areequivalent;thatis,theycanbebroughtintoone-to-onecorrespon-denceinsuchawaythata◦bwillcorrespondwitha
◦b
wheneveraandbinMcorrespondwitha
andb
inM
,respectively[Huntington,1902,277].Wesawabove(§2.3)thatVeblen’snotionofcategoricityissemantic.Veblenalsoexplicitlydistinguishesdefinitioninthesenseofapredicate’sdefiningasetfromdefinitioninthesenseofexplicitdefinition:E.V.Huntington,inhisarticleonthepostulatesoftherealnumbersystem,expressesthisconceptionbysayingthathispostulatesaresuf-ficientforthecompletedefinitionofessentiallyasingleassemblage.Itwouldprobablybebettertoreservetheworddefinitionforthesubsti-tutionofonesymbolforanother,andtosaythatasystemofaxiomsiscategoricalifitissufficientforthecompletedeterminationofaclassofobjectsorelements[Veblen,1904,346–7].10Considerablylater,both[HilbertandAckermann,1928]and[Fraenkel,1928],citedoftenbyTarski,workwiththenotionsofsatisfactionandasetofobjectsbeingamodelofasetofsentences,treatingthemasunremarkable.Indeed,aswillbeatissuelater(§7.3.2),HilbertandAck-ermannareabletoposethequestionsolvedbyGödel’sCompletenesstheorembecausetheyattendtothequestionofwhichsentencesaretrueinmodelsofallcardinalities[HilbertandAckermann,1928,68].Onthemathematicalside,then,thepictureisclear:mathematicianswereworkingcomfortablywiththesemanticnotionofasystemofobjectssat-isfyingsomeaxiomsinthedecadesbeforeTarski’swork,andachievingfruitfulresults.Thereisnoevidencethatthesefiguresharboredsomemisgivingsaboutthesemanticnotionsinvolved(cf.[Feferman,2008,80]).11Asaresultofthesetrends,manyauthorsatthetimeweren’tpushingthesubstitutionalaccountsofquantificationthatTarski’sclosestphilo-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01sophicalinfluenceswere.ForexampleHilbertandAckermann,whomTarskicites,haveapassablyclearsemantictreatment:Westilllackasymbolicexpressionforthegeneralityofstatements.Toarriveatone,inthemannerofmathematicsweintroducealongsidethesignsfordeterminateobjects(thepropernames(Eigennamen))alsovariablesx,y,z...withwhichwecanlikewisefilltheemptypositions10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE93—#10\n94AlfredTarski:PhilosophyofLanguageandLogicinafunction-sign.Adeterminatecompletionofanemptypositioniscalledavalueofthevariableinquestion.Thevaluesofavariableareingenerallimitedtodeterminatetypesofobjects,whicharethemselvesdeterminedthroughthemeaning(Bedeutung)ofthefunction-sign.Forexample,thebasicrelationofplanegeometry:“derPunktxliegtaufderGeradeny”isrepresentedbyafunction-signwithtwoargumentsL(x,y)Hereonlypointsareeligibleasvaluesforxandonlylinesasvaluesofy[HilbertandAckermann,1928,45](mytranslation).Thepassagedisplayssomepeculiarities—e.g.thereappearstobeatleastsomeuse–mentionconfusionofCarnap’ssortintheaccountofthevalueofavariableattheendofthefirstparagraph—butoveralltheaccountistherightone:thevaluesofthevariablesinthesecondparagrapharepointsandlines,nottheirnames.Thusrangingoverobjectsandreferringtothemareimportantrelationsintheaccount.Suchanexplanationcouldnotbeofferedbyanyonewho,likeCarnap,thoughttherewassomethingwrongwithsemanticconceptsandwhobelievedthatthejobcouldbedonewithoutthem.3.3IntuitionisticFormalismin“OnDefinableSets”Ontheoneside,wehaveCarnap,Le´sniewskiandothersinsistingthatsemanticnotionssimplycan’tplayaroleinserioustheory;ontheother,wehaveSkolem,thepostulatetheoristsandothersengagedinagooddealofserioustheorythatmakesessentialuseofsemanticnotions[Vaught,1974,161].ItissometimeswonderedwhatsetTarskioffontheprojectofdefiningtruth[Feferman,2008,72–3],butsurelythestateofaffairsoccasionedjustthereconciliationheattemptedtoeffect.Com-parehereGödel,whosimilarlywaspayingattentionbothtoskepticslikeCarnapanddogmatistslikeSkolemandwhowasmovedtoelimi-natetheappealtointuitivesemanticnotionsin[Gödel,1967b]fromthe“rigorous”proofofhistheorem.IthusthinkthatFefermanunderstatesthecaseat[Feferman,2008,72]whenheholdsthattherewasno“com-pellinglogicalreason”forTarski’sdefinitionsandthatTarskihadonlyCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01“psychologicalandprogrammaticreasons”forthedevelopmentofhistruthdefinitions.Tarski’saimwasn’tclarityforitsownsake(contra[Feferman,2008,80]);hisaimwastoestablishthatsemanticnotionswereimmunefromcertainsortsofskepticaldoubt.Importantly,notethatTarskineversaysthatsemanticconceptsareunclear;onthecontrary,herepeatedlyemphasizesour“intuitiveknowledge”ofthem[Tarski,1983a,154]and10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE94—#11\nSemantics95thefactthat,ofthem,wecan“giveanaccountthatismoreorlesspreciseinitsintuitivecontent”[Tarski,1983f,111].IntuitionisticFormalismmeantthatsemanticconceptscouldbeexpressedeffectivelyjustincasetheiranalysescouldbeexpressedinthetheoremsofadeductivetheory,whiletheworriesaboutparadoxrequiredthatthisbedonebydefinitionratherthanaxiomatizationinordertoguaranteerelativeconsistency.ThusTarskihadtotackletwoproblemsatonce:(a)thedevelopmentofanIntuitionisticFormalistcon-ceptionoftheconditionsunderwhichadeductivetheoryhadtheoremsthatconstrainedoneofitsexpressionstoexpresssomesemanticconcept,and(b)thedevelopmentoftechniques,acceptableinmetamathematicsasheconstruedit,fortheintroductionofsuchatermviaadefinition.WecanrecognizeherealreadythetwodesiderataofstatedinConven-tionT:theneedfor(a)istheneedfora“materiallyadequate”treatmentofsemanticconcepts,whiletheneedfor(b)istheneedfor“formallycorrect”definitions.12However,Tarskiwasquickertoseetheneed(b)forexplicitdefini-tionandtodeveloptechniquesformeetingit,thanhewastoworkout(a)anIntuitionisticFormalistconceptionofwhattheoremsneededtobeprovableinadeductivetheoryinorderforthattheorytohaveatermthatexpressestheconceptoftruth.Introducinganexplicitdefini-tionofatermtheintendedinterpretationofwhichissemanticintoanon-semantictheoryforcestheintuitiveinterpretationofthetermtobewhateverisassociatedwiththedefiniensbythedefinition.Butitisonethingtoseethis,andanotherthingtohaveaviewaboutwhatittakesforaterm(primitiveordefined)tohaveaformalrolethatsuitsittoexpressasemanticconcept.3.3.1TheintuitivenotionofdefinabilityThefirstsemanticworkinLogic,Semantics,Metamathematicsis1930’s“OnDefinableSetsofRealNumbers”.Herethequestionforegroundedseemstobeinthespiritofthepostulatetheorists:givenatheoryandtheReals,whichsetsofRealsaredefinedbyopensentencesofthetheory?13Importantly,someaspectsofthetruth-definitionappearinthearticle,butwithoutthefanfareaccordedthemin“TheConceptofTruthinFor-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01malizedLanguages”.Thisisalreadysignificant:Tarskitakeshimselftobedoingsomethingimportantinthelatterpaper,butitcannotbemerelysettingoutthetruth-definition,sincethisappearsin“OnDefinableSets”withonlybriefcommentary;indeed,henotestheequivalenceoftruthwithsatisfactionbyanarbitrarysequenceinasinglesentence[Tarski,1983f,117].10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE95—#12\n96AlfredTarski:PhilosophyofLanguageandLogic“OnDefinableSetsofRealNumbers”isamorepuzzlingarticlethanitmightatfirstappear.Superficially,Tarskipointsoutthatsemanticdefin-tionistheconverseofsatisfactionandintroducessomeofthemechanicsofthetreatmentofthelatterfamiliartousfromCTFL(though,aswillbeestablishedbelow,farlessthancommentatorsoftenclaim),andthenprovessomeresultsaboutwhatsetsofRealsaredefinableinparticu-lartheories—e.g.,inTheorem2,thatfirst-orderarithmeticallydefinablesetsofrealsarefinitesumsofintervalswithrationalendpoints[Tarski,1983f,134].ThepuzzleaboutthearticleconcernshowTarskicouldpos-siblythinkthatdoingwhathedoesinthearticleamountstodoingwhathesaysattheoutsetheisgoingtodo.Whathesayshewilldoisthis:Thedistrustofmathematicianstowardthenotioninquestion[defin-ability]isreinforcedbythecurrentopinionthatthisnotionisoutsidetheproperlimitsofmathematicsaltogether.Theproblemsofmakingitsmeaningmoreprecise,ofremovingtheconfusionsandmisun-derstandingsconnectedwithit,andofestablishingitsfundamentalpropertiesbelongtoanotherbranchofscience—metamathematics.InthisarticleIshalltrytoconvincethereaderthattheopinionjustmentionedisnotaltogethercorrect.Withoutdoubtthenotionofdefinabilityasusuallyconceivedisofametamathematicalori-gin.IbelievethatIhavefoundageneralmethodwhichallowsustoconstructarigorousmetamathematicaldefinitionofthisnotion.Moreover,byanalyzingthedefinitionthusobtaineditprovestobepossible(withsomereservationstobediscussedattheendof§1)toreplaceitbyadefinitionformulatedexclusivelyinmathematicalterms.Underthisnewdefinitionthenotionofdefinabilitydoesnotdifferfromothermathematicalnotions[Tarski,1983f,110–111].ThepuzzleisthatthoughTarskicertainlysetsupacorrelationbetweenhistheoryoffirst-orderarithmeticandthesettheoreticconstructionwithwhichhepairsit,theredoesnotatfirstreadingappeartobeanythinginthearticlethatwouldjustifytheclaimthathehascap-turedthe“meaning”(“sens”[Tarski,1986g,519])of“definablesetofrealnumbers”—for,surely,thatmeaningis“metamathematical”,whileCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Tarski’s“mathematical”definitionofacertainsetofsetsofRealsastheclosureofcertainothersundercertainoperationspreciselyisnot.Fur-thermore,aswewillseeshortly,crucialaspectsofthesemantictreatmentinCTFLaresimplymissingfromthearticle,therebymakingitques-tionablewhetherinthetheoryitself—asopposedtothemindofthemathematiciancontemplatingit—anythingsemanticisexpressedatall.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE96—#13\nSemantics97Thedeterminationtointrouduceatermthatexpressesasemanticcon-ceptintoadeductivetheoryinIntuitionisticFormalistfashioncanbeseeninoneofthemostphilosophicalpassagesinTarski’scorpus:Theproblemsetinthisarticlebelongsinprincipletothetypeofproblemswhichfrequentlyoccurinthecourseofmathematicalinves-tigations.Ourinterestisdirectedtowardsatermofwhichwecangiveanaccountthatismoreorlesspreciseinitsintuitivecontent(contenuintuitif),butthemeaningofwhichhasnotatpresentbeenrigor-ouslyestablished,atleastinmathematics.Wethenseektoconstructadefinitionofthistermwhich,whilesatisfyingtherequirementsofmethodologicalrigor,willalsorender(établie)adequatelyandpre-ciselytheactualmeaningoftheterm[Tarski,1983f,111–112][Tarski,1986g,521].Noteseveralthingsaboutthispassagesofar.First,wehavehereaphilo-sophicaltakeonmathematicsandmathematiciansbeingconcernedwithsomethinglikeconceptualanalysis:clarifyingan“intuitivecontent”isamathematicalgoalinthispassage.SinceweknowfromKotarbi´nskithatanyonewhostudieddeductivetheorieswasa“mathematician”,andsinceweknowfromLe´sniewskithatoneofthemaingoalsofadeduc-tivetheoryistoallowtheexpressionofthoughtsinvolvingtheconceptsexpressedbyitsprimitives,thepassagecanbetakentomeanthattherehasbeendoubtastowhethersemanticnotionscouldbeexpressedinadeductivetheoryinIntuitionisticFormalistfashion.TarskisurelyhasLe´sniewskiinmindhere.Second,notetheclearappearanceoftheIntu-itionisticFormalistgoalofexpressinganintuitiveconceptinadeductivetheoryinthepassage:thereisaconceptweintendtoexpressbyaterm;wehaveagoodideaofwhatitis,yetthissomehowcompatiblywithalackof“rigor”.Wehavethenotionof“adequatelyandprecisely”(avecjustesseetprécision[Tarski,1986g,521])capturing“intuitivecontent”,andalsoanalliedrequirementthatwedoso“whilesatisfyingtherequirementsofmethodologicalrigor”—precursorsofConventionT’s“materiallyade-quateandformallycorrect”.Thegapbetweenourintuitivenotionandthissought-forrigoristobeclosedbyconstructingaformaldefinition.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01SofarthisisallorthodoxIntuitionisticFormalism.Letuscontinuewiththepassage:Itwasjustsuchproblemsthatthegeometerssolvedwhentheyestab-lishedthemeaningoftheterms‘movement’,‘line’,‘surface’,or‘dimension’forthefirsttime[Tarski,1983f,112].10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE97—#14\n98AlfredTarski:PhilosophyofLanguageandLogicThelate19thandearly20thcenturygeometersTarskiisthinkingofwerethosewhogaverigorousaxiomatictreatments.AsweknowfromourdiscussionofIntuitionisticFormalism,capturing“intuitivecontent”inthewaythatconcernsTarskiisamatterofsettingupadeductivetheorythatimpliestheoremsinvolvingasymbolthatconstrainittoexpressthiscontent.Continuing:HereIpresentananalogousproblemconcerningtheterm‘definablesetofrealnumbers’[Tarski,1983f,122].Givenallthis,thetasksetforthearticleistheexpressiveoneTarskihasbeeninterestedinformetamathematicalconcepts,andformaldefinitionwithinadeductivetheorywillbethefocusofattention.Althoughthearticleisaboutsemanticdefinition,itsstatedgoalisanexplicitformaldefinitionthatcapturesacertainconcept.Wecansee,hereaselsewhere,theextenttowhichTarski’smaininterestintheearly1930sisinthisexpressivetask,withthesemanticcontentofthearticle,suchasitis,onlycomingininserviceofit.Continuingagain:Strictlyspeakingthisanalogyshouldnotbecarriedtoofar.Ingeome-tryitwasaquestionofmakingprecisethespatialintuitionsacquiredempiricallyineverydaylife,intuitionswhicharevagueandconfusedbytheirverynature.Herewehavetodealwithintuitionsmoreclearandconscious,thoseofalogicalnaturerelatingtoanotherdomainofscience,metamathematics[Tarski,1983f,122].Theannouncedtaskistoanalyzeaconceptfrommetamathematics—notmerelytodosomemetamathematicsbysettinguptheformalseman-tics.Rather,theformalsemanticsservestheconceptuallyanalyticgoals.Continuing:Tothegeometersthenecessitypresenteditselfofchoosingoneofseveralincompatiblemeanings,butherearbitrarinessinestablishingthecontentoftheterminquestionisreducedalmosttozero[Tarski,1983f,122].ThereferencehereispresumablytoaxiomaticstudiesofgeometrysuchCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01asthoseofHilbert[Hilbert,1971]andVeblen[Veblen,1904].Itakethepointofthepassagetobethatingeometryintuitiveconceptsofspaceacquiredineverydaylifedon’tselectamongvariousgeometriesandthathencetheconceptualanalysisofgeometricnotionsiscontrover-sial.Bycontrastthereisnosimilardisputeabout“definable”andhencewehavegoodhopeforanuncontroversialconceptualanalysis,leaving10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE98—#15\nSemantics99usonlythetaskofcraftingadeductivetheorythatconstrainsoneofitsprimitivetermstoexpresstheconceptsoanalyzed.Ishallbeginthenbypresentingtothereaderthecontentofthisterm,especiallyasitisnowunderstoodinmetamathematics.TheremarksIamabouttomakearenotatallnecessaryfortheconsiderationsthatwillfollow—anymorethanempiricalknowledgeoflinesandsurfacesisnecessaryforamathematicaltheoryofgeometry[Tarski,1983f,122].Sincethetopicdiscussionofwhichis“notatallnecessary”hereisthesameonethatis“ofcapitalimportance”at[Tarski,1983f,129],thepointofthispassagemustbethestandardIntuitionisticFormalistonethatadeductivetheorycanbeunderstoodbysomeonewhodoesn’tknowwhatconceptsitsprimitivetermsareintendedtoexpress.Thisisclearinthenextsentence:Theseremarkswillallowustograspmoreeasilytheconstructionsexplainedinthefollowingsectionand,aboveall,tojudgewhetherornottheyconveytheactualmeaningoftheterm.Thecompletedtheorywillsupposedlyexpressaswellaspossiblethe“intuitivecontent”of“definablesetofrealnumbers”;thepointhereisthatitdoessowhetherornotoneknowsthatitdoes,butthatofcourseonecannotknowthatitdoesifonedoesn’thavethisintuitivecontentinmind,andsothereaderwillberemindedofthiscontentbeforetheconstructionbegins.WecanpauseheretonoteanissuethatwediscussedinChapter2:therelativizationofTarski’sdefinitionstoaparticularobjectlanguage.Aswiththenotionsofsentenceandconsequence,Tarski’streatmentofdefinabilityisrelativetoaparticulardeductivetheoryofarithmetic,buthesystematicallysuppressesthisrelativizationandishappytospeakofdefining“the”conceptofdefinability,ratherthanofdefiningtheconceptofdefinabilityinthatformulationofarithmetic.Thepointsmadetherecarryoverhere:onTarski’sconceptionofaconceptlinguisticCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01conceptsmustberelativizedtoalanguagetohaveconditionsofappli-cabilityandthusextensions.Thereisonlyoneconceptofadefinablesetofrealnumbers,butitisappliedonlyrelativetoanobjectlanguage.Tarskipushesthisrelativizationintothecommentaryonhismetathe-orybecausethenotionofbeinginalanguageisn’tmetamathematicallytractable.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE99—#16\n100AlfredTarski:PhilosophyofLanguageandLogic3.3.2Definingdefinablesetsvsdefining“Defines”ReadersofCTFLwillexpectcertainthingsfromhereforwardinODSthatthearticlesimplydoesn’tsupply.Inparticular,onewouldexpect1.Arigorousdistinctionbetweenobjectlanguage,metalanguage,andmetametalanguage2.Acriterionofadequacyforasuccessfuldefinition,formulatedinthemetalanguage,‘xisadefinablesetofrealnumbers’fortheobjectlanguage3.Meticulousnessaboutuseandmention;inparticular,acleardistinc-tionbetweenexpressionsofthemetalanguagethatrefertoordescribeobjectlanguagesentences,andthesesentencesthemselves.4.Aconceptionofdefinitionsofsemantictermsonwhichtheyareinsomesense“logicalsums”of“partialdefinitions”thattradebetweenuseandmention[Tarski,1983a,264].ODS,however,containsnoneofthesethings.Theonlynodtothemisthispassage:Letusnowconsiderthesituationmetamathematically.Foreachdeductivesystemitispossibletoconstructaparticularscience,namelythe‘metasystem’,inwhichthegivensystemissubjecttoinvestigation.Hence,tothedomainofthemetasystembelongallsuchtermsas‘variableofthenthorder’,‘sententialfunction’,‘freevariableofasententialfunction’,‘sentence’,etc.Ontheotherhandnothingforbidsourintroducingintothemetasystemarithmeticalnotions,inparticularrealnumbers,setsofrealnumbers(desnotionsd’Arithmetique,doncenparticuliercelledenobreréel,d’ensembledenom-bresréels),etc.Byoperatingwiththesetwocategoriesofterms(andwithgenerallogicalterms)wecantrytodefine(préciser)thesenseofthefollowingphrase:‘Afinitesequenceofobjectssatisfiesagivensententialfunction’[Tarski,1983f,116][Tarski,1986g,523].Notethat“define”for“préciser”obscurestheimportantIntuitionisticFormalistdistinctionbetween“makingprecise”anintuitivemeaningandintroducingadefinitionthataugmentsadeductivetheorytocon-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01tainatermthathasthatmeaningsoprecisifiedandthatWoodgerhasdroppedtheinterestinintroducingthe“notions”ofrealnumbersandsetsofthem.Theconstructionin§2goes,insomerespects,inanow-familiarway.First,Tarskiaugmentssimpletypetheorywith“thespecifictermsofthearithmeticoftherealnumbers”,sententialfunctionsυ,µand10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE100—#17\nSemantics101(k)(k)(l)(k)σ,receivingtheinterpretations“x
=1”,“x

x
”and“x
=(l)(m)x
+x
”,thesubscriptsexpressingorderandthesuperscriptsenu-meratingthevariableswithinanorder[Tarski,1983f,114].Tarskithenpointsoutthateachoftheprimitivesententialfunctionsdeterminesasetofsequencesofobjectsthatsatisfyitandheusesthesetabstrac-tionoperatortoformexpressionsforthesesets;wecanrepresentthesehere,ignoringthepointthatreallysetsoforderedpairsaredeterminedasTarskidoesthingsforone-placesententialfunctions([Tarski,1983f,117];thesequencesarejustbook-keepingfortheenumeratedvariables),as{x|x=1},{(x,y)|x
y}and{(x,y,z)|x=y+z}.Correspodingtotheclassicallogical“fundamentaloperations”areoperationsonthesesetsofthefamiliarsorts—unionfordisjunctionandsoon.Thefirst-orderdefin-ablesetsRealsinSTTaugmentedbytheprimitivesententialfunctionsarethensimplytheclosureoftheprimitivesetsundertheoperations(Def.9,128)andthesetsofRealsdefinablearethosetobefoundinanyone-termsequence(Def.10,128).14Itissometimesclaimed(e.g.[Hodges,2008,125])thatthearticlecon-tainsthefirstcompositionalsemanticsforaformallanguage.15Thoughitistruethatsomeofthetechnicalapparatusforthisisinplaceinthearticleitislessclearwhetherwhathappensinthetheorydiscussedinthearticlereallyamountstoasmuch.Forratherthanintroducingsomesystematicmeansofreferencetosentencesofhisdeductivearithmetictheoryinhismetatheory,alongwithtranslationsofitssentencesintothelanguageoftheset-theoreticconstructionandthenexplicitlydefin-ing“xisthesetofrealnumbersdefinedbyy”inthismetatheory,Tarskisimplyintroducesset-theoreticvocabularyintowhatappearstobethearithmetictheoryitself,definescertain“primitivesets”and“operations”onthem,andthendefinesasetDofsetsofrealnumbersastheclosureoftheprimitivesetsundertheoperations[Tarski,1983f,128](ignor-ingsomedetails).Whenweturntothequestionofwhetherthesenseof“definable”hasbeencapturedintheextendedtheory,wegetthisannouncement:Nowthequestionariseswhetherthedefinitionsjustconstructed(theformalrigourofwhichraisesnoobjection)arealsoadequatemateriallyCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01(justeaupointdevuematériel);inotherwords,dotheyinfactgrasp(saisit)thecurrentmeaningofthenotionasitisknownintuitively(lesenscourantetintuitivementconnudelanotion)?Properlyunderstood,thisquestioncontainsnoproblemofapurelymathematicalnature,butitisneverthelessofcapitalimportanceforourconsiderations[Tarski,1983f,129][Tarski,1986g,538].10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE101—#18\n102AlfredTarski:PhilosophyofLanguageandLogicAnticipatingthestructurefoundinCTFL,wewouldexpectatthispointsomeargumenttotheeffectthat“∈D”hasbeenintroducedinsomewaythatimpliesclearlysemantic“partialdefinitions”insomeplausiblyexhaustiveway;herethepartialdefinitionswouldbeoftheform:xisdefinedbyϕiff∀y(y∈x⇐⇒ϕ(y))This,however,isnotwhatweget.Tarskifirststatesthefollowing:Wehaveinfactalreadyestablishedthatthereisastrictcorrespon-dencebetweenprimitivesetsofsequencesandthefundamentaloperationsonthesesetsontheonehand,andtheprimitivesententialfunctionsoforder1andthefundamentaloperationsontheseexpres-sionsontheother…Fromthesefacts,Def.9,andthedefinitionsof§1ofasententialfunctionanditsorder,wecanshowwithoutdiffi-culythatthefamilyDfisexactlythefamilyofsetsofsequenceswhicharedeterminedbysententialfunctionsoforder1.ItfollowsalmostimmediatelythatthefamilyDcoincideswiththatofthedefinablesetsoforder1inthesenseof§1[Tarski,1983f,129].Whatthissaysisthatwecanquiteeasilyprovethattheclosureoftheprimitivesetsunderthefundamentaloperationsisthesetofsetsdefinedbysentencesopenonasinglefirst-ordervariable.Nothing,however,inthe“metamathematical”§1ofthearticleshowsushowtoeliminate“definable”intermsofsomethingnon-semantic.Thedescribedproof,inturn,simplyshowsusthatthe(first-order)definablesetsmetamath-ematicallyconstruedareexactlythesetsintheclosureoftheprimitivesetsunderthefundamentaloperationsof§2.Thatshowsthatthe“math-ematical”Definition10getstheextensionright.Itdoesnot,however,showwhatTarskiclaimsinthearticlewillbeshown,namely,thattheconceptofdefinabilitywillbeexpressedinatheorywithmathematicalprimitives.Thesketchedproofthusdoesnot,sofar,addresstheissueof“capitalimportance”.Todothatwouldrequiresomeargumentthat“∈D”iscaughtupintheoremsoftheintroduceddeductivetheorythatconstrainittoexpresstheintuitiveconceptofdefinability.Amereproofofextensionaladequacydoesn’tdothisjob.16Tarskithereforecontinues:Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Ifwewishtoconvinceourselvesofthematerialadequacy(justessematérielle[Tarski,1986g,538])ofDef.10andofitsconformitywithintuitionwithoutgoingbeyondthedomainofstrictlymathematicalconsiderations,wemusthaverecoursetotheempiricalmethod.17Infact,byexaminingvariousspecialsetswhichhavebeenarithmeti-callydefined(intheintuitivesenseofthisterm),wecanshowthat10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE102—#19\nSemantics103allofthembelongtothefamilyD;conversely,foreveryparticularsetbelongingtothisfamilyweareabletoconstructanelementarydefini-tion.Moreover,weeasilynoticethatthesamecompletelymechanicalmethodofreasoningcanbeappliedinallthecasesconcerned.Thefollowingshowswhatthismethodinvolves:LetAbeanysetofnumberselementarilydefinedwiththehelpoftheprimitivenotions‘1’,‘
’,and‘+’.ThedefinitionofthesetAmaybeputintheform:A=E(k)φ(x(k))xThesymbol‘φ(x(k))’hererepresentsacertainsententialfunctioncon-tainingthevariable‘x(k)’oforder1asitssolefreevariable.Itmayalsocontainaseriesofboundvariables‘x(l)’,‘x(m)’,etc.,providedthatallthesevariablesarealsooforder1(sinceifthiswerenotthecasethedefinitioncouldnotbecalledelementary).Asweknowfromlogic,thefunction‘φ(x(k))’canbeconstructedinsuchawaythatitcontainsnologicalconstantsotherthanthesignsofnegation,oflogicalsumandproductaswellastheuniversalandexistentialquantifiers.Inthesamewaywecaneliminateallthearithmeticalconstantsexceptthesigns‘υ’,‘µ’,and‘σ’,correspondingtothethreeprimitivenotions.Itisnotdifficulttoshowineachparticularcasethatthefor-mula[displayedabove]canbetransformedinthefollowingway.Thesententialfunctions‘υ(x(k))’,‘µ(x(k),x(l))’,and‘σ(x(k),x(l),x(m))’arereplacedbysymbolsoftheform‘Uk’,‘Mk,l’,and‘Sk,l,m’,respectively;thesignsoflogicaloperationsbythoseofthecorrespondingopera-tionsonsetsofsequences,introducedinDefs.5–7;andfinallythesymbol‘Ex(k)’by‘D’.Bythistransformationtheformula[above]takestheformA=D(S)whereinplaceof‘S’thereisacompositesymbol,thestructureofwhichshowsatoncethatitdenotesasetofsequencesofthefamilyDfwithcounterdomainconsistingofasingleelement.ByapplyingCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Def.10weconcludeatoncethatthesetAbelongstothefamilyD.Ishallnowgivesomeconcreteexamples:1.LetAbeasetconsistingofthesinglenumber0.WeseeatoncethatA=Ex(x=x+x),i.e.thatA=Exσ(x,x,x).BytransformingthisformulainthewayjustdescribedweobtainA=D(S1,1,1),sothatA∈D.[Tarski,1983f,129–131].10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE103—#20\n104AlfredTarski:PhilosophyofLanguageandLogicTarskiaddsseveralmoreexamplesandthenfinisheswithaflourish:“Asaconsequenceoftheseconsiderations,theintuitiveadequacyofDef.10seemstobeindisputable”[Tarski,1983f,132].Howdoesthishelptoanswerthequestionof“capitalimportance”ofwhether“∈D”expresses,withinthedeductivemetatheory,theconceptsemanticdefinition,therebymakingit“indisputable”thatDef.10isintuitivelyadequate?Notethattheidentitiesprovenherebythe“com-pletelymechanicalmethod”ofreplacingarithmeticvocabularywithappropriatelyselectedset-theoreticvocabularyandtheset-abstractionoperatorwith“D”involvenointerplaybetweenuseandmention.Theyarealloftheform{x|ϕ(x)}={x|ψ(x)}whereϕisanarithmeticpredicateandψisaset-theoreticone.Theidenti-tiesallestablishthatasetofrealnumbersspecifiedbyabstractionfromanarithmeticconditionistherange(officially,inthearticle,the“domain”[Tarski,1983f,121note†])ofasetofone-termedsequencesspecifiedintermsofTarski’sprimitivesetsandfundamentaloperations.PartoftheproblemhereisthatTarskiseemstohavebamboozledhimselfwithhisownmetalinguisticexplanationofhisset-abstractionoperatorwhich,heclaims,whenprefixedtoasentenceφopenononefirst-ordervariable,“denotesthesetofallobjectsthatsatisfythecondi-tionφ”[Tarski,1983f,121].(ThisexplanationcorrespondstoCarnap’sat[Carnap,1929,16].)Theoperator,however,doesn’tactuallydothat:itsimplyformsareferringexpressionfromanopensentence.Whathap-pensinthearticle,however,isthatforthephilosophicalpurposesof“convincingourselvesofthematerialadequacyofDefinition10”thesetabstractionoperatorisviewedasintroducingaquotationalcontext—thisdespiteTarski’squalmsaboutquotation[Tarski,1983a,159ff].Viewedthisway,theidentitiesprovablebyTarski’s“completelymechanicalmethod”dohavemetalinguisticcontent;ratherthantheabove,theyareoftheform:E(ϕ(x))={x|ψ(x)}Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01WhatisreallygoingonhereisactuallysomethingsomewhatrelatedtothetransformationtowhichTarskisubject’sPadoa’smethodin[Tarski,1983k].Justasbefore,inTarski’s“justification”ofthemethodbyformulatingPadoa’sconditionasadeductiveone,thecrucialmetamath-ematicalcontentisinthemindofthemathematicianwhocontem-platesadeductivetheorywhile“varyingthemeaning”oftheprimitive10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE104—#21\nSemantics105terms—inODStoothesemanticsisstillhappeninginthemindofthemathematician:contemplatingadeductivetheorythatcontainsbotharithmeticandset-theoreticvocabulary,onecanviewtheset-abstractionoperatorasintroducingaquotationalcontextandtherebytaketheidentitiesprovenbythe“mechanicalmethod”ashavingmetalinguis-ticcontent.Butwithinthecontemplatedtheoryitself,wesimplyhavemathematicalequivalences—indeed,thisisassumedbythe“mechanicalmethod”,whichwouldotherwiseinvolveobviouslyinvalidsubsti-tutionsofexpressionswithinquotationalcontexts.Itisthisdoubleinterpretationofthesetabstractionoperatorthatunderliestherathersurprisinglackoffastidiousnessaboutissuesofobjectlanguage,meta-languageandmetametalanguage,aswellasconcomitantissuesaboutuseandmention,in§2ofODS.Inparticular,thereaderwillnotethatwhileCTFLcarefullydistinguishes,withinthemetalanguage,betweentrans-lationsofobject-languageexpressionsandstructuraldescriptivetermsthatrefertothem,thisapparatusisentirelymissinginODS,sinceitsjobhasbeentakenupbyuse–mentionambiguityintheinterpretationofthecontextintroducedbytheset-abstractionoperator.Nevertheless,anearly,inchoategrasponConventionTseemstobeatworkinTarski’s“completelymechanicalmethod”,sincetheideaisthatoncewetakeaone-placefirst-orderarithmeticopensentenceinnormalform,itsspecificallyarithmeticvocabularycanbesystematicallyreplacedbyrelatedset-theoreticvocabulary:theprimitivesetsfortheprimitivesententialfunctionsandtherelevantset-theoreticoperationsforthelog-icaloperationsonsententialfunctions.ThethoughthereisclearlythecorrectonethatwilllaterinformConventionT:“{1}”translates“υ(x(k))”,“+˚”occurringbetweentwoexpressionsthatdenotesetstranslates“or”asitoccursbetweentwoone-placeopensentences,andsoon.However,Tarskihasn’tyetfullygraspedthelaterideathattheessenceofcaptur-ingasemanticconceptisacertaininterplaybetweenuseandmention.Instead,whatwegethereisaninvitationtothinkaboutadeductivethe-oryinwhichcertainidentitiescanbeproved,andtoviewtherelevantidentitiesashavingmetalinguisticcontent.TheadvanceofCTFLwillbetobringthisinterplayintothetheoryitself,justasTarski’streatmentofPadoa’smethodbringsthe“variationinmeaning”itinvolvesoutoftheCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01mindandintotheformalizedaspectsofthedeductivetheoryitself.Indeed,sincegenuinemetamathematicalcontentisreallymissingfromthesection,whatwegetisnotadefinitionofatermthatexpressesarelationofdefinabilitybetweenasetofrealsandanarithmeticthe-ory,butratherasetofrealsthathappenstobethesetofrealsdefinablewithinaparticulararithmetictheory.18ODSthereforesimplydoesnot10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE105—#22\n106AlfredTarski:PhilosophyofLanguageandLogicsucceedincapturingtheintuitivenotionofdefinabilityadumbratedin§1withinthemathematicaltheoryof§2.Therelevanttopic,captur-ingconceptsofword–worldrelations,simplyhasn’tcomeclearlyintoviewyet.Tarskiisgropingtowardtheideathattheoremswiththenow-familiarinterplayofuseandmention(oruse,translationandmention)arerequired,buttheroleoftheappealtotranslationandthecrucialplaceoffastidiousnessaboutuseandmention,andhenceobject-,meta-andmetametalanguage,isn’tyetinproperlyinfocus.Thisisn’ttosaythatTarskiwasn’telsewherecarefulaboutobject-andmetalanguageissues,oruseandmentionbefore1929.Whathedidn’tclearlyconceiveatthetimeofcomposingODSwasthatproperlycapturingasemanticcon-ceptappliedtosomeobjectlanguagewithinadeductivemetatheorywasgoingtorequirehimtoincorporatetheapparatusneededforsuchclarityintothemetatheoryitself,andthusmoreclearlytodistinguishthemetametalanguagefromthemetalanguage.AfewyearslaterTarskiputsthethoughtassomethingquiteclearandobvious:Foreachofthese[semantic]conceptsweformulateasystemofstate-ments,whichareexpressedintheformofequivalencesandhavethecharacterofpartialdefinitions;asregardstheircontents,thesestate-mentsdeterminethesenseoftheconceptconcernedwithrespecttoallconcrete,structurallydescribedexpressionsofthelanguagebeinginvestigated.Wethenagreetoregardawayofusing(oradefinitionof)thesemanticalconceptinquestionasmateriallyadequateifitenablesustoproveinthemetalanguageallthepartialdefinitionsjustmen-tioned.Bywayofillustrationwegiveheresuchapartialdefinitionoftheconceptofsatisfaction:JohnandPetersatisfythesententialfunction‘XandYarebrothers’ifandonlyifJohnandPeterarebrothersItshouldalsobenotedthat,strictlyspeaking,thedescribedcon-ventions(regardingthematerialadequacyoftheusageofsemanticalconcepts)areformulatedinthemetametalanguageandnotinthemetalanguageitself[Tarski,1983b,404–5].Thesethoughtsnotyetinview,ODSbreaksdownatjustthepointwhereCTFLsucceeds.Tarskihasn’tyetdoneforsemanticswhathedidforCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Padoa’smethodby1926:bringtherelevantconceptsintothededuc-tivetheoryintheformofaconditiononwhatmustbederivableforitsexpressionstoexpressthetargetmetamathematicalconcepts.TheworkinODSdatestosometimein1929[Hodges,2008,124].TheFrenchversionofthearticleappearedin1931,butKuratowskireportedonitscontentsinmid-1930.Sowhatwecanconcludeisthis.Bythe10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE106—#23\nSemantics107springof1930,Tarskiunderstoodhistaskasbeingtoconstructdefi-nitionsthatintroducedtermsthatexpressedsemanticconceptsintoamathematicaltheoryforafixedobjectlanguageandhehadsomeofthetoolsrequiredforthisinhand.Lessclearbeforemid-1930wasaspeci-ficationoftheconditionsunderwhichthishadbeenaccomplished.ItisclearfromtheopeningpagesofODSthathehadsomeideathathewantedtocapturethe“intuitivemeaning”ofsemanticnotionsinmath-ematicaltheories,butitisalsoclearfromthetreatmentofthismatterof“capitalimportance”inthearticlethatatthetimeofwritinghehadn’tfullyspecifiedeitherwhatthistaskamountedto,orhowhewastoshowthatithadbeenaccomplished.Sincethedatingplaces“OnDefinableSets”rightaroundtheperiodwhenKotarbi´nski’sElementyappeared,wecansaythatTarskihadanunclearconceptionofaphilosophicalprojectthatwaswaitingtobedeveloped.WiththetreatmentoftruthintheElementyTarskiwasabletomoveforward[Hodges,2008,129].Nevertheless,theprojectembarkeduponwithODSisclear:craftadeductivetheorywithastructuresuchthatcertainprimitiveordefinedexpressionsexpresssemanticconceptsinIntuitionisticFormalistfash-ion.Theissueof“capitalimportance”isjustthis,andTarskitrieshisbesttoshowthathisdefinitionof“definableset”isinthatsense“intu-itivelyadequate”.Inlate1929,whenTarskiwasbeginningtodeveloptheformaltechniquesforwhichheisnowfamous,theirpurposewastocontributetotheexpressionofsemanticconceptsbytheexpressionsofadeductivetheory.SemanticexpressionsinODSarepreciselynotpartoftheimplicit“theoryofmeaning”ofthearticle;thattheoryofmeaningisthesameoneTarskiadheredtothroughoutthe1920sandupuntil1934,theIntuitionisticFormalistonethatstressedtheexpressionofintuitivelyvalidthoughtsandthecontentsofconceptsindeductivetheories.WecanalsonoteforlaterusethatinODSthenotionofconsequenceremainstheproof-theoreticonespecifiedbyintuitivelyvalidinferences[Tarski,1983f,116].Semanticsin1930forTarskihadnothingtodowiththeanalysisofmeaningortherelationoflogicalconsequence.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-0110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP03”—2011/11/29—14:25—PAGE107—#24\n4TruthTarski’sinterestintheearly1930sisintheexpressionofmetamathemat-icalconceptswithinmathematicaldeductivetheories.Havingexpressedtheconceptoflogicalconsequenceasheunderstooditatthetime,Tarskimovedontosemanticconcepts,motivatedbytheclashbetweenphilo-sophicalscruplesandmathematicalpracticewithwhichweopenedthepreviouschapter.SinceIntuitionisticFormalismrequiresthatanexpres-sionthatexpressesasemanticconceptbeboundupintheoremsthatconstrainittoexpressthatconcept,theexerciserequiresaconceptionofwhatthesetheoremsare.Thisrequirementappearsinaninchoatewayinearly1930astherequirementthattheseeliminativedefinitionsbe“intuitively”or“materially”adequate,butinlate1929orearly1930Tarskilacksanyparticularlyclearideaofwhatthisamountstointhecaseofsemanticconcepts,thoughheshowssomesignsofgropingtowardtherightideainhistreatmentofthe“completelymechanicalmethod”oftranslatingarithmeticformulaeintoset-theoreticones—aclearprecursorofCTFL’sappealtotranslation.Wethushavetwoprimaryexpectationsfor“TheConceptofTruthinFormalizedLanguages”.First,Tarski,asisshownbythecontinuationofthisinterestaslateas1934’s“DefinabilityofConcepts”,willinthearticlebeinterestedinthefirstinstanceintheexpressionofintuitiveseman-ticconceptsinadeductivetheory.Second,themagnumopusshouldextendandcorrecttheworkof“OnDefinableSets”bydevelopingamoreCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01articulateconceptionofwhatittakestoexpressintuitivesemanticcon-ceptswithinadeductivetheory,whethertheexpressionsinquestionaredefinedornot.Thetwomostcelebratedaspectsofthearticlefulfillthesetwinexpectations:ConventionTistheclearconceptionthatreplacesthefumblingtreatmentofintuitiveadequacyin“OnDefinableSets”,whilethemethodofprovidingadefinitionthatisadequateaccordingto10810.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE108—#1\nTruth109ConventionTprovidesdefinedtermsthathavethesought-afterfeatures.Extensionalsemantics,ontheotherhand—strangelyenough,giventhereceptionofTarski’sworkbyposterity—willremaininthebackseatthroughoutthearticle,itsonlyinterestbeingitsroleintheexplicitdefinitionofsemantictermsinanintuitivelyadequateway.Thefirstthreeparagraphsofthework,puzzlingwithoutanunder-standingofTarski’sIntuitionisticFormalism,thusreadexactlyasweshouldnowexpect:thegoalistoshowhowtocraftadeductivethe-orycontainingacertaintermsothatthattermisconstrainedtoexpressafamiliarconcept.OntheusualreadingofTarski,onwhichadefini-tionissupposeditselftoexpressa“thoroughanalysis”ofaconcept,itisamysterywhyTarskiwouldbothsetouttodefinetruthanddenythatheintendsathoroughanalysisthereof.Thepassage,however,insti-tutesjustthestructurewehaveseenintheearlypagesof“OnDefinableSets”.Aformaltheory,understandableinitsownterms,istobegivenacertainstructure.Metatheoretically,thisstructurecanbeseentosuitoneofitssignsfortheexpressionofanintuitiveconcept.Someremarkswillbemadeonthisintuitiveconceptinorderto“remindthereader”ofitsbasicfeatures.However,thattheformaltheoryhastherequisitestructureisamatterofitshavingcertainsentencesastheorems;ifthetheoryachievesthisdefinitionally,thisdoesn’tmeanthatthedefinitioninquestionitselfstatesananalysisofthetargetconcept.4.1ConventionT4.1.1TerminologicalnotesTheannouncedaimof“TheConceptofTruthinFormalizedLan-guages”is,inWoodger’stranslation,thedevelopmentofa“materiallyadequateandformallycorrectdefinitionoftheterm‘truesentence’”[Tarski,1983a,152].“Materiallyadequate”hereisWoodger’srenditionofBlaustein’s“sachlichzutreffende”,whichinturnrendersthePolish“mery-torycznietrafnka˛”.Thisintroducesanimportantissueinthetranslations.“Merytoryczny”iscognatewith“merit”.RudolfHallerrendersTarski’s“merytoryczny”inGermanas“meritorisch”[Tarski,1992,3],makingcleartheetymologicalconnectionwith“meritorious”,butBlausteinrendersCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01“merytoryczny”as“sachlich”.ContrasthereTarski’sownFrenchof“OnDefinableSets”,inwhichheisperfectlyhappytouse“intuitif”(e.g.[Tarski,1986g,521,538])repeatedlytoexpressthesameidea:capturingtheordinarymeaningofaterm.FollowingTarski’sFrench,theGerman“intuitiv”andthencetheEnglish“intuitive”wouldhavebeenperfectlyacceptable.110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE109—#2\n110AlfredTarski:PhilosophyofLanguageandLogicWoodgerthenbringsBlaustein’s“sachlich”intoEnglishas“materi-ally”.ThistransformationthenlaysthefoundationforalongstandingconfusionaboutTarski’saims,andinparticularaboutthepurposeofConventionT:theconfusionaccordingtowhichthepointofCon-ventionTistoensurethatthedefinedtruth-predicateisextensionallyadequate,since“material”suggests“materialconditional”whichinturnbringstomindconceptionsofextensionallogic.Oneofmyprimaryaimsherewillbetoarguethatthisisnotright.ConventionTcontributestoTarski’sIntuitionisticFormalistproject:itstatesaconditionunderwhichanintuitiveconcepthasbeenproperlyexpressedinadeduc-tivetheory.Thatsaid,Woodgerhashisreasons:[Tarski,1983f]treats“intuitif”and“matériel”assynonymousinseveralpassages,e.g.[Tarski,1986g,538]wherethequestionofwhetherDef.10possesses“justessematérielle”,answeredaffirmatively,givesrisetotheclaimthatthedefi-nitionhas“justesseintuitive”.Moreover,Woodger,workinginthe1950s,had[Tarski,1986e]athisdisposal,andTarski’sownEnglishuses“mate-rial”forthenotion,so“material”itwasforthe1933workaswell.Nevertheless,theimportantnotionisbeingintuitivelyadequate(recallherealsothatadditionalappealstointuitioninTarski’sPolisharehiddenbytheGermanandEnglishtranslations’substituionoflinguisticusageforintuition(§1.4.3)).Wewilldobesttokeepthisinmindbystickingto“intuitive”forthenotion.ThisbringsustoasecondissuewithBlaustein’stranslationandWoodger’stranslationofit[Hodges,2008].Adefinitionoftruthshouldbeintuitivelyadequateandformallycorrect.Therearethustwoevalua-tivenotionsatwork:beinglogicallyinorder(poprawny),andcapturingtheconcept(trafny).Unfortunately,Blausteinvariesthetranslationofthelatter,andWoodgerthengetsconfusedaboutwhichtwoofBlaustein’sthreetermsgotogether:Blausteintranslatespoprawnyaskorrekt,whilehetranslatestrafnysometimesaszutreffendandsometimesasrichtig.Woodgerrealizesthatthereareonlytwoconceptsinvolved,buthemistakenlythinksthatrichtigisavariantofkorrektandnotofzutreffend.Sohetranslatesbothrichtigandkorrektas‘correct’;inoneplacewhereBlausteinhasCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01korrekteundrichtigeDefinition,Woodgerjustwrites‘correctdefinition’[Hodges,2008,117].HodgesgivesafulllistoftheplaceswhereWoodger’stranslationneedstobecorrected;I’llnotetheseplacesbelowwheretheyarerelevanttoourconcerns.Theproblemisquiteserious:innearlyeveryoneofthecentraldiscussionsof§3and§4whereintuitiveadequacyisatissue,10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE110—#3\nTruth111Woodgerhas“correct”whereTarski’sPolishhas“trafny”(pages214and224areprimeexamples).ThisgreatlyobscurestheextenttowhichTarskirepeatedlyreturnstothethemeofcapturingtheintuitiveconceptoftruth,leavinghimlookingmuchmoreconcernedwithmattersthataresomehow“formal”thanheactuallyissincethereaderisprimedtohear“correct”as“formallycorrect”.Thatsaid,Blausteinseemstohaveanexcuse,havingsimplyadoptedTarski’sownindifferencebetween“richtig”and“zutreffend”asGermantranslationsfor“trafny”:InthePostscripttoCTFLTarskihas“sachlichzutreffend”at[Tarski,1986a,192]and“richtig”at[Tarski,1986a,191].2Likewise[Tarski,1986c,262,264]has“sachlichzutreffend”,butTarskiswitchesto“sachlichrichtig”at[Tarski,1986c,263].4.1.2TruthintheLvov–WarsawschoolInanencyclopedictreatmentatthispointIwoulddiscussanalysesoftruthbeforeTarski’stimeinordertosettheanalysisoftheconceptoftruththatTarskitakesasthetargetforexpressioninadeductivetheoryinfullhistoricalcontext.ThisisnotpossiblehereatacceptablelengthandsoIwillturndirectlytothetraditionthatleadstoConventionT.3TwoideaswerepresentinthePolishtraditioninWarsawduringTarski’stime:endorsementofcertainformulationstakentobeformulationsofthecorrespondencetheoryoftruth,andtheviewthattheconceptoftruthissomehowcapturedbytheT-sentences.Endorsementofthecorre-spondencetheory,itsassociationwith1011b1ofAristotle’sMetaphysics,andcriticismofconceptionsofcorrespondenceintermsoffactsandpropositions—familiarfeatures,all,ofTarski’sdiscussions([Tarski,1983a,155],[Tarski,1983b,404])—stemfromTwardwoski,quotedat[Wolenskietal.,2008,21–4323]:Anaffirmativejudgmentistrueifitsobjectexists,anegativejudg-ment,ifitsobjectdoesnotexist.Anaffirmativejudgmentisfalseifitsobjectdoesnotexist;anegativejudgment,ifitsobjectexists.Likewise,Ajdukiewiczremarkson:thepopulardefinitionofatruesentenceaccordingtowhich:asen-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01tenceistrueifthereissomethinginrealitycorrespondingtothatsentence,scil.that,whichisassertedbythesentence[Ajdukiewicz,1966,40].The“correspondencetheory”intheseformulations,4then,wasthebasicanalysisoftheconceptoftruthinWarsawwhileTarskiwasastudent.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE111—#4\n112AlfredTarski:PhilosophyofLanguageandLogicThesecondtraditioninvolvedtakingitthattheT-sentencesexpresstheconceptoftruth.WolenskiandMurawskiattibuteafocusontheT-sentencestoCze˙zowski5:Truthisapropertyofspecialsignificance.IfacertainsentenceAistrue,thesentenceAistrueisalsotrue,ifoneofthemisfalse,thensoisthesecond:thesentencesAandAistrueareequivalent[Cze˙zowski,1918,106](mytranslation).Strictlyspeaking,though,thispassageadvocatessomethingliketheredundancytheory,discussedandrejectedbyKotarbi´nski[Kotarbi´nski,1966,107ff]inhisdiscussionoftheinadequacyofthe“verbal”interpre-tationoftruth,andexplicitlyattributedbyWole´nskiandMurawskitoZygmuntZawirskionthesamepage,andnotexplicitlytheideathattheT-sentencesaretrue.6However,sincedistinguishingequivalencefromthetruthofabiconditionalrequiresconsiderationoflogicsforwhichthedeductiontheoremfails,Wole´nskiandMurawski’sattributionoftheviewtoCze˙zowskicanbeacceptedasfairenough.Tarskihimself,ofcourse,attributestheideathattheT-sentencesareoffundamentalimportanceinthetheoryoftruthtoconversa-tionswithLe´sniewskiandtonobodyelse[Tarski,1983a,154].Hisowndiscussionoftruthbeginswithacontrastbetweenthe“classi-cal”and“utilitarian”conceptionsthatstemsfromKotarbi´nski[Tarski,1983a,153].Kotarbi´nskifavorstheclassicalconception,hisobjectionstotheutilitarianconceptionbeingstandardobjectionstopragmatistaccountsoftruth[Kotarbi´nski,1966,106–7].7OftheclassicalconceptionKotarbinskiwrites:Intheclassicalinterpretation,trulymeansinaccordancewithreal-ity…Letus…askwhatisunderstoodby“accordancewithreality”.Thepointisnotthatatruethoughtshouldbeagoodcopyorsimileofthethingofwhichwearethinking,asapaintedcopyoraphotographis.Abriefreflectionsufficestorecognizethemetaphoricalnatureofsuchacomparison.Adifferentinterpretationof“accordancewithreality”isrequired.Weshallconfineourselvestothefollowing:“JohnthinksCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01trulyifandonlyifJohnthinksthatthingsaresoandso,andthingsinfactaresoandso.”Ifanyonefindsthemeaningofthatformulainsufficientlyclear,lethimconsiderthefollowingexamplesofitsapplication:forinstance,thecentralideaoftheCopernicantheoryistrue;itconsistsintheassertionthattheEarthrevolvesaroundtheSun;nowCopernicus10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE112—#5\nTruth113thoughttruly,forhethoughtthattheEarthrevolvesaroundtheSun,andtheEarthdoesrevolvearoundtheSun[Kotarbi´nski,1966,106–7].Tarskiobviouslyhasthispassageinmindintheearlypagesof“TheConceptofTruth”,bothbecausehereferstoKotarbi´nskiandbecauseheexplicitlymakesKotarbi´nski’sdistinctionbetweenthe“classical”and“utilitarian”conceptionsoftruth.TarskihasshiftedfromKotarbi´nski’streatmentof“thinkingtruly”to“true”asappliedtosentences,butgiventheIntuitionisticFormalistbackgroundthisisbutasmallchange;cf.[Kotarbi´nski,1966,105]wheretheviewisthat“true”asappliedtosen-tencesderivesfrom“truly”asappliedtothinking.GivenTarski’sremarksat[Tarski,1983a,174]heprobablyagrees,butleavesthematterofftherecord.SinceTarskitakeshimselfsimplytobeshowinghowtointroduceanexpressionconstrainedbytheoremstoexpresstheconceptoftruthsoconstrued,heacceptsKotarbi´nski’sanalysisoftheconceptwholesale,andlikewisesimplytakesonboardtheidea,probablyfromLe´sniewski,thattheT-sentencessomehowexpresstheconceptoftruth.Thisissignif-icantbecausedoingboththingsatonceseemstoinvolvetheconflationoftwoideasthataretodaythoughttobequiteopposed:theideathattruthissomekindofcorrespondenceandthebasicdeflationistideathattruthissomehowanalyzedbytheT-sentences[David,1994][Patterson,2003].Thisofcoursesignificantlycomplicatesthequestion,towhichwewillreturn,ofwhetherTarskiwasadeflationist.Ofmoreproximateimportancehere,though,isthesimplefactthattheT-sentencesdon’tinanyobviouswayexpressthe“classical”conceptionoftruthcommontopredecessorslikeTwardowskiandKotarbi´nski,yetTarskitreatsthetwoideasasobviouslybearingsomecloserelationship:Amongstthemanifoldeffortswhichtheconstructionofacorrect(korrekt,poprawnej)definitionoftruthforthesentencesofcolloquiallanguagehascalledforth,perhapsthemostnaturalisthesearchforasemanticaldefinition.BythisImeanadefinitionwhichwecanexpressinthefollowingwords:(1)atruesentenceisonewhichsaysthatthestateofaffairsissoandso,andthestateofaffairsissoandsoCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Fromthepointofviewofformalcorrectness,clarity,andfreedomfromambiguityoftheexpressionsoccurringinit,theaboveformu-lationobviouslyleavesmuchtobedesired.Neverthelessitsintuitivemeaningandgeneralintentionseemtobequiteclearandintelligible.Tomakethisintentionmoredefinite,andtogiveitacorrectform,ispreciselythetaskofasemanticaldefinition.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE113—#6\n114AlfredTarski:PhilosophyofLanguageandLogicAsastarting-pointcertainsentencesofaspecialkindpresentthem-selveswhichcouldserveaspartialdefinitionsofthetruthofasentenceormorecorrectlyasexplanationsofvariousconcreteturnsofspeechofthetype‘xisatruesentence’.Thegeneralschemeofthiskindofsentencecanbedepictedinthefollowingway:(2)xisatruesentenceifandonlyifp.Inordertoobtainconcretedefinitionswesubstituteintheplaceofthesymbol‘p’inthisschemeanysentence,andintheplaceof‘x’anyindividualnameofthissentence[Tarski,1983a,155–6].NoteherethatforonceWoodger’s“correct”isactuallycorrect:theGer-manandPolishare“korrekt”and“poprawny”.Thisisveryimportant:inthetransitionfrom(1)to(2)thesemanticaldefinitionisnotrejectedforbeingintuitivelyinadequate,butforbeingformallyincorrect.Thesemanticaldefinitioncannot,onitsown,serveasadefinitionoftruthwithoutbringingparadoxinitswakeand,moreimportantlyforourpurposesnow,itis,thoughadefinition,onethatdoesn’teliminatemeaning-theoreticvocabulary,sinceitexplicitlyappealstothenotionofsaying.Sodespiteitsnaturalnessasadefinitionoftruth—despite,thatis,thefactthatitdoesaccuratelystatethecontentoftheconceptoftruth—itcannotbeincorporateddirectlyintoaseriousdeductivetheoryinmetamathematics.Tarskithinksthatthesemanticaldefinitionactu-allydoesexpressthecontentoftheconceptoftruth;hejustdoesn’tthinkthatitcanbedirectlyincorporatedintoapropermetamathematicalthe-ory.HisattentionthusturnstothesecondaspectofPolishthoughtabouttruth:theT-sentences.Theproblemwiththepassagefrom(1)to(2)isthat(2)doesn’tinanyobviouswaysimplysaywhat(1)says.Yeta“semanticaldefinition”issupposedtoworkouttheideaexpressedin(1),its“startingpoint”isthetruthoftheT-sentences,andConventionTissimplytheideathatanintuitivelyadequatedefinition—onethatexpressesthecontentoftheconceptoftruth,whichTarskiexplicitlyholdsisexpressed,thoughnotwithrigor,by(1)—sumsuptheT-sentences.Oneofthemainpuzzlesaninterpreterneedstosolvehereishowthe“logicalsum”oftheT-sentencescouldbesomehowcloselyrelatedtothe“semanticaldefinition”.WhenCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01weareabletodothiswewillunderstandhowTarskisawthetwostrandsofPolishthoughtabouttruthfittingtogether.4.1.3SemanticconceptsinamathematicaltheoryAfteracoupleofexamplesofT-sentencesformedbothquotation-allyandwithstructural–descriptivenamesofsentences,Tarskinotes10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE114—#7\nTruth115thatparadoxicalsentencescanbeformedincolloquiallanguage.Hewrites:Acertainreservationisnonethelessnecessaryhere.Situationsareknowninwhichassertionsofjustthistype,incombinationwithcertainothernotlessintuitivelyclearpremises,leadtoobviouscontradictions,forexampletheantinomyoftheliar[Tarski,1983a,157].Hethenfollowswithasimplecaseof“contingent”paradox,usingasentencethatsaysthatthesentenceprintedinacertainlocationisnottrue,wherethesentenceinquestionisthissentenceitself,andcontinues:Thesourceofthiscontradictioniseasilyrevealed:inordertoconstructassertion(β)wehavesubstitutedforthesymbol‘p’inthescheme(2)anexpressionwhichitselfcontainstheterm‘truesentence’(whichtheassertionsoobtained…cannolongerserveasapartialdefinitionoftruth).Neverthelessnorationalgroundcanbegivenwhysuchsubstitutionsshouldbeforbiddeninprinciple[Tarski,1983a,158].NoticethatatthispointTarskihasnotsuggestedsomehowquantifyingoverthevariablesin(2)toproduceadefinition,sotheantimonyoftheliardoesn’traiseaproblemjustforanattemptedsemanticaldefinitionoftruthforcolloquiallanguage.TheparadoxasformulatedcallsintoquestionnotmerelywhethertruthcanbedefinedbytheT-sentences,butwhethertheT-sentencesthemselvescanevenbejointlytrue.Thisisimportantforourunderstandingofhisgeneralremarks,asishiscom-mentthat“norationalground”canbegivenforforbiddingsubstitutionsintotheT-schemaofsentencesthemselvesthatcontaintheword“true”.Onemightthinkthatavoidanceoffalsehoodandparadoxwouldberationalgroundsforsuchathing,sothecommentmustpertaintothe“spirit”ofeverydaylanguage,ashecallsitinhisgeneralremarks:forbid-dingsentencesthathadastheirtruthconditionsthatcertainsentencesbetruewouldnotbeinthespiritofeverydaylanguage,asitwouldbedirectlycontrarytoitspresumeduniversality[Tarski,1983a,164].Settingasidetheworryaboutsemanticparadoxforthenonce,TarskinextconsidershowonemightgeneralizetheT-sentencesintoadef-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01inition.Heconsiderssimplybindingthesententialvariableinthedisquotationschemawithasententialquantifier,producing:(5)forallp,‘p’isatruesentenceifandonlyifp[Tarski,1983a,157]butnotesthatsuchadefinitionwouldn’teliminate“istrue”inallcon-texts.Inordertoovercomethis,thesuggestedsemanticaldefinitionis10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE115—#8\n116AlfredTarski:PhilosophyofLanguageandLogic(6)forallx,xisatruesentenceifandonlyif,foracertainp,xisidenticalwith‘p’andp[Tarski,1983a,159].Thisisthenfollowedbyhisratherinfamousremarksonquotation.Tarskiconsiderstworeadingsofquotationmarks.Onthefirst,“quotation-marknamesmaybetreatedlikesinglewordsofalanguage,andthuslikesyntacticallysimpleexpressions”[Tarski,1983a,157].Assuch,hemaintains,theoccurrenceof“p”insideofquotationmarksisn’tboundbythequantifierin(5)and(6),withtheresultsthatitisaconsequenceof(6)thattheletter“p”istheonlytruesentenceandthat(5)isinconsistent,sincetheletter“p”willbeassignedeverytruthcondition.Theremarkshereareobjectionableonanumberoffronts,buttosavespaceIwillnotcommentonthematter;seeforinstance[Soames,1999,86–90].Theotherproposedreadingofquotationmarksisintermsof“quota-tionfunctions”Thequotationmarksthenbecomeindependentwordsbelongingtothedomainofsemantics,approximatingintheirmeaningtotheword‘name’,andfromthesyntacticalpointofviewtheyplaythepartoffunctors.Butthennewcomplicationsarise.Thesenseofthequotation-functionandofthequotationmarksthemselvesisnotsuf-ficientlyclear.Inanycasesuchfunctorsarenotextensional;thereisnodoubtthatthesentence“forallpandq,incase(pifandonlyifq),then‘p’isidenticalwith‘q’”isinpalpablecontradictiontothecustom-arywayofusingquotationmarks.Forthisreasonalonedefinition(6)wouldbeunacceptabletoanyonewhowishesconsistentlytoavoidintensionalfunctorsandisevenoftheopinionthatadeeperanalysisshowsittobeimpossibletogiveanyprecisemeaningtosuchfunctors[Tarski,1983a,161].Furthermore,asTarskinotes,promiscuoususeofquotationmarkssoconstruedwillallowustoreconstructtheliarparadox.Tarski’sworryaboutintensionalityiscertainlyapposite:thereferenceofaquotationisnofunctionofanysemanticpropertyofwhatitnames.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Havingconsideredthesetwointerpretationsofquotationmarksthathefindsunsatisfactory,Tarskigivesupandmovesontoaperfunctorydiscussionoftheobviouslydoomedstrategyofprovidingastructuraldefinitionoftruthofthesortwehaveencounteredbeforeintermsoftheclosureofsomesetofsentencesundercertaintransformations;outsideofanaxiomaticcontexttheprocedureissenseless.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE116—#9\nTruth117§1isadvertisedasanargumentthattheverypossibilityofaconsistentuseoftheexpression‘truesentence’whichisinharmonywiththelawsoflogicandthespiritofeverydaylanguageseemstobeveryquestionable,andconsequentlythesamedoubtattachestothepossibilityofconstructingacorrectdefinitionofthisexpression[Tarski,1983a,165].Thisitis,andwewillreturntotheissueinthenextchapter.SinceonTarski’sviewtheT-sentencesexpresstheconceptoftruthasgraspedinthesemanticaldefinition,buttheT-sentencesincolloquiallanguagecan-notevenbejointlytrue,Tarskicannotfollowthestrategyofexpressingthecontentexpressedbythesemanticaldefinitionbytakingthetheinstancesof(2),theT-sentences,tobeoneandallassertiblesentencesofcolloquiallanguage.Theideaisthenthis,thatifwedowanttoexpresstheconceptoftruthasgraspedinthesemanticaldefinition,andifwearecommittedtotheviewthatwecandosobymakingtheT-sentencesforthesentencesofalanguageoneandallassertibleinourtheory,thistheorycannotitselfbestatedincolloquiallanguage;norcanittakecolloquiallanguageasitsobjectlanguageeither,sincetheuniversalityofcolloquiallanguagemeansthatthelanguageofthetheorycouldbetranslatedintotheobjectlanguage,creatingtheconditionsforthepara-dox.Wethusturntoexpressingtheconceptoftruthonlyin(andfor)non-universalformalizedlanguages[Tarski,1983a,167],andtheworkofthepaperbeginsinearnestwith§2anditssyntactictreatmentofthelanguageofthecalculusofclasses.4.1.4T-sentencesAttheoutsetof§3,afternotingtheweaknessofadefinitionoftruthintermsofprovability,Tarskisuggests:returningtotheideaofasemanticaldefinitionasin§1.Asweknowfrom§2,toeverysentenceofthelanguageofthecalculusofclassestherecorrespondsinthemetalanguagenotonlyanameofthissentenceofthestructural–descriptivekind,butalsoasentencehav-ingthesamemeaning.Forexample,correspondingtothesentence

Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01‘x
x

AIx
x

Ix

x
’isthename‘12(ι1,2+ι2,1)’andthesentence‘foranyclassesaandbwehavea⊆borb⊆a’.Inordertomakeclearthecontentoftheconceptoftruthinconnexionwithsomeoneconcretesentenceofthelanguagewithwhichwearedealingwecanapplythesamemethodaswasusedin§1informulatingthesen-tences(3)and(4)(cf.p.156).Wetakethescheme(2)andreplacethe10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE117—#10\n118AlfredTarski:PhilosophyofLanguageandLogicsymbol‘x’initbythenameofthegivensentence,and‘p’byitstrans-lationintothemetalanguage.Allsentencesobtainedinthisway,e.g.‘12(ι1,2+ι2,1)isatruesentenceifandonlyifforanyclassesaandbwehavea⊆borb⊆a’,naturallybelongtothemetalanguageandexplaininapreciseway,inaccordancewithlinguisticusage,themeaningofphrasesoftheform‘xisatruesentence’whichoccurinthem.Notmuchmoreistobedemandedofageneraldefinitionoftruesentencethanthatitshouldsatisfytheusualconditionsofmethodologicalcor-rectnessandincludeallpartialdefinitionsofthistypeasspecialcases;thatitshouldbe,sotospeak,theirlogicalproduct[Tarski,1983a,187].Thissetofideasprovidesfortruththeclearconceptionofwhatistobeaccomplishedthatwaslackingfordefinabilityin“OnDefinableSets”,onepreconditionofwhichwassufficientresourcesinthemetalanguageforapropertreatmentofuseandmentionofsentencesintheobjectlanguageandtheirtranslationsintothemetalanguage.Thecontentoftheconceptoftruthis,accordingtothe“semanticaldefinition”,thepropertyofsayingthatso-and-sowhenthingsareso-and-so.Thetaskistocraftadeductivetheorythatforcesanexpression“∈Tr”tohavethiscontentasitsmeaning,andthetaskwillbetakentobeaccomplishedwhenthetheoryismadetoimplytheT-sentencesfor“∈Tr”.“∈Tr”willexpresstheconceptoftruthjustincaseitisassociatedbytheconventionsofthemetalanguagewiththecontentexpressedbythesemanticaldefinition.However,unlikethecaseofconsequenceasconstruedaround1930,whereTarskiwasablesimplytorequirethattheoremsthatstatetheconceptionassociatedwiththeconceptofconse-quencebeprovableinthemetatheory,nosimilarrequirementwilldointhecaseofthesemanticconceptoftruth,sincethesemanticaldefinitioncannotbepartofaformallycorrectmetamathematicaltheory.Thusmeansneedtobefoundtoconstraintheintuitiveinterpretationof“∈Tr”downtothepointwhereitexpressestruthanalyzedaccordingtothesemanticdefinitionwhileguardingagainsttheproblemsdirectincorporationofthesemanticaldefinitionintoatheorywouldbringinitswake.Sincetheswitchtonon-universalformalizedlanguagesallowstheT-sentencesfortheobjectlanguagetobeconsistent,wecanmakeCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01useofthem.ButthequestionremainsofhowhavingtheT-sentencesastheoremsmakes“∈Tr”expresstheconceptoftruthasgraspedinthesemanticaldefinition.ThiswastheinterpretivepuzzleaboutthesemanticaldefinitionandtheT-sentencesnotedabove.ThegeniusofConventionTisthatincorporatingtheT-sentencesintothemetatheoryforcestheinterpretationof“∈Tr”toaccordwiththe10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE118—#11\nTruth119semanticaldefinition,whileleavingthedefinitionitselfunstated.This,Ipropose,istheconnectionbetweenthesemanticaldefinitionandtheT-sentences,thetwoelementsofPolishthoughtabouttruththatTarskibringstogether.Toputitanotherway,iftheT-sentencesarethe-orems,“∈Tr”expressestheirreduciblysemanticcontentoftheconceptoftruth,thecontentexpressedinthe“semanticaldefinition”,andiftheT-sentencesaretheoremsbecausethemetatheoryhasbeenextendedbyacertaindefinition,wetherebyforceatheorywithonlymathematicalprimitivestoexpressasemanticconcept,therebymeetingtheelimina-tivegoalstatedat[Tarski,1983a,154]thatundefinedsemantictermswillnotappearwithinthemetatheory.Hereishowthetrickworks.Whenweputtheelementstogether,whatwefindisthatanexpressionforwhichtheT-sentencesaretheoremsis,giventheconventionsgoverningtheobjectlanguageandthemetalan-guage,forcedtoexpressthecontentoftheconceptoftruthasconstruedinthesemanticaldefinition.Sincethelanguageofthemetatheoryhasintuitivemeaningjustasmuchasdoestheobjectlanguage(seethedis-cussionoftranslationbelowin§4.3.1),eachT-sentenceismeaningfulaccording,asKotarbi´nskiwouldsay,totheadoptedconventionsor,asTarskihasitinhischangetotheGerman(§1.4.3)inaccordancewithlinguisticusage.Inparticular,aT-sentencelike:12(ι1,2+ι2,1)istrueifandonlyifforanyclassesaandbwehavea⊆borb⊆ameansthat12(ι1,2+ι2,1)istrueifandonlyifforanyclassesaandbwehavea⊆borb⊆a.Now,thoughtheT-sentenceitselfsaysnothingofitsownsayingthatsomethingisthecase,apartytotheconventionsgoverningthemetalanguagewillrecognizethatinfacttheT-sentencedoessay(inKotarbi´nski’sterms,stateindirectly)exactlythat.Likewise,apartytotheconventionsgoverningtheobjectlanguagewillrecognizethat12(ι1,2+ι2,1)saysthatforanyclassesaandbwehavea⊆borb⊆a.Hence,apartytobothsetsofconventionswillrecog-nizethatwhattheT-sentencesaysisthat12(ι1,2+ι2,1)istrueifandonlyifso-and-so,whereso-and-soispreciselywhat12(ι1,2+ι2,1)says.Theconcept“istrue”istherebyconstrainedbythetheoremhoodCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01oftheT-sentencetoapplyto12(ι1,2+ι2,1)ifandonlyifwhat12(ι1,2+ι2,1)saysisthecaseisinfactthecase,andsomeonewhoacceptstheconventionsgoverningbothobjectandmetalanguagewilltherebyacceptthis.Asaresult,sincetheT-sentenceisatheorem,“∈Tr”asappliedto12(ι1,2+ι2,1)hastoexpressthecontentoftheconceptoftruth.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE119—#12\n120AlfredTarski:PhilosophyofLanguageandLogicGettingthehen-scratchesoutofthepicturemayhelpthereader.ConsidertheFrenchsentenceLaneigeestblanche.anditsEnglishT-sentence,withadummylettersubstitutedfor“true”:“Laneigeestblanche”isFifandonlyifsnowiswhite.ConsidernowapartytotheconventionsofFrenchwhoisalsoapartytotheconventionsofEnglishwhobeginsbyassociatingnointuitivecontentwithF.Suchapersonwillacceptthat“Laneigeestblanche”isassociatedbytheconventionsofFrenchwiththethoughtthatsnowiswhite.sinceacceptingthisispartofherbeingpartytotheconventionsofFrench.Equivalently,shewillacceptthatBytheconventionsofFrench“Laneigeestblanche”saysthatsnowiswhite.NowsupposethispersonconsidersthemodifiedEnglishT-sentence,wonderingwhatshewouldbecommittedtobyit.Whatshedoesknowisthatwereshetoacceptit,shewouldbeacceptingthat“Laneigeestblanche”fallswithintheextensionofFifandonlyifsnowiswhite.Shealsoknowsthat“Laneigeestblanche”saysthatsnowiswhite.Sosheknowsthat“Laneigeestblanche”saysthatthestateofaffairsisthatsnowiswhite,andsheknowsthattheT-sentencesaysthat“Laneigeestblanche”isFifandonlyifthestateofaffairsisthatsnowiswhite.Sosheknowsthat,whenitcomesto“Laneigeestblanche”,thereisastateofaffairsthatso-and-sosuchthat“Laneigeestblanche”saysthatthestateofaffairsisso-and-soand(bythemodifiedT-sentence)itisFifandonlyifthestateofaffairsisso-and-so.Hence,ifshedoespossesstheconceptoftruthasanalyzedbythesemanticaldefinition,shewillrecognizethat“Laneigeestblanche”fallswithintheextensionofFjustincaseithasthepropertythatisthecontentoftheconceptoftruth,namely,sayingthatthestateofaffairsisso-and-sowhenthestateofaffairsisso-and-so.ShemaythereforerightlyconcludeCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01thatatleastwhenrestrictedto“Laneigeestblanche”,asentencefallswithintheextensionofFjustincaseitfallsundertheconceptoftruth.SimilarthoughtaboutT-sentencesfortherestofthelanguagewillleadtotheconclusionthatFappliestoasentencejustincaseitsaysthatthestateofaffairsisso-and-so,whenso-and-so.Fistherebyconstrainedtoexpresstheconceptoftruthasanalyzedbythesemanticaldefinition,yet10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE120—#13\nTruth121noneoftheproblematicaspectsofthesemanticaldefinitionhavebeenexplicitlyincorporatedintothemetatheoryforFrenchstatedinEnglish:thenotionsof“saying”,“stateofaffairs”andtheschematic“so-and-so”allremainwithinthemindofonewhocontemplatesthemetatheoryandtheobjectlanguagewhileacceptingtheirIntuitionisticFormalistmeaning-determiningconventions.Wethusacceptthatthesemanticaldefinitioniscorrectandforcetheconceptso-analyzedtobeexpressedby“∈Tr”withouttheformalincorrectnessthatwouldcomefromdirectinclusionofthesemanticaldefinitioninthemetatheory.ApointshouldbeenterednowontherespectinwhichTarskiaimsto“reduce”semanticconceptstootherconcepts[Tarski,1983a,153].Thereadingofthisthatcomesmosteasilytothemindofthecontem-poraryreaderisthatwithinTarski’stheories,semanticconceptswillbeexplicitlydefinedinotherterms,andthattherebyitwillsomehowbeshownthattheconceptoftruthcanbeanalyzedinnon-semanticterms.ThisisnotwhatTarskiisupto.Preciselybecausehehasnointentionofprovidingananalysisoftheconceptoftruthatall,Tarskiafortiorihasnointentionofgivingareductiveanalysisoftheconcept.TarskisimplyassumesKotarbi´nski’sanalysis,andKotarbi´nski’sanalysisdoesn’teliminate“says”.ForallTarskieverdoesorshows,thetargetconceptitselfisirreduciblysemantic;indeed,sincethe“semanticaldefinition”isthe“mostnatural”,itseemsthatTarskithoughtasmuch.Hisgoalisn’ttogetridofthis;hisgoal,rather,istoshowthataproperlystructuredmathematicaldeductivetheorycanconstrainoneofitstermstoexpressasemanticconcept.ItistothisendthatConventionTisdirected.WecanmakeageneralpointaboutIntuitionisticFormalistmetamath-ematicsasTarskidevelopsithere.ThereisnothingintrinsicallyreductiveaboutTarskianmetamathematics.Theaimisn’ttoshowthatsemantic(orothermetatheoretic)conceptscanbedefinedaway,thattheyaresafeformathematicsorphysics,oranythingelseofthesort.8Theaimistoexploretheextenttowhichthepropertiesofsentencesanddeductivetheoriesattributedinintuitivemetatheoreticthoughtcanbetreatedofintheoriesinwhichtheformsofexpressionsgoproxyfortheseproperties.ConventionTanddefinitionsthatareintuitivelyadequateaccordingtoitarethecrowningachievementofthisprogram:giventheconventionsCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01oftheobject-andmetalanguages,thetheoremhoodoftheT-sentencesgivesadeductivetheoryaformthatconstrainsoneofitsexpressionstoexpressatargetsemanticconcept.(Inthemetametalanguage,ofcouse,wedon’tspecifytherelevantformentirelysyntacticallyforitisherethatintuitivemeaningsandconventionsoflanguageareappealedtoinConventionT’snotionoftranslation.)10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE121—#14\n122AlfredTarski:PhilosophyofLanguageandLogicTarskistatesConventionTasfollows:ConventionT.Aformallycorrectdefinitionofthesymbol‘Tr’,formulatedinthemetalanguage,willbecalledanadequatedefinitionoftruthifithasthefollowingconsequences:(α)allsentenceswhichareobtainedfromtheexpression‘x∈Trifandonlyifp’bysubstitutingforthesymbol‘x’astructural–descriptivenameofanysentenceofthelanguageinquestionandforthesymbol‘p’theexpressionwhichformsthetranslationofthissentenceintothemetalanguage;(β)thesentence‘foranyx,ifx∈Trthenx∈S’(inotherwords,‘Tr⊆S’)[Tarski,1983a,187-8].ConventionTfixestheproblemwefoundinODS:itstates,inthemetametalanguage,acriterionofadequacyforadefinedexpressionofasyntacticdeductivetheoryforsomeobjectlanguagetoexpressthecon-ceptofbeingatruesentenceofthatlanguage.Therelativizationoftheconceptexpressedtotheobject-languageisleftimplicitinthesamewaysasitisleftimplicitinallofTarski’sotherdefinitionsoflinguisticnotions,e.g.hisdefinitionsofsentencehoodinCTFLorintheearlypapersontheconsequencerelation.Again,thisisnotananalysisoftheconceptoftruth,norisitacriterionofadequacyontheanalysisofthisconcept.ForTarskithephilosophers,inparticularKotarbi´nskiandAristotle,havetakencareofthat.Itisacriterionofadequacyonanexpression’sbeingconstrainedbythestructureofadeductivetheorytoexpresstheconceptoftruthsoanalyzed.4.2Tarski’sdefinitionsPeopleoftenspeakof“Tarski’sdefinitionoftruth”.Thisisamisnomer.Thedefinitionsetoutin§3isaformaldefinitionof“∈Tr”forthespecificlanguageofaspecificdeductivetheory,namelythefirst-ordertheoryofthesubsetrelationexpressedinPolishnotationwithcertainspecificaxioms.Theremainderof§4thenconcernshowthemethodsemployedinsettingoutthatdefinitioncanbemodifiedandextendedtoconstructsimilardefinitionsforotherdeductivetheoriesinotherformallanguages.NoneoftheproposeddefinitionsarethemselvesanalysesoftheconceptCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01oftruth;thisanalysis,borrowedfromKotarbi´nski,takesplaceinthemetametalanguageasthemotivationforConventionTjustdiscussed.4.2.1TruthforthelanguageofthecalculusofclassesWecanaffordtobebriefandinformalindiscussingTarski’sexam-pleitself.Thelaborof§2goesintodevelopmentofstringtheory,10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE122—#15\nTruth123formalsyntaxandproof-theoreticconsequence,theonlynotionofcon-sequenceatworkin1931[Tarski,1983a,166],forTarski’sfirst-ordertheoryofthesubsetrelation.Definition22definessatisfactionforthelanguageinthefamiliarway.Sincetheonlynon-logicalprimitiveis⊆,representedasan‘I’precedingtwovariables,anatomicopensen-tenceIxixjissatisfiedbyasequenceifftheithelementisasubsetofthejth;negation,disjunctionandthequantifiergetthenow-obvioustreatment.Itfollowsfromthisdefinitionthatsentenceswithnofreevariablesaresatisfiedeitherbyallsequencesorbynone.Definition23thensaysthatxisatruesentenceifandonlyifeverysequenceofclassessatisfiesit.9WhatclosesthegapbetweenmathematicsandsemanticsthatwasleftopeninODSiscareaboutuseandmentionfacilitatedbythestructural–descriptiveexpressionsfortheobjectlanguage,plustheenumerativedefinitionofsatisfactionforatomicpredicatesoftheobjectlanguage(ofwhichthereisonlyoneinthecaseathand).Belowwewillconsidertheextenttowhichthisreallyclosesthegapbetweensemanticsandmathematicsinawaythatwouldmeetvariousdemands.TarskithensetsouttoshowthatthedefinitionsatisfiesConventionT.HeprescindsfromdoingthisindetailonthegroundsthatdoingitwithrigorwouldrequiretheformalizationofthemetatheoryandaproofinthemetametatheorythatDef.23,againstthebackgroundtheoryandprecedingdefinitionsdoesinfactimplyalloftheT-sentencesforthelanguageofthecalculusofclasses[Tarski,1983a,195].RatherwearetoconvinceourselvesoftheintuitiveadequacyofDef.23giventhebackgroundtheoryandpriordefinitionsby“theempiricalmethod”.10SinceitisthecasethatthedefinitionplusthepriortheoryanddefinitionsimplytheT-sentences,wecanmoveon.Tarskialsopointsoutthat“somecharacteristicgeneraltheorems”canbederived,inparticularthatnosentenceanditsnegationaretrueandthateverysentenceoritsnegationistrue,thatconsequences(inthesenseof§2)oftruesentencesaretrue,andsoon[Tarski,1983a,197–8].Thesetheoremsarerequiredto“fixtheconvictionoftheintuitiveadequacy”(“sachlichenRichtigkeit”,“merytorycznejtrafnosci´”)ofDef.23becauseTarskiholdsthat“nodefinitionoftruesentencewhichisinCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01agreementwiththeordinaryuseoflanguageshouldhaveanyconse-quenceswhichcontradicttheprincipleoftheexcludedmiddle”[Tarski,1983a,186]—and,presumably,similarlyfortheotherpoints.ThisisunderemphasizedinTarski’sremarksontheideabehindasemanticaldefinitionin§1,buttheideahastobethatforanysentencethatsayssomething,whatitsaysiseitherthecaseornot.Ofcourse,thisultimately10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE123—#16\n124AlfredTarski:PhilosophyofLanguageandLogicassumesthelawoftheexcludedmiddleinthemetametalanguage,butthissortofconsiderationdoesn’tbotherTarski—afterall,heishappytoenshrineasTheorem5of§3theclaimthatthetheoremsoftheobjecttheoryaretrue,whichisjustaconsequenceofthefactthatthetheorystatedintheobject-languageisasub-theoryofthemathematicalthe-oryacceptedinthemetalanguage.Tarskiremarksonthissortofthing,withreferenceto[Ajdukiewicz,1966],inconnectionwithprovingtheconsistencyoftheobjecttheory,thatthisdoesn’tmaketheproofsofthetheoremscircular,thoughtheyfailto“addmuchtoourknowledge”[Tarski,1983a,237].UnfortunatelythisdiscussionmakesclearthatTarski’streatmentofintuitiveadequacyisinconsistent.ConventionTitselfstatesthatatheory’simplyingtheT-sentencesissufficientaswellasnecessaryforintuitiveadequacy.But[Tarski,1983a,186]statesthat“nodefinitionoftruesentencewhichisinagreementwiththeordinaryuseoflan-guageshouldhaveanyconsequenceswhichcontradicttheprincipleofexcludedmiddle”andTarskiholdsthatprovinggeneralizedexcludedmiddle(Theorem2)issomethingdone“inordertofixtheconvictionoftheintuitiveadequacy(“sachlichenRichtigkeit”,“merytorycznejtrafnosci´”)ofthedefinition”[Tarski,1983a,197](corrected).SoConventionTandotherpassagesmakeimplyingtheT-sentencessufficientforintuitiveade-quacy,while186and197seemtoimplythatitisn’t,sinceimplyingvariousgeneralizationssuchasnon-contradictionandexcludedmiddlearenecessary.Idon’tgettheimpressionthatTarskinoticedthisprob-lem,butsincebothConventionTandthelaterclaimthataxiomatictruththeoryisintuitivelyadequate,whenTarskiexplicitlynotesasaweaknessofitthatitdoesn’timplythedesiredgeneralizations,indicatethatimplyingtheT-sentencesistheheartofthematter.Moreover,theprovability,say,ofexcludedmiddleobviouslydependsonthelogicinthemetatheoryanddoesn’treallyhaveanythingtodowiththeconcep-tionoftruth.Sinceaterminologicaldistinctionbetween,say,“intuitiveadequacy”and“completeintuitiveadequacy”orsomethingofthesortwouldclearthingsupIwillnotpursuethematterfurther.4.2.2HigherorderandpolyadicityCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Withoneexampleworkedoutindetail,Tarskiturnsin§4toageneraldiscussionoftheapplicationofhismethodfordefiningtruth.Muchofthediscussionis,again,familiarenough:addmattersufficientforformalsyntaxplusatranslationoftheobjectlanguagetosomesufficientback-groundlogic,andthenproceedtothedefinitionofsatisfaction[Tarski,1983a,210–11].Whatrequiresfurtherdiscussioninthewayinwhich10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE124—#17\nTruth125satisfactionneedstobehandledforlanguagesofvaryingexpressivepower.Theissuesherearebothtechnicaland,forTarski,philosophical.Thetechnicalissuesmainlyconcernmathematicalbookkeeping:whenweallowforvariablesofdifferentordersorrelationsofvaryingadicityweneedneedtokeeptrackoftheminoursequences.Orderitselfishandledrathereasily;theintroductionofsecondordervariablessimplyrequires“two-rowedsequences”,onerowcomprisingobjectsoverwhichfirst-ordervariablesrangeandtheotherobjectsoverwhichsecond-ordervariablesrange[Tarski,1983a,237].Higherbutfiniteordergoesthroughwiththeadditionofmorerows.Variableadicitycausesmoretrouble.Ifitisn’tboundedabove,wedon’thaveafinitenumberofrowsinourmany-rowedsequencesanymore,sinceamanyrowedsequenceneedsafirstrowofobjects,asecondrowofsets,thatis,monadicrelations,athirdrowofdyadicrelations,afourthrowoftriadicrelations,andsoon[Tarski,1983a,232].Thisproblemcanbesolved,however,bytakingsatisfactiontobearelationbetweenopensentences,infinitesequencesofobjects,andinfinitesequencesofsetsoffinitesequencesofobjects[Tarski,1983a,234].InallthesecasesTarskialsonotesthatbyexploitingone-onecorre-lationsbetweenobjectsandcertainrelations(orthesetsofsequencestheysemanticallydefine)wecangetbyinthecasesdiscussedwithasingleinfinitesequenceofappropriatelyselectedobjects[Tarski,1983a,228,235].Thisiswhathecalls‘themethodofsemanticalunificationofthevariables’[Tarski,1983a,227].Asimpleexampleofthemethodisprovidedat[Tarski,1983a,228]:sinceeveryindividualxcanbeone-onecorrelatedwiththerelationthatholdsbetweenyandzifandonlyify=xandz=x,wecansimplytakefirst-ordervariablestorangeovertheserelations.Nothingmorethanalittlearithmeticisrequiredtokeeptherangesofthefirst-andsecond-ordervariablesdistinguishedwithinthesequences([Tarski,1983a,229])andwecaninthismannertakesat-isfactionagaintobearelationbetweenanopensentenceandasingleinfinitesequence.Tarskinotesonehitchthatwillmatterlater:theuni-fyingcategorycannotbeoflowerorderthanthehighest-ordervariableinthelanguage[Tarski,1983a,230].Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Thephilosophicalissuesareofmorerelevancetous.TheycenteronwhatTarskicalls“thetheoryofsemanticalcategories”,whichhecred-itstoHusserl,withreferencealsotoLe´sniewskiandAjdukiewicz(whoseversionwaslaterpublishedas[Ajdukiewicz,1935]),whichherelatestothetypetheoryofPrincipiaMathematicaandthesimplifiedtypethe-ory(STT)ofChwistekandCarnap’sAbriss.11Unfortunatelynowritten10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE125—#18\n126AlfredTarski:PhilosophyofLanguageandLogicpresentationofitbyLe´sniewskiexists,thoughitgetssomementionat[Le´sniewski,1992e,421,477–8].However,Ajdukiewiczpresentsaversionofitin[Ajdukiewicz,1978d]andtheideasmaybefamiliartothereaderfrom[Lewis,1970].Appliedtoalanguagetheviewassignsasemanticcategorytoeveryexpression,takingsometwocategories(sen-tenceandsomethingelse,usuallyname)asbasic,andexpressingothercategoriesasfunctionsfromn-tuplesofcategoriestosomecategory(e.g.aoneplacepredicateisanS/N,anadverbisan(S/N)/(S/N)andsoon).Tarskicommentsonitssignificance:Whilstthetheoryoftypeswasthoughtofchieflyasakindofprophy-lactictoguardthedeductivesciencesagainstpossibleantinomies,thetheoryofsemanticalcategoriespenetratessodeeplyintoourfun-damentalintuitions(fundamentalenIntuitionen,podstawoweintuicje)regardingthemeaningfulnessofexpressions,thatitisscarcelypos-sibletoimagineascientificlanguageinwhichthesentenceshaveaclearintuitivemeaning(inhaltlichenSinn,intuicyjnysens)butthestruc-tureofwhichcannotbebroughtintoharmonywiththeabovetheory[Tarski,1983a,215].(CompareLe´sniewskiontheconceptofasemanticalcategory:“Frankly,Iwouldstilltodayfeelobligedtoacceptthisconcepteveniftherewerenoantinomiesatall”[Le´sniewski,1992e,421].)Thecategoriesareequiv-alenceclassesofexpressionsintersubstitutablewithinsentences(openorclosed)preservingsentencehood.Thephilosophicalissuesdepend,forTarski,ontheviewwehavealreadymentionedthatintuitivemean-ingistiedtosemanticalcategory.Withrespecttoaformulationofsecond-orderquantificationovertwo-termedrelations,hewrites:Toobtainacorrectandadequate(korrekteundrichtige,poprownaitrafna)definitionofsatisfactioninconnexionwiththelanguageweareconsideringwemustfirstextendourknowledgeoftheconcept.Inthefirststageofoperatingwithitwespokeofthesatisfactionofasententialfunctionbyone,two,threeobjects,andsoon,accord-ingtothenumberoffreevariablesoccurringinthegivenfunctionCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01(cf.pp.189ff.).Fromthesemanticalstandpointtheconceptofsatis-factionhadthereastronglyambiguouscharacter;itincludedrelationsinwhichthenumberoftermswasdiverse…Strictlyspeakingweweredealingnotwithoneconcept,butwithaninfinitenumberofanal-ogousconcepts,belongingtodifferentsemanticalcategories.Ifwehadformalizedthemetalanguageitwouldhavebeennecessarytouse10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE126—#19\nTruth127infinitelymanydistincttermsinsteadoftheoneterm‘satisfies’.Thesemanticalambiguityoftheconceptincreasesstillmorewhenwepasstolanguagesofmorecomplicatedlogicalstructure…thesemanticalcategoryofarelationnotonlydependsonthenumberofdomains,i.e.thenumberoftermsstandingintherelationtooneanother,butalsoonthecategoriesofthesedomains…Tofunctionswhichbelongtotwodistincttypestwosemanticallydifferentcategoriesalwayscorrespond[Tarski,1983a,224–5,mycorrection].Onewaythepointemergeshereisthis:afullyformalizedtreatmentofalanguagethatconformstotherequirementsofthetheoryofsemanticalcategoriesrequiresametalanguagethatalsosoconforms,butthenthefactsofthecasewillmeanthatnoexpressioncanexpress“the”satisfac-tionrelationthatobtainsbetweenboth,say,first-orderone-placeopensentencesandsequencesofobjectsandsecond-ordersentencesopenonatwo-termedrelationalexpressionandsequencesoforderedpairs.Theconceptexpressedbyanoccurrenceof“satisfies”isdeterminedinpartbythenumberandorderoftheexpressionsrequiredtoformasen-tencewithit,sothatinourtwocaseswehavetheformssat(x,Y)andsat(x,y).Furthermore,forTarski,thetheoryofsemanticalcategoriesistoconstrain“independentmeaning”(selbständigeBedeutung,samodzielnegosensu),whichisalliedwithhisnotionsofconceptandintuitivemean-ing.Histreatmentofsemanticalcategoryinthetext,henotes,includesexpressionswithout“independent”meaning,suchas“itisnotthecasethat”,“Forallx”and“eitherxisasubsetofyor”[Tarski,1983a,217].Apropertreatmentwouldconstrainthetheorydowntocoverexpres-sionsonlywithindependentmeanings,whichmeansthatthetheoryofsemanticalcategories,properlyconceivedandfullydeveloped,isatheoryoftheconditionsunderwhichexpressionswithindependentmeaningcancomposetoformawholewithindependentmeaning.Theseremarksmakeclearthatthetheoryofsemanticalcategoriesisultimatelyconcernedwithourcapacitytoattachintuitivemeaningstoexpressions,inaccordwith“ourfundamentalintuitionsregardingthemeaningfulnessofexpressions”[Tarski,1983a,215].Butifthetheorythusconstrainstheseintuitivemeanings,wegettheconclusionthatCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01thereisnosingleconceptofsatisfaction.Wecanseeinthispassagethatevenin1931theconstraintsofSTTarebeginningtochafe.AsTarskinotesinseveralplacesinCTFLandelse-where(e.g.[Tarski,1983k,310n.1])manyofhisformulaeareactuallyschematicoverthesimpletype-theoretichierarchy.Whenheencoun-teredthisproblem,hehadnoqualmsaboutengagingin“systematically10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE127—#20\n128AlfredTarski:PhilosophyofLanguageandLogicambiguous”schematicgeneralizationovertypes,yetTarskiandhisread-erstreattheresultingformulationsasperfectlyintelligible.ItisinsuchmomentsthatwecanseetheseedsofTarski’sshifttoamoretolerant(inCarnap’ssense)attitudetowardlogicalsystems,therelaxationoftherestrictionsofSTTandthetheoryofsemanticalcategoriesinthe1935Postscript(§6.1)andhislatermovetofirst-orderlogicformanypurposes.Wewillreturntotheseissuesbelow.CTFLthusbearsaninterestingandasfarasIknowpreviouslyunnoticedresemblancetoWittgenstein’sTrac-tatus:thebodyoftheworkconsistssubstantiallyofexpressionsthatarenonsensebythework’sownstandards.4.2.3DomainrelativizationandconsequenceTarski,bothinhisspecificexampleofthelanguageofthecalculusofclassesandalsointhegeneraldiscussionof§4alsotreatsatopicthatlooksmuchmoretobeaforerunnerofcontemporarymodeltheory:“thenotionofcorrectortruesentenceinanindividualdomaina”[Tarski,1983a,199].Themodificationstothedefinitionsoftruthandsatisfactionarestraightforwrard:wesimplybuildourinfinitesequencesfromobjectsinthespecifiedsetandeverythingelserunsasbefore[Tarski,1983a,200].Tarskialsomakesthepointthatthedeductivetheoryatissuecan’tdis-tinguishdomainsofthesamecardinality,sothattheonlyimportantnotionisthatoftruthofasentenceinadomainofcardinalityk(LemmaG[Tarski,1983a,202])andhealsodefinestruthinalldomains[Tarski,1983a,201].TarskisetsoutapresentationoftheLöwenheim–Skolemtheorem[Tarski,1983a,204–5]andthediscussionculminateswithan“accidental”structuraldefinitionoftruth(Th.28)forthelanguageofthecalculusofclasses.ItissometimesworriedthatTarskionlyimaginesrestrictingthedomainandthathisdefinitionsdon’ttakeintoaccounttruthindomainsthatincludeobjectsthatdon’texist.SinceTarski’sinter-estswereentirelymathematicalandhistheoriesextensionalthisisnosurprise;moreover,asMcGeesuggests[McGee,1992,278]aslongas,asisplausible,anypossiblestructureisisomorphictosomepureset-theoreticstructure,thereisnoharminthis.12CTFLremainsold-fashionedinitsunderstandingoflogicalconse-quence.By1931thetruth-definitionhadbeenborn,butwithrespectCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01toconsequencenothinghadchanged.Definitions15and16in§2simplyworkouttheideathat“amongtheconsequencesofagivenclassofsentencesweincludefirstallthesentencesbelongingtothisclass,andallthesentenceswhichcanbeobtainedfromthesebyapply-ing,anarbitrarynumberoftimes,thefouroperationsofsubstitution,detachment,andinsertionanddeletionoftheuniversalquantifier[Tarski,10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE128—#21\nTruth1291983a,181].This,asdiscussedintheintroduction,tiesthenotionofconsequencetoasetofintuitivelyvalidrulesofinferenceandhencedoesn’tprovidewhatisoftheessenceoflogicasweknowittoday:theinterplaybetweenproof-theoreticandsemanticconceptionsofconse-quence.AsTarskinotes,italsodoesn’tcapturetheconsequencerelationintuitively.ThisissuewillloomlargeinChapter7.SinceTarski’ssemanticscanbeputtothisuse,ashehimselfputitin1935,weseehere,aswedidwithhistreatmentofPadoa’smethod,whatlookslikeanotherblindspotforTarskiintheearly1930s.AswithPadoa’smethod,however,whatthisreallyshowsisn’tthatTarskiwasn’tseeingsomethingthathewassomehowlookingfor,butthathewasn’tlookingforwhattodaywetakeitwasstaringhimintheface.In1931TarskihadnointentionofreplacingtheLe´sniewskianideathatthepurposeofadeductivetheoryistoexpressintuitivelygraspedthoughtsandtheintuitiveinferentialconnectionsbetweenthemwithsemantics;seman-ticswas,rather,atoolforproducingdefinitionsthatintroducedintodeductivetheoriessoconstruedtermsthatexpressedmetamathematicalconcepts.4.3EvaluatingTarski’saccountTarskioccupiesanintriguingplaceinphilosophicalmythology.Althoughheisuniversallyacknowledgedasimportant,allofhissubstan-tivephilosophicalviewsarewidelyagreedtobefailures.Inthissectionwewilldebunkanumberoffamiliarcriticismsofwhatisknowngener-allyas“Tarski’sTheoryofTruth”.Aswewillsee,themainthemewillbetheconfusiononwhichadefinitionconstructedalongTarski’slinesisitselfsomehowTarski’s“theory”oftruth.4.3.1FamiliarquestionsIbeginwithanumberoffamiliarquestionsandfalseassumptionsaboutTarski’sviewsthatareoflesserphilosophicalimportthantheissuestobediscussedbelow.Firstandperhapsforemost:ConventionTisnot,inthefirstinstance,acriterionofextensionaladequacy,asisveryoftenclaimed.13Itis,rather,astatementoftheconditionsunderwhichCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01“∈Tr”willbesoconstrainedbythestructureofthemetatheorythatitexpressestheintuitiveconceptoftruth(relativetotheobjectlanguagewithitsmeaning-determiningconventionsassociatingthoughtswithsentences).Confusiononthispointisuniversal:nearlyeverytreatmentofTarski’sviewsthatonecanfindinprinttreats“material”asmeaning“extensional”.Extensionaladequacyfollowsfromthetheoremhoodof10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE129—#22\n130AlfredTarski:PhilosophyofLanguageandLogictheT-sentences(assumingthatthemetatheoryistrue),butthepointofConventionTisnotmerelytointroduceapredicatewiththerightextension.Inadditiontotheplainobviousnessofthepointthatitcouldalwaysturnout,onewayoranother,thatthesetoftruthsofalanguagehappenedtobetheextensionofsomeextraneouspredicate,Tarski’sdis-cussionofTh.28makesclearthatheappreciatedthepoint.Theorem28statesthat,forthelanguageofthecalculusofclasses,x∈Trifandonlyifxfollowsfromtheusualaxioms,plustheclaim(“sentenceα”)thateverynon-emptyclasshasasingletonclassasasubclassandthenegationsofallsentencesγlstatingthatthereexistexactlylthingsforallnaturall.AsTarskinotes[Tarski,1983a,208],Theorem28couldjustaswellbeadefinitionof“∈Tr”.ButTh.28,treatedasadefinitiononitsown,isn’tintuitivelyadequate:addingjustit(andthedefinitionsitdependson)tothe“morphologyoflanguage”forthelanguageofthecalculusofclasseswouldyieldanextensionallyadequatedefinitionof“Tr”butnotonethatwouldallowtheproofoftheT-sentences.Clearly,then,sinceTarskiexplicitlynotesthematter,hedoesn’tequateextensionalandintuitiveadequacy.(HealsonotesthattheprovabilityofTh.28dependson“spe-cificpeculiarities”ofthelanguageofthecalculusofclassesaswellasthe“strongexistentialassumptions”adoptedinthemetatheory.Theacci-dentalcaseofastructuralcharacterizationofthetruesentencesaffordedbyTh.28innowayunderminesthepointthatingeneralthesetoftruthsofalanguagecan’tbecharacterizedstructurally.)Movingon,wecannowsettlethefamouslyvexedquestionofwhatTarskithinks“translation”betweenobjectandmetalanguagesentencesis.Onesentencetranslatesanotherjustincasetheconventionsgovern-ingbothassociatethesamethoughtwiththemand,moregenerally,oneexpressiontranslatesanotherwhentheconventionsassociatethemwiththesamethought-constituents.At[Tarski,1983a,166]itisclearthattheexpressionsoftheobjectlanguagearemeaningfulinthat“wealwaysascribeconcreteand,forus,intelligiblemeanings”tothem,whileat[Tarski,1983a,170]thetranslationofobjectlanguageexpressionsbythoseofthemetalanguageisduetothefactthatthey“havethesamemeaning”.ForTarskitheresimplycannotbeanyquestionaboutwhetherCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01onesentencetranslatesanother,becausetheresimplycannotbeanyquestionwhetherthethoughtsassociatedwithtwosentencesbytheconventionsoftheirlanguagesarethesameornot.ArelatedobjectionthatisoftenheardisthatinConventionTTarskisomehowbreakshispromise“nottomakeuseofanysemanticalcon-cept”,sinceheappealstothenotionoftranslation.Indeed,itis10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE130—#23\nTruth131sometimessuggestedthatthewholeaccountisimplicitlycircularif,inturn,twosentencestranslateoneanotheronlyiftheysharetheirtruthconditions.Tworepliesareinorderhere.First,thenotionoftranslationistiedtoTarski’saccountofintuitivemeaning,nottothenotionofatruth-condition,sothattranslationisn’tintherelevantsense“semantical”[Raatikainen,2008,254].Second,eveniftranslationwereunderstoodbyTarskiintermsoftruth-conditionstheaccountwouldn’tbecircu-lar;indeed,theaccountwouldn’tbecircularifTarskihadsimplysaidthatagooddefinitionoftruthforagivenobjectlanguagewasonethatimpliedallofthetrueT-sentencesoftheobjectlanguage.Statingacri-terionofadequacyonadefinitioninametalanguageusingthedefinedtermdoesn’trenderthedefinitionintheobjectlanguagecircularand,infact,asIhavestressedelsewhere,properlyunderstoodnoevaluationofausage-reportingdefinitionascorrectcanproceedwithoutdeployingthedefinedterm([Patterson,2007],[Patterson,2008b]).Nextwecometooneofthecentralpointsofcontentionintheliter-ature:thefactthatTarski’sdefinitionsapplytoonlyonelanguageatatime:Theextensionoftheconcepttobedefineddependsinanessentialwayontheparticularlanguageunderconsideration.Thesameexpressioncan,inonelanguage,beatruestatement,inanotherafalseoneorameaninglessexpression.Therewillbenoquestionatallhereofgivingasingledefinitionoftheterm.Theproblemwhichinterestsuswillbesplitintoaseriesofseparateproblemseachrelatingtoasinglelanguage[Tarski,1983a,153].ThisissuehasalreadybeenaddressedinChapter2:onTarski’sconcep-tionoflinguisticconceptstheyhaveapplicationconditionsandhenceextensionsonlyrelativetoalanguagewithitsmeaning-determiningcon-ventions.Tarskisuppressestherelativizationtoalanguagebecausethenotionofbeing“in”alanguagedoesn’tadmitofametamathematicaltreatment.Theaimofmetamathematicsistotreatlinguisticnotions,inthepresentcasesemanticones,onlyinsofarastheformsofsentences,asdeterminedbyformationrules,canstandinforthesenotions.TarskiCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01iswillingtospeakof“the”conceptoftruthbecausehethinksthereisonlyonesuchconcept,whichappliestoasentenceonlyrelativetotheconventionsofitslanguage.Thesematters,however,notbeingmathe-maticallytreatable,retreatintothemetametalinguisticcommentary,tobekeptinmindinjudgingwhetherthemetatheory’streatmentoftheobjectlanguageisintuitivelyadequate.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE131—#24\n132AlfredTarski:PhilosophyofLanguageandLogicItissometimeswonderedwhyTarskicallsConventionTa“convention”.Varioushypothesesarefloatedaboutthis.OurdiscussionofKotarbi´nskiandLe´sniewskimakesthematterclear:itisthroughtheconventionsoflanguagethatexpressionsgettheirmeaning.ConventionTisaconventionlikeanyotherinthisrespect:itstatesawayofusingacertainexpressionthatsuitsittoexpressacertainintuitivemeaning.RecallherethatLe´sniewskiisatpointsexplicitthatadefinitionisakindofconventionfortheuseofaterm.Tarski’swordingitwith“if”hasoccasionedspeculationtotheeffectthatheintendsConventionTtostateonlyasufficientcondition,butthisissquelchedbyhishabitualuseof“if”,oftenwithhelpofphraseslike“willbecalled”,toexpressdefinitions,andhisexplicitendorsementofthispracticeinthelogictextbookat[Tarski,1995,36].14Footnote1at[Tarski,1983a,188]alsomakesclearthatConventionTissupposedtoamounttoadefinition,whichrequiresnecessityaswellassufficiency.15(TheEnglishwronglystatesittobeadefinitionofthe“meta-theory”;theGermancorrectlyhas“Meta-Metawissenschaft”[Tarski,1986a,99].)Anotherissueconcernstheclaimthatissometimesmade(e.g.[Etchemendy,1988,55])thatalthough,supposedly,TarskirequiresthatadefinitionoftruthimplytheT-sentences,hisowndefinitionsdon’tinfactimplytheT-sentences,sinceotherresourcesofthemetatheoryarerequiredtoprovethem.WhatwecannowsayisthatforanIntuitionis-ticFormalistthefundamentalnotionofintuitiveadequacyappliestoanexpression–theorypair,nottoadefinition.WhatisintuitivelyadequateforTarskiis,inthefirstinstance,the“linguisticusage”imposeduponatermbythedeductivestructureofanaxiomatictheory.Iftheusageisright,theroleoftheterminthetheoryisintuitivelyadequate.Deriva-tively,adefinitioncanbeadequaterelativetoatheoryif,whenaddedtothattheory,itrendersitintuitivelyadequate.ESSisexplicitonthematter:Nowifwesucceedinintroducingtheterm‘true’intothemetalan-guageinsuchawaythateverystatementoftheformdiscussedcanbeprovedonthebasisoftheaxiomsandrulesofinferenceofthemetalanguage,thenweshallsaythatthewayofusingtheconceptofCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01truthwhichhasthusbeenestablishedismateriallyadequate.Inpartic-ular,ifwesucceedinintroducingsuchaconceptoftruthbymeansofadefinition,thenweshallalsosaythatthecorrespondingdefinitionismateriallyadequate[Tarski,1983b,404].Tarskithusisn’tconfusedinthewaysuggested;herealizesthatthedefinitionsdon’timplytheT-sentencesallbythemselves,andheasks10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE132—#25\nTruth133fornomorethanthattheydosoinconjunctionwiththerestofthetheory.16InterpreterssometimesmakemuchofTarski’suseofstructuraldescrip-tivenamesforsentences,suggestingoftenthattheyservesomeepistemicfunction,e.g.,makingitthecasethatasentencecanalwaysbe“recov-ered”fromitsname.Thisisaredherring.Thereasonstructuraldescriptivenamesareneededissimplythatwithoutthemtherewouldbenowaytostatetherecursionclausesinthedefinitionofsatisfaction.AsidefromthatneedTarskiisperfectlyhappywithanysortofwayofnamingasentence.Thisoughttohavebeenobviousfromthemerefactthatforafinitelanguagealist-likedefinitionassuggestedat[Tarski,1983a,188]couldperfectlywelluseunstructurednames.4.3.2TarskiandefinitionsandTarski’s“theory”AlargefamilyofstandardcriticismsderivesfromthecombinationofthefactthatagivenTarskiandefinitionoftruthappliesonlytoasin-glelanguageasitisactuallyinterpretedwiththeideathataparticularTarskiandefinitionitselfissomehowTarski’s“theory”oftruth.Afewclassicexamplesare[Black,1948,60],Davidson’sclaimthat“NothinginTarski’struthdefinitionshintsatwhatitisthatthesedefinitionshaveincommon”[Davidson,1990,288],andPutnam’sfamousclaimthat“Tarski’stheoryfailsasbadlyasitispossibleforanaccounttofail”[Putnam,1994,333].Listingthevariousobjectionsofthisvariety,wehavethefollowing:171.Tarski’saccountdoesn’ttelluswhattodowhenanewtermisaddedtoalanguageforwhichwehavegivenatruthdefinition[Black,1948,57–8],[Davidson,1990,287].2.Tarski’saccounttellsusthatweneedacompletelynewdefinitionoftruthwhenanewtermisaddedtoalanguageforwhichwehavegivenatruthdefinition[Black,1948,57–8].3.Tarski’saccountdoesn’ttelluswhatpredicatesdefinedinhiswayhaveincommon[Black,1948,60][Davidson,1990,288,296].4.Tarski’saccountimpliesthat“snowiswhite”wouldbetrueevenif“snow”meantwhat“coal”currentlymeans[Putnam,1994,333].5.Tarski’saccountimpliesthat“‘Snowiswhite’istrueinLifanonlyifCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01snowiswhite”isalogicalormathematicaltruth[Putnam,1994,333],[Etchemendy,1988,57],[Heck,1997,537],[Soames,1984,422–4],[Davidson,1990,288].AllfiveobjectionstakeTarski’s“theory”oftruthsomehowtobeexpressedinhisdefinitions.InadditionthefourthunderstandsTarski’s10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE133—#26\n134AlfredTarski:PhilosophyofLanguageandLogicdefinitionsasnecessarytruthsandthefifthtakesitthatadefinitionoftruthinamathematicalorlogicaltheoryisitselfamathematicalorlogicaltruth.Thecommonthemecanbedealtwitheasily:Tarski’sdefinitionsarenothis“theory”oftruth.Tarskihasatheoryoftruth:the“semanticaldefinition”of[Tarski,1983a,155]hailingfromKotarbi´nskiandotherPolishpredecessorsandultimately,astheysawit,fromAristotle.Ifithadbeenpossibletointroducethesemanticaldefinitionintoameta-mathematicaltheoryinaformallycorrectwaythatwouldhavebeentheendofthematter.Thisiswhy,asfarastheconceptoftruthitselfgoes,Tarskisimplydeferstothephilosophers.Tarski’sprojectwasnottoprovidea“theoryoftruth”inthesenseofaconceptualanalysisorsomethinglikeametaphysicalaccountofthenatureoftruth.Itwastoprovide,inaccordwithIntuitionisticFormal-ism,(i)anaccountoftheconditionsunderwhichatermofadeductivetheoryhasarolethatconstrainsittoexpresstheconceptoftruth,and(ii)themeansofintroducingtermswithsuchroles,perhapswhilemeet-ingfurtherdesiderata—inparticularthatthetermsbeintroducedviaexplicitdefinitionsoastoguaranteerelativeconsistency.Tarski’saccountof(i)isthat“∈Tr”expressestheconceptoftruthasappliedtotheobjectlanguageifandonlyifallT-sentencesformedwithitaretheorems.Hisaccountof(ii)isthemethodofrecursiononsatisfaction.But,aswehaveseenindetailabove,nomorethaninanyothercase(e.g.inthedefini-tionofsubstitutionofavariableat[Tarski,1983a,180])doesTarskithinkthataparticulardefinitionof“∈Tr”statesananalysisoftheconceptoftruth.Indeed,thedefinitiondoesn’tevenstateananalysisofthecon-ceptoftruthrestrictedtotheobjectlanguageathand,sinceforTarskitheonlyconstraintonadefinitionintendedtoextendatheorytoonecontainingatermexpressingatargetconceptisthattheresultantthe-oryhavetheoremsthatconstraintheinterpretationofthedefinedtermdowntothetargetconcept.Tarski’sdefinitionsareneitheranalysesoftheconceptoftruthnor“theoriesoftruth”.ItdoesfollowfromTarski’swayofproceedingthatifanewtermisaddedtoalanguagethedefinitionoftruthneedstobealtered.Ifthedefinitionwereintendedasaconceptualanalysis,thiswouldbebizarre,Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01justascommentatorswhoassumethatthedefinitionisananalysishavetakenittobe.Butsincethedefinitionisn’taconceptualanalysis,thepointsimplythatrecursionbysatisfactionneedstobeginfromlexicalbaseclauses.Isit,however,afailingofTarski’saccountthatittellsusnothingabouthowtoextendadefinitionwhentheobjectlanguageisextendedwith10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE134—#27\nTruth135anewexpression?TheanswerhereisthatTarski’saccount,takenasawhole,tellsusexactlyasmuchasitshouldtellus,neithermorenorless.WhatTarski’saccounttellsusisthatthenewdefinitionfortheextendedobjectlanguagehastobesuchthat,whenaddedtoformalsyntax(which,note,likewiseneedstobeextendedinawaynotindicatedbythepriorsyntactictheory)theT-sentencesfortheextendedlanguagebecomethe-orems.Whattheaccountdoesn’ttellusisexactlyhowtodothis.Thisisexactlyright,becausetheaccountoftruthshouldn’ttellushowtodothis.Ifwhatweaddisanexpressionthatbelongstoanalreadynon-emptycategoryofexpressions—e.g.anewnamealongsidethosealreadyinthelanguage—whatTarskitellsusnothingaboutiswhatthedenota-tionofthisnameis.Thisisexactlyasitshouldbe:whyonEarththinkthatadefinitionoftruthshouldimplywhynamesrefertowhat?Thepointisevenmoreobviousifwhatweaddisanewtypeofexpression(e.g.atwo-placefunctorsuchas“thesonofxandy”)previouslylack-ing.Neithertheaccountoftheconceptoftruthnortheaccountoftheconditionsunderwhichanexpressionofadeductivetheoryexpressesitshouldeverhavebeenexpectedtotellusanythingofthesort.Tarski’saccountoftruthdoes,contrarytoobjection3,telluswhatthedefinedpredicateshaveincommon:theyallexpresstheintuitiveconceptoftruthbyhavingthetheintuitivecontentstheyexpresscon-strainedbythetheoremhoodoftheT-sentences.Onewhoknowstheconventionsassigningmeaningstoobject-andmetalanguagesentencestherebyrecognizesthat“∈Tr”expressestheconceptoftruthasanalyzedinthesemanticaldefinition.Onlythinkingthataparticulardefinitionissomehow“the”theoryespousedbyTarskicouldmakeitlookotherwise.DavidsonisabsolutelyrightthatthereissomethingmoretotheconceptoftruththanTarski’sdefinitionstellus[Davidson,1990,285].WhatiswrongistoconcludefromthisthatTarskihasnothingtosayaboutwhatthedefinedtermshaveincommon,andnothingtosayabouttheconceptoftruth.Howaboutthefourthobjection,oftencalledthe“modal”objection[Raatikainen,2008,247]?Accordingtothesemanticaldefinition,asen-tenceistrueiffwhatitsaysisthecase.Now,if“snow”hadmeantwhat“coal”currentlymeans,“snowiswhite”wouldhavebeentrueiffwhatitCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01said,namely,thatcoaliswhite,wasthecase.AdeductivetheoryalongTarski’slinesintendedtoapplytoalanguagewiththatintuitiveinterpre-tationwouldberequiredtohave‘“snowiswhite”istrueiffcoaliswhite’asatheorem.18Tarski’saccount,takenasawhole,simplydoesnotimplythattruthisapropertythat“snowiswhite”wouldhaveiffsnowwerewhite,evenif“snow”hadmeantwhat“coal”currentlymeans.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE135—#28\n136AlfredTarski:PhilosophyofLanguageandLogicWhatistrueisthatadefinitionalongTarski’slinesforalanguagecon-taining“snowiswhite”withtheintuitivemeaningthatithasinEnglishassignsthetruthconditionthatsnowbewhitetothesentence“snowiswhite”withnoregardtothedependenceoftheaccuracyofthistruthconditionontheintuitiveinterpretationimposedonthesentencebytheconventionsgoverningtheobjectlanguageandhencenosensitivitytochangesinthatintuitiveinterpretation.Iftheintuitivemeaningofanexpressionoftheobjectlanguagechanges,Tarski’sdefinitionneedstobereworkedtocapturetheconceptoftruthforthethelanguagesoreinter-preted.Thisisaconsequenceofthefactthatintuitivemeaningisan“offtable”mattertobekeptinthemindoftheusersofthetheory,or,toputitmoreintermsoftheunderlyingphilosophyoflanguage,becausetheconventiongoverningtheuseoftheobjectlanguagethatmaps“snowiswhite”tothethoughtthatsnowiswhatissomethingthatisassumedratherthanstatedbythedefinitionoftruthfortheobjectlanguage.ThatTarski’stheoriesareextensionalisofcourserelevanthere.Tarski’sdefinitions,formulatedwithinextensionaltheories,saynothingaboutwhatthetruthconditionsofsentencesmustorcanbe.Tarski’sexten-sionaltheoriesaresimplysilentonthematter,andhisdefinitionsareintendedsimplytoextendextensionaltheories.Theymakenoclaimtocaptureanysupposedintension,inthecontemporarysenseofafunctionfromworldstoextensions,ofthetruth-predicate.ThestandarddefensesofTarskiagainstthemodalobjectionclaimthathislanguagesareindividuatedbythemeaningsoftheirexpressions(e.g.Raatikainen,2008).Isthistherightresponse?IntuitionisticFormalismunderstandsbothobjectandmetalanguageasgovernedbyconventionsthatmapsentencessyntacticallyindividuatedtothethoughtsthattheyexpress.Alanguageidentifiedbyitssetofsentencesisn’tindividuatedbythemeaningsofitsexpressions,butbythesesentences.Alanguageplusasetofconventionsmappingexpressionstoconceptsandsentencestothoughtsis,inturn,ofcourseindividuatedbytheintuitivemeaningsofitsexpressions.Iamthusinclinedtosplitthedifferencewiththestandardresponsetothemodalobjection.WhatisrightinthemodalobjectionisthatTarski’sdefinitionsthemselvessaynothingaboutthedependenceoftruthconditionsonintuitivemeaning.ThedependenceCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01oftruthconditionsonmeaning,however,isn’tabsentfromTarski’soverallaccount;itispresentintheassumedconventionsoftheobjectlanguage,conventionsthatarethemselvesnotstatedinthemetatheorytowhichthetruthdefinitionisadded.Butjustthisfactbluntstheforceofthemodalobjection:Tarskiinnosenseseriouslysuggeststhat“snowiswhite”wouldbetrueiffsnowiswhitenomatterwhat“snowiswhite”meant.Hisviewsimplyimpliesthatthisisachangeinconventions10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE136—#29\nTruth137requiringacorrespondingchangeinextensionalsyntacticcumsemantictheory.Tothatextentmyreadingimpliessomethinglikethestandarddefense:thewaythedefinitionislaidoutassumesthatconventionsfix-ingthemeaningoftheobjectlanguageareheldfixed.WheremyviewdiffersfromthestandarddefenseisthatIdonotattributetoTarskitheviewthatlanguagesareindividuatedbythemeaningoftheirexpressions,sincethisisincompatiblewithhisremarksaboutextendingalanguageat[Tarski,1983a,164].Movingontothefifthobjection,onecommonlineofcriticismofTarskiisthatbyaddingdefinitionsofsemanticconceptstomathemati-calorlogicaltheorieshetherebymakestheT-sentences(andinstancesofassociatedschemataforothersemanticconcepts)intomathematicalorlogicaltruths.Nowwhatisclearisthatgivenstandardtruth-conditionalsemanticsandthetheory-relativeconceptionofdefinition,substitutioninaccordwithagooddefinitiondoesn’tpreservelogicalandmathe-maticaltruthandhencethefactthatTarski’sdefinitionsareaddedtomathematicaltheoriesisnoreasontothinkthattheyonhisviewstatemathematicaltruths.Thiswastheargumentof[Patterson,2008b].19(Quickexample:“anunmarriedmanisanunmarriedman”isalogi-caltruth,but“abachelorisanunmarriedman”isn’t,eventhoughthesecondisderivedfromthefirstbysubstitutioninaccordwithagooddefinition—namely,itself.)ButIntuitionisticFormalismmuddiesthewaterssignificantlyheresincepartofTarski’sviewisthatinadefinitiondefiniensanddefiniendumsharetheirintuitivemeaning.OnethoughtmightbethatamathematicaltheoryexpressesonlymathematicalintuitiveconceptsbeforeaTarskiandefinitionoftruthforsomeobjectlanguageisaddedtoit;sincetheexpressionintermsofwhich“∈Tr”isdefinedexpressedamathematicalconceptbeforethedefinitionwasadded,addingthedefinitionforces“∈Tr”toexpressamathematicalconcept.Butthisisnogood,sinceitassumesthattheaccountofintuitivemeaningiscompositional,whichitisn’t(cf.§2.2.1).Onthecontrary,asIhavearguedinthischapter,thepointofConventionTistoforceanexpressionofatheorywithonlymathe-maticalprimitivestoexpressasemanticconcept.Inaccordwiththetheory-relativeconceptionofdefinitioncombinedwithTarski’saccountCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01oftheassignmentofintuitivemeaningstoexpressionsconstrainedbytheorems,onceweaddtheTarskiandefinitiontoamathematicaltheory,whatwehaveisanewtheorythatdemandsnewinterpretationsforsomeofitsexpressions;inparticularboth“∈Tr”anditsdefiniensexpresstheconceptoftruth.HenceonTarski’sviewtheT-sentencesexpresssemantictruths,notmathematicaltruths.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE137—#30\n138AlfredTarski:PhilosophyofLanguageandLogicNowifTarskihadgivensomeindependentargumentthatsemantictruthswereactuallymathematicaltruthsitwouldbeanothermatter,butsincehedidn’t,wehavenothingtogoonbuttheobviousnessofthepointthatsemantictruthsaren’tmathematicaltruths,andthefactthatTarskihadnoneed,inordertosupportanyotherpositionheheld,tomaintainthatsemantictruthsaremathematicaltruths.Thus,Ithink,eventakingIntuitionisticFormalismintoaccount,thestrongerviewisthatTarskididn’tholdthattheT-sentencesthatfollowfromoneofhisdefinitionsstatelogicalormathematicaltruths.4.3.3ReductionandphysicalismThereisnowasignificantliteratureonwhetherTarski’sviewsare,asheappearstoclaim([Tarski,1983b,406]wherehealsomakesreferencetoanotherViennaCircledoctrine,the“unityofscience”),compatiblewithphysicalism—whetherheintendedhisdefinitiontosomehowsupportphysicalismandwhat,accordingtohim,physicalismamountedto.Pas-sagesotherthanthecentralonefromESSindicatereductivedesignsofsomesort,e.g.theintroductiontoCTFL:InthisconstructionIshallnotmakeuseofanysemanticalconceptifIamnotablepreviouslytoreduceittootherconcepts[Tarski,1983a,153].Yet,famously,Field[Field,1972],secondedbyothers[Soames,1984],arguesthatTarski’sdefinitionsdonotsucceedinreducingsemanticconceptstophysicalisticallyacceptableones.TheproblemliesagaininTarski’senumerativedefinitionofsatisfaction(andothersub-sententialsemanticnotions,ifthelanguagehasnon-logicallexicalexpressionsotherthanpredicates).Agenuinelyphysicalistreductionofthenotionofsatifaction,forinstance,wouldtaketheformSat(s,x)↔ϕ(s,x)wherexisasequenceandϕissomegenuinelyphysicalpredicate.20Field’spreferredcandidate,ingood1970sstyle,isthecausaltheoryofreference[Field,1972,366–7],21butTarski’s“reduction”isobviouslymerelyextensional[Field,1972,362].22AsFieldquips:Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Bysimilarstandardsofreduction,onemightprovethatwitchcraftiscompatiblewithphysicalism,aslongaswitchescastonlyafinitenumberofspells:forthen‘castaspell’canbedefinedwithoutuseofanyofthetermsofwitchcrafttheory,merelybylistingallthewitch-and-victimpairs[?,369].10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE138—#31\nTruth139Thefirstthingtonotehere,ofcourse,isthatbyTarski’sownstandardsthiswassufficient,sincethetheorieswereextensional.However,inlightofField’scriticismasubstantialdiscussionhasdevelopedconcerningtheextenttowhichTarskireallyintendedanyphysicalisticreductionofsemanticconceptsorproperties.ManyauthorsdiscountTarski’scommitmenttophysicalismbynotingthatonlyin[Tarski,1983b]isitsuggestedthathisprojectaimstomakesemanticssafeforphysicalism[García-Carpintero,1996,117],[Frost-Arnold,2004,267–272]andthatinthispassageheisclearlytryingtoingratiatehim-selftotheaudienceattheUnityofScienceCongress,Paris1935.23As[Rojszczak,1999,123]pointsout,inthePolishversionofthearticle,Tarskisays,asintheGermanandWoodger’sEnglishtranslationofit,thatiftruthwereaxiomatizedratherthandefinedtherewouldbeathreatthephysicalismandtheunityofscience,butgoesontoadd,asthoseversionsdonot,thatthesearedoctrines“propagatedbyagreatmanyphilosophersfromtheso-calledViennaCircle”.ThisconfirmstheviewthatTarskihimselfdidn’thaveanyparticularlysignificantreduc-tivegoalsforhissemantics,butwassimplyingratiatinghimselftotheViennaCircle.Ontheviewofferedhere,Tarskiintendednoreductionofanykindoftheconceptoftruth:theintensionor“content”ofthisconceptisexpressedintheirreduciblysemantic“semanticaldefinition”of[Tarski,1983a,155].Whathewantedtodowastoconstructdeductivetheorieswithonlysyntacticandlogical(andhencemathematical,byarith-metization)primitivesthat,havingtheT-sentencesfor“∈Tr”fortheobject-languageastheorems,constrainedittoexpressthissemanticalcontentoftheconceptoftruth.Tarskihadnospecificconcernsatallaboutthecontentoftheconceptoftruthitself.Hetookittobefamiliarbothfromoureverydayunderstandingoflanguageandfromphilosoph-icalworksontruth.Theissuewastheconditionsunderwhichatermofadeductivetheorythatwasitselfconceptuallyunproblematiccouldexpressitconsistently.Explicitdefinition,andhenceascenttoastrongermetalanguage,wererequiredforthislastdesideratumandthegeneralthrustofmetamathematicsashehadunderstoodthatfromwellbeforethesemanticperiodsetthelimitstotheacceptabletheories.ThefirstCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01paragraphofCTFLisclearonthispoint:thoughthecontentofthecon-ceptseemsclear,“manyinvestigations”oftheproblemofdefiningthecontentthatbeginfrom“apparentlyevidentpremises”breakdowninparadox.Theissuewasn’tgettingstraightabouttheconcept,itwasget-tingstraightaboutthewayinwhichitcouldconsistentlygetexpressedinadeductivetheory.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE139—#32\n140AlfredTarski:PhilosophyofLanguageandLogic4.3.4CorrespondenceanddeflationismInterpretersaresplitonwhetherTarskiwasacorrespondencetheoristoradeflationist.Popper,famously,waswonoverimmediatelybythepageproofsofCTFLinViennaandmaintainedthenceforththatTarskihad“rehabilitatedthecorrespondencetheoryofabsoluteorobjectivetruth”[Popper,1963,223].MorerecentfigurestomaintainthatTarski’sviewisacorrespondencetheoryinclude[Sher,2004],[Sher,1999],[Schantz,1998],[Niiniluoto,1999a],[Weingartner,1999]and[Fernández-Moreno,2001].Mostimportantly,ofcourse,Tarskihimselfsaysthatthecon-ceptoftruthinits“classicalinterpretation”involvestheideathat“‘true’signifiesthesameas‘correspondingwithreality’”[Tarski,1983a,153],[Tarski,1983b,401].24OfcoursewhetherTarskiisacorrespondencethe-oristdependsonwhatacorrespondencetheoryisandsoinduecoursewewillhavetosayalittleaboutthat.TherearetwobadreasonsandonebetteronetotakeTarskitobeacorrespondencetheorist.ThebadreasonsaretheroleoftheT-sentencesinTarski’sdefinitionsandtheroleinthesedefinitionsoftherelationofsatisfactionthatholdsbetweenopensentencesandsequences.Onthefirstcountthereisafairlywellestablishedtraditionofdistinguishing“weak”from“strong”correspondencetheories,wherea“strong”the-orypositssomesortofstructuralrelationshipbetweensentencesandtheotherrelataofthecorrespondencerelation,andaweakonemerelyinsiststhatwhetherornotasentenceistruedependsonwhetherwhatitsaysisthecase[Wole´nskiandSimons,1989,418]and,indifferentterminology,[Kirkham,1992,119],[Vision,2004,223].SincetheroleoftheT-sentencesaloneinTarski’sviewgivesonenoreasontothinkthathisviewisacorrespondenceviewinthestrongsense—theT-sentences,afterall,aren’tevenoftherightformtoascribearelation,muchlessoneofstructuralsimilarity[Patterson,2003]—theresultantreadingsinsistonTarski’sbeingaweakcorrespondencetheorist.(Kotarbi´nskihimselfcrit-icizesstrongcorrespondencetheoriesjustbeforeintroducinghisversionofTarski’ssemanticaldefinitionat[Kotarbi´nski,1966,106–7].)Thisiscorrectasfarasitgoes,butonlythecommonconfusion[Jennings,1983],[Vision,2004,ch.6],[Sluga,1999,40],[Sher,1999,136]onwhichdefla-tionistssomehowhaveanentirely“immanent”or“intralinguistic”con-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01ceptionoftruthpreventstherecognitionthat“correspondence”theoriesintheweaksensearen’tevendistinguishedfromdeflationarytheories.Asforsatisfaction,itisoftenpressedintotheserviceofcorrespon-dencereadingsofTarski’sdefinitions[Davidson,2002b],[Wole´nski,1989,178],[Niiniluoto,1999a],[Fernández-Moreno,2001,139].The10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE140—#33\nTruth141standardobjectionhereisthatbyDef.23truthissatisfactionbyallsequencesandhence,byagivenTarskiandefinition,alltruesentencescorrespondtothesamething,namely,thesetofallsequences[Davidson,1990,303-4].Anobjectionheardlessoften,butequallyimportant,isthatforTarskithereis,strictlyspeaking,norelationofsatisfactionintermsofwhichtruthisdefined,aswediscussedin§4.2.2.Addtothisthatn-placeopensentencesdon’tsharea“form”withinfinitesequencesofobjectsinanyinterestingsenseandwehaveanabundanceofreasonsthattheappealtosatisfactionbycorrespondencereadingsofTarskiisfruitless.ThebetterreasontotakeTarskitobeacorrespondencetheorististhathesaysthattheideabehindhisviewistocapturetheintuitionsbehindthe“semanticaldefinition”[Tarski,1983a,155]oftruthandheiswillingtoequatetheseformulationswithvariouscorrespondenceformulations[Tarski,1983a,153],[Tarski,1983b,401].ThesourceofthepreviouspoorconstrualsofTarskiasacorrespondencetheoristisreallyagainjusttheconflationofTarski’s“theory”withaparticulardefinition,forwiththisconflationinplacetakingTarskiathiswordthathisviewisacorre-spondenceviewmandatesfindingsomethinginaparticulardefinitionoritsconsequencesthatamountstoacorrespondenceview.Properlydistinguishingthecontentoftheconcept,thetheoremsofadeductivetheorythatforce“∈Tr”toexpressit,andthedefinitionthat,addedtothetheory,resultsinitsimplicationofthesetheoremsimmediatelysortstheissueout.Neithertherecursiononsatisfactionnorthethe-oremhoodoftheT-sentencesmakesTarski’s“theory”acorrespondencetheory.Tarskiisacorrespondencetheoristinthesensethatheacceptsthesemanticaldefinitionandholdsthatthesemanticaldefinitiondefinesacorrespondencenotion.WeturnnexttotheideathatTarskiwasanearlydeflationist.Therearemanycharacterizationsofdeflationismintheliterature.Withoutenter-ingintothedebateoverhowtodefinedeflationismitself,25IwillsimplystickwiththeviewIhavemaintainedforsometimenowthattheonlythesisintheareaclearenoughtoadmitofmeaningfuldisputeisthethe-sisthattheT-sentencesforalanguagedefinetruthforthatlanguage[Pat-terson,2002],[Patterson,2008b].26Thuscharacterizingdeflationism,Tarski’stheorylooksessentiallydeflationaryaslongasonestressestheCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01ideathatadefinitionoftruthisthe“logicalproduct”[Tarski,1983a,187]oftheT-sentences,aslongasonestickstocommonassumptionsaboutwhatadefinitionis,andaslongasonethinksthatTarski’sdefinitionishistheory.Quine[vanOrmanQuine,1960,24]attributestoTarskithe“clas-sicdevelopment”oftheinsightthattosaythat“BrutuskilledCaesar”istrueisthesameastosaythatBrutuskilledCaesar.[Leeds,1978,120–122]10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE141—#34\n142AlfredTarski:PhilosophyofLanguageandLogictakesitthattheT-sentences“axiomatize”thenotionoftruthandthatTarski’sachievementwastodiscoverasmuch.Horwich[Horwich,1982,191]distinguishesdeflationistsfrom“metaphysicalrealists”bytheirholdingthattheconceptoftruthis“exhausted”byTarski’scharacter-izationintermsoftheT-sentences.MichaelDevitttakesitas“generallyagreed”thatTarski’s“theory”isdeflationaryat[Devitt,2001,73].Nevertheless,deflationistsareoftencauseddiscomfortbytherecur-siononsatisfactioninastandardTarskiandefinition,sotherearealsomanyattemptstodistinguishTarskifromdeflationismintheliterature—e.g.Horwich’slaterwork[Horwich,2003].However,surelytheoverallemphasisondefiningtruthandrelatednotionsintermsoftheinstancesoftheT-schemalooksconsonantwithwhatmanycontemporarydefla-tionistschampion.IfonlyTarskitookhisdefinitionsandtheirconse-quencesasconceptuallyanalyticoftruth,deflationaryreadingsofTarskiwouldbeinbusiness.But,asarguedheretothispoint,thisisn’tTarski’sview.TheT-sentencesconstraintheinterpretationof“∈Tr”downtobethecontentoftheconceptoftruth,buttheyaren’tthemselvesconcep-tualanalyses,andneitheristhedefinitionthatbringsthemastheoremsinitswake.Ontheviewofferedhere,thefamiliarclashofinterpretationsisentirelyexplicable.TherequirementthattheT-sentencesbetheoremsiftheconceptoftruthistobeexpressedby“∈Tr”looksdeflationary.Tarski’s“semanticaldefinition”,coupledwithTarski’sexplicitclaimsthatitisacorrespondencetheory,makeTarskilooklikeacorrespondencetheorist.ThemuddleintheliteratureisunclarityovertherelationshipbetweenTarski’sconceptionoftruthandthetheoremsthatmakethe“∈Tr”ofaparticulartheoryexpressit.Sortingoutthismuddleisoneoftheprimarybenefitsoftheinterpretiveworkdoneinthesefirstfourchapters.TheT-sentencesdon’tdirectlystateconceptualtruthsabouttruth;givenTarski’sIntuitionisticFormalistaccountofintuitivemeaningintermsofconventions,thetheoremhoodoftheT-sentencesconstrains“∈Tr”toexpressthecontentoftheconceptoftruthasexplicitlystatedinthesemanticaldefinition.Tarskiwasacorrespondencetheoristabouttheconceptoftruth,grantingtheclaimthatthesemanticaldefini-tionisequivalenttovariouscorrespondenceformulations.However,Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Tarskiisn’tacorrespondencetheoristinthesensethathepositssomesortofisomorphismorpicturing-relationbetweentruesentencesandfactsorsomethingofthesort.TheupshotofhisIntuitionisticFormalistviewsisthatatheorywiththeT-sentencesastheoremsissufficienttoexpressthatconcept,andthistheorylookslikeadeflationarytheory.ButTarskidoesn’tholdthatthetheorydefinestheconcept,merelythat10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE142—#35\nTruth143itconstrains“∈Tr”toexpressit.Hencehisviewlacksmostoftheconse-quencesofdeflationaryviews,e.g.theimplicationthatstrictlyspeakingtherearedifferentconceptsoftruthfordifferentlanguages.SoisTarskiacorrespondencetheorist,atleastofthecontentoftheconceptoftruththathisvariousdefinitionsaimtocapturerelativetotheirobjectlanguages?Thisdependsonwhether“xistrueiffxsaysthatp,andp”isitself,asTarskisaiditwas,equivalenttocorrespondenceconceptionsoftruth.Itcertainlyisn’tequivalenttoa“strong”suchconception,sinceitmakesappealtononotionofsimilarityinform.Itdoes,as“weak”correspondencerequires,allowthattruthdependsonextralinguisticfact,butasnotedabovethisdoesn’tdistinguishitfromdeflationarytheories.Formymoney[Patterson,2008b],[Patter-son,2005],thesemanticaldefinitionisdeflationarysinceitassumessomemeaning-theoreticnotionandexplainstruthintermsofit,whileonly“strong”correspondencetheoriesreallydeservetobecalled“cor-respondencetheories”.Inanycase,thepointofthissub-sectionstands:Tarskitakesthesemanticaldefinitiontobeacorrespondenceconcep-tionoftruth,andaimstocaptureitindeductivetheoriesthathavetheT-sentencesfortheirobjectlanguagesasconsequences.Aslongaswedon’tconflateTarski’sparticulardefinitionswithhis“theory”andaslongaswerecognizethataparticulardefinitionisn’tsupposedtostateanythingconceptuallyanalyticabouttruth,andthattheT-sentencesactasconstraintsthat,inconjunctionwithTarski’sIntuitionisticFormal-istconceptionofintuitivemeaning,hedge“∈Tr”aboutwithsufficientrestrictionstoforceittoexpressthecontentoftheconceptoftruthasconstruedinthesemanticaldefinitionratherthandirectlyanalyzingthiscontent,wecanappreciatetheoverallstructureofconcept,definitionandtheorythatgivesrisetothevaryingclaimsabouttruththathavehithertocausedinterpreterssomuchdifficulty.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-0110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP04”—2011/11/29—14:25—PAGE143—#36\n5IndefinabilityandInconsistencyOurdiscussionofTarski’sIntuitionisticFormalisttreatmentoftruthin§§2–4ofCTFLconcernedmaterialthatseemstohavebeendevelopedby1931([Hodges,2008,120-6])andmerelypolishedforthe1931sub-missionofthePolishoriginalofthework.(Cf.Tarski’sown“HistoricalNote”wheretheadditionoftheGödeliantreatmentofindefinabilityisdescribedas,ofalltheresultsinthework“theonlyonesubsequentlyaddedtotheotherwisealreadyfinishedinvestigation”[Tarski,1983a,278].)Wehavethusbeenabletodiscussthematerialof§§2–4withlittleconcernforouroverallstoryaboutTarski’sdevelopmentfromthelate1920suptotheParisUnityofScienceconferencein1935.NowthatweturntoTarski’sremarksontheconditionsunderwhichtruthcan-notbedefinedwewillbeunabletoignoretheissuesraisedbyTarski’sevolvingviews.Inparticular,threedifferentviewsontheimpossibilityofconstruct-inganintuitivelyadequatedefinitionoftruthfor“expressivelyrich”languagescanbefoundinCTFLaswehaveit:apre-Gödelianargu-mentthatanintuitivelyadequatetruth-definitionisimpossiblebasedonconsiderationsfromthetheoryofsemanticalcategories,abasicallyGödelianargumentthatappearsinthe“proof-sketch”accompanyingTheoremIof§5—thoughonestillconceivedintermsofTarski’soriginalIntuitionisticFormalistmetamathematics—andareassessmentofthe§5resultssetoutinthepostscripttotheGermantransla-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01tionof1935,whichbeginstopavethewayforanabandonmentofIntuitionisticFormalismthatbeginsinearnestwiththetreatmentofconsequencein1935.Eventhiscategorizationistoosimple:theargumentforTheoremIisamessthatcombinesaspectsofGödel’spresentationwithbitsofTarski’sinterestinformaldefinability,andthetextofthepassageitselfisapatchworkattempttopastethe14410.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE144—#1\nIndefinabilityandInconsistency145laterresultintoaworkthatwasalreadywrittentoleaduptotheearlierone.5.1Indefinability5.1.1Indefinabilitybefore1931WhatTarskisaysaboutthetheoryofsemanticalcategoriesin§4alreadyimpliesakindofindefinabilitytheorem:nolanguagetheorderofthevariablesofwhichhasnofiniteupperboundcanreceiveadefinitionoftruthalongthelinessofardeveloped,becausethesatisfactionpred-icatemustbeofhigherorderthananyvariableinthelanguage.In§5Tarskiintroduces“thegeneraltheoryofclasses”,whichhasvariablesofthisdescriptionandatomicopensentences,Tarski’sintuitivereadingofwhichis“theclassX(ofn+1thorder)hasasanelementtheobjectY”or“TheobjectYhasthepropertyX”’[Tarski,1983a,242–3].Tarskistatestheargumentforthrightlyafterintroducingthelanguageandtheaxiomsforthetheory:Whenwetrytodefinetheconceptofsatisfactioninconnexionwiththepresentlanguage,weencounterdifficultieswhichwecannotover-come.Inthefaceoftheinfinitediversityofsemanticalcategorieswhicharerepresentedinthelanguagetheuseofthemethodofmany-rowedsequencesisexcludedfromthebeginning,justasitwasinthecaseofthelogicofmany-termedrelations.Butthesituationhereisstillworse,becausethemethodofthesemanticalunificationofthevariablesalsofailsus.Aswelearntin§4,theunifyingcategorycan-notbeoflowerorderthananyoneofthevariablesinthelanguagestudied.Sequenceswhosetermsbelongtothiscategory,andstillmoretherelationofsatisfaction,whichholdsbetweensuchsequencesandthecorrespondingsententialfunctions,mustthusbeofhigherorderthanallthosevariables.Inthelanguagewithwhichwearenowdeal-ingvariablesofarbitrarilyhigh(finite)orderoccur:consequentlyinapplyingthemethodofunificationitwouldbenecessarytooper-atewithexpressionsof‘infiniteorder’.Yetneitherthemetalanguagewhichformsthebasisofthepresentinvestigations,noranyotherofCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01theexistinglanguages,containssuchexpressions.Itisinfactnotatallclearwhatintuitivemeaningcouldbegiventosuchexpressions[Tarski,1983a,243–4](altered).The1931projectscreechestoahaltatthelimitsofintuitivemeaning.1Thisisboundbythetheoryofsemanticalcategories,andthissimplyleavesnoroomfortheideaofanexpressionof“infinite”10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE145—#2\n146AlfredTarski:PhilosophyofLanguageandLogicorder.Tarskibrieflyconsidersdoingthingsinapiecemealfashion[Tarski,1983a,244–5]:sinceeachparticularsententialfunctionhasfreevariablesofsomefixedfiniteorder,onemighthopeforsomesortofrecursionontheorderofopensentences.However,thishopeisdashedbythepointthatopensentencesofacertainordermaycontainboundvariablesofarbitrarilyhighorder[Tarski,1983a,245].Tarskiconcludesonalownote:“Thereforewecansay—consideringthefailureofprevi-ousattempts—thatatpresentwecanconstructnocorrectandintuitivelyadequate(sachlichzutreffende,zgodnejzintuicjka˛)definitionoftruthforthelanguageunderinvestigation”[Tarski,1983a,245–6].Myhypothesisisthatwhentheworkoriginallywenttopressthiswasitsfinalsentence.Thepublishedwork,however,doesnotendthere.TarskinotesthatTheoremIandthesketchofitsproofwereaddedtothework“afterithadalreadygonetopress”.Giventhatotherconcernsintherestof§5—theω-rule,thepossibilityofdevelopinganaxiomatictheoryoftruth—arealsoforeigntotheconcernsoftheearliersections,Ithinkthebestguessisthateverythingin§5afterthetopofpage246wasaddedtowhatwassupposedtohavebeenafinishedworkandthatthesecond,longerproofsketchforTheoremIat249–50wasaddedevenlater.Aswewillsee,theresultsofthispiecemealcompositionshow.Ashehimselfnotes,Tarskiwasmovedtoextendtheworkbytheappearanceofthe1930reportannouncingGödel’s1931articlepresent-ingtheincompletenesstheorem.ThoughTarskihimselfhadindepen-dentlyrecognizedthebasicpossibilityofarithmetizationandmentioneditat[Tarski,1983a,184],Gödel’sreportpromptedTarskitoextendhisworkalongGödel’slinesandtoamendtheworkinpresstogettheresultsintoprint.ThefirstthingTarskidoesintheaddedmaterialistoaskaquestioninageneralformthatisclearlyraisedbyhisoriginaldownbeatconclusion,butthatthematerialuptothatpointintheworkputhiminnopositionatalltoaddress:“whetheronthebasisofthemetatheoryofthelanguageweareconsideringtheconstructionofanadequate(richtige,trafnej)definitionoftruthinthesenseofConventionTisinprinciplepossible”[Tarski,1983a,246](corrected).BeforeencounteringGödel’stechniques,allTarskiwasinapositiontosaywasthathistechniquescouldbeappliedwithintheCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01boundssetbythetheoryofsemanticalcategoriesonlytolanguagesoffiniteorder.Ifnothingelsewerepossible,onewouldliketoknowthat,butnothingTarskihadbeforelate1930puthiminapositiontoaddressthequestion.ThissetsthestagealsoforTarski’searlyadaptationofGödel’sresults.Tarski’squestionallalonghadbeenwhethersemanticconceptscould10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE146—#3\nIndefinabilityandInconsistency147beformallyexpressedbydefinitionallyeliminableterms.ThismeansthatwhenGödelfirstallowedhimtoextendhisinvestigationsthetech-niqueswereviewedentirelyinlightofthequestionTarskihadalreadybeenasking:cantheintuitiveconceptoftruthbeexpressedbyanyexplic-itlydefinedexpressionofthemetalanguageundertheconditionssetoutin§5?Inparticular,infirstappropriatingGödel’swork,TarskiviewsitandtheresultthatnowbearsTarski’snamenotasitisviewedtoday,asasemanticresultaboutwhetheranexpressioncansemanticallydefinethesetoftruesentences,butasaformalresultaboutwhetheranyexpressionofthemetalanguagecanbeintuitivelyadequatetoexpressthecontentoftheconceptoftruththroughtheT-sentencesforallsentencesoftheobjectlanguagebeingtheorems.IttookTarski’sthinkinganumberofyearstoemergefromthe1929–1930projectandseetheresultsinafullycontemporarysemanticlight,andalongthewaytheearlycon-ceptionofmeaning,towhichthesemanticsforwhichTarskiisnowrememberedwasamereauxiliary,hadtobesupersededbythissamesemantics.Theversionoftheresultspresentintheworkasweknowitis,bycontrast,thoroughlycoloredbyhisconceptionofitasacontribu-tiontohisprojectofformaldefinitionand,ultimiately,theexpressioninadeductivetheoryofintuitivesemanticconceptsasheunderstoodthem.25.1.2TheoremI:textualissuesTheusualviewofTheoremIandtheproof-sketchesthatfollowitisthatthereinTarskistatesanearlyformof“Tarski’sTheorem”,accordingtowhichnoone-placeopensentenceinthevocabularyofatheorythatextendsaweaksystemofarithmeticcanhaveasitsextensionexactlythesetofGödelnumbersofthetheory’struths.Sometimesitisrecognizedthatthetextisoutdatedtosomeextent,e.g.initsformulationinSTT,butbyandlargethisistheextentoftheappreciationofthepassage.“Tarski’sTheorem”aswenowknowitisatheoremaboutsemanticdefinability.Consideraweaksystemoffirst-orderarithmeticlikeRobin-son’sQandanytheoryTthatextendsit.GödelcodethelanguageofT,representingtheGödelcodeofanexpressioneas
e,andconsiderthesetofGödelcodesofsentencesssuchthatT|=s.SupposethatψCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01semanticallydefinesthissetrelativetoT,thatis,that(REP)T|=sifandonlyifT|=ψ
sNow,bythediagonallemmaforextensionsofQ,foreveryopensentenceFxthereisasentencessuchthatTs↔F
s.So,withrespecttoψthere10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE147—#4\n148AlfredTarski:PhilosophyofLanguageandLogicexistsasentences0suchthat(DIAG)Ts0↔¬ψ
s0Now,supposethatT|=s0.Then,bythesoundnessofrelativeto|=andclosureofthelatterundermodusponens,ifT|=s0andTs0↔¬ψ
s0thenT|=¬ψ
s0.Thus:T|=s0ifandonlyifT|=¬ψ
s0However,by(REP):T|=s0ifandonlyifT|=ψ
s0Therefore:T|=ψ
s0ifandonlyifT|=¬ψ
s0Hence,if(DIAG)and(REP)aretrue,Tissemanticallyinconsistent.SoifTisasemanticallyconsistentextensionofQ,(REP)isnottrue.ThereisthusnoψinthelanguageofTthatsemanticallydefinesthesetofsentencesssuchthatT|=s.IfwefurtherassumethatTistrue(andthat|=preservestruth),wehaveTarski’sindefinabilitytheorem:notheorythatextendsQcansemanticallydefinethesetofitsowntruths.Thisargumentmakesnouseofanyassumptionthatψhasanexplicitdefinitionthat,whenaddedtoT,impliestheT-sentencesψ
s↔s.Theassumptionforreductiois,rather,(REP).Thoughbasicallythisargumentisoftenattributedtothepassage,thereareseriousquestionstoberaisedastowhetheritisactuallytobefoundthere.In[Patterson,2008a]Icontrastedtworeadingsofthepassage,whatIcalleda‘semantic’onewhichatleastfindssomeanalogueofthecon-temporaryargumentinthepassage,aversionofwhichcanbefoundin[Patterson,2006],anda‘syntactic’onewhichisclosetothatofferedby[Gómez-Torrente,2004]andwhichfindsnothinglikethecontemporaryargumentinthepassage.([Vaught,1986,871–2]givesasimilarrecon-structionmuchmorebriefly,claiminginaddition,asGómez-Torrentedoesaswell,thatthenotionoftruthreallyplaysnoroleintheargu-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01ment).Atthetime,asstatedthen,Iwassimplyunsurehowtomakesenseofthepassage,sinceneitherreadingseemedabletotoaccountforallofitsfeatures.Ihavesincebecomesomewhatmoresympathetictothesyntacticreading—notenoughsotothinkthatitcandeclarethepassageaswehaveitcoherent,butenoughtothinkthatthesyntacticreading’sargumentmustbewhatTarskihadinmindandthatotherfeaturesof10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE148—#5\nIndefinabilityandInconsistency149thepassageareprobablyjusttobechalkeduptohastycompositionandTarski’sperhapsimperfectgraspontheissuesatthetime.Theresulttobeestablishedisstatedthus:TheoremI(α)Inwhateverwaythesymbol‘Tr’,denotingaclassofexpres-sions,isdefinedinthemetatheory,itwillbepossibletoderivefromitthenegationofoneofthesentenceswhichweredescribedinthecondition(α)oftheConventionT;(β)assumingthattheclassofallprovablesentencesofthemetatheoryisconsistent,itisimpossibletoconstructanadequatedefinitionoftruthinthesenseofConventionTonthebasisofthemetatheory[Tarski,1983a,247].Noticethatthisassumesthat“Tr”hasanexplicitdefinition.Themainargumentthenrunsasfollows.Tarskipointsoutthatthenaturalnum-berscanbedefinedinthegeneraltheoryofclassesandthatarithmeticcanbeexpressedthere;hecommentsontheidentificationofnumberswithclassesofequinumerousclassesandthenremindsthereaderofthepossibilityofenumeratingthesentencesoftheobjectlanguageandtherebyarithmetizingthemetalanguage’sformalsyntax[Tarski,1983a,249].Thencomestheargument:LetussupposethatwehavedefinedtheclassTrofsentencesinthemetalanguage.Therewouldthencorrespondtothisclassaclassofnaturalnumberswhichisdefinedexclusivelyintermsofarithmetic.
3Considertheexpression‘(ιn.φn)∈Tr’.Thisisasententialfunction1ofthemetalanguagewhichcontains‘n’astheonlyfreevariable.Fromthepreviousremarksitfollowsthatwiththisfunctionwecancorrelateanotherfunctionwhichisequivalenttoitforanyvalueof‘n’,butwhichisexpressedcompletelyintermsofarithmetic.Weshallwritethisnewfunctionintheschematicform‘ψ(n)’.Thiswehave:
3(1)foranyn,(ιn.φn)∈Trifandonlyifψ(n).1Sincethelanguageofthegeneraltheoryofclassessufficesforthefoundationofthearithmeticofthenaturalnumbers,wecanassumethat‘ψ(n)’isoneofthefunctionsofthislanguage.Thefunction‘ψ(n)’Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01willthusbeatermofthesequenceφ,e.g.thetermwiththeindexk,‘ψ(n)’=φk.Ifwesubstitute‘k’for‘n’inthesentence(1)weobtain:
3(2)(ιk.φk)∈Trifandonlyifψ(k).1
3Thesymbol‘(ιk.φk)’denotes,ofcourse,asentenceofthelanguage1underinvestigation.Byapplyingtothissentencecondition(α)oftheConventionTweobtainasentenceoftheform‘x∈Trifandonly10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE149—#6\n150AlfredTarski:PhilosophyofLanguageandLogicifp’,where‘x’istobereplacedbyastructural-descriptiveoranyother
3individualnameofthestatement(ιk.φk),but‘p’bythisstatement1itselfforbyanystatementwhichisequivalenttoit.Inparticularwe
3cansubstitute‘(ιk.φk)’for‘x’andfor‘p’—inviewofthemeaning1ofthesymbol‘ιk’—thestatement‘thereisannsuchthatn=kandψn’or,simply‘ψk’.Inthiswayweobtainthefollowingformulation:
3(3)(ιk.φk)∈Trifandonlyifψk.1Thesentences(2)and(3)standinpalpablecontradictiontooneanother;thesentence(2)isinfactdirectlyequivalenttothenegationof(3).Inthiswaywehaveprovedthefirstpartofthetheorem.Wehaveprovedthatamongtheconsequencesofthedefinitionofthesymbol‘Tr’thenegationofoneofthesentencesmentionedinthecondition(α)oftheConventionTmustappear.Fromthisthesecondpartofthetheoremimmediatelyfollows[Tarski,1983a,250–1].Thesyntacticreadingtakesthepassagetobearelativelystraightfor-wardargumentforthetheoremasstated,thoughithastopositsomeexpositoryhiccups.Itrepresentsthepassageasarguingsomethinglikethis:Wecanenumeratetheexpressionsoftheobjectlanguageinthemet-alanguage.Nowconsiderthemetalanguageopensentence‘thenthsentenceisnotF’,forarbitraryF.arithmetizationtellsusthatthisisequivalentinthemetalanguagetosomearithmeticopensentenceψ(n).Hencewehave‘ThenthsentenceisnotF↔ψ(n)’.Alsobyarith-metization,ψ(n)isinourenumeration;letussupposeitisthekthexpression.Instantiatingntok,wehave(2)‘thekthsentenceisnotF↔ψ(k)’.Now,supposethatFistheexpressionwe’veintroducedbyadefinitionthatsatisfiesConventionT.ThentheT-sentence(3)‘thekthsentenceisF↔ψ(k)’followsfromthedefinitionoftruthandtherestofthemetatheory.But(2)isthenegationoftheT-sentence(3),andsoourtheoryisrenderedinconsistentbytheassumedintuitivelyadequatedefinitionof‘F’.Thisargumentmakessense,showsroughlywhattheexplicitstatementofthetheoremsaysistobeshown,andrunsasitshouldrunifthepointCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01isanapplicationofageneralpointaboutarithmetizationbeingsufficientforthemetatheorytoimplythenegationofaT-sentence.However,thereisagreatdealaboutthepassagethatdoesn’tsitwellwiththereading.Tarskibeginstheargumentbyassumingthatthe“wehavedefinedtheclassTrofsentencesofthemetalanguage”.Ifthepointweresim-plythesyntacticoneaboutdiagonalizationandtheT-sentences,there10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE150—#7\nIndefinabilityandInconsistency151wouldbenoneedforthis:thepointneedsimplybethatforeverypredicateFthereisasentencessuchthats↔F
sisatheorem.Gómez-Torrente’s“anachronisticreconstruction”representstheargumentasdoingexactlythis[Gómez-Torrente,2004,31]—Gómez-Torrentesim-plydoesn’taddressthefactthatTarskinotonlyassumesthat“Tr”isatissue,butthattheassumptionisthattheclassTrofsentenceshasbeendefinedinthemetalanguage(theGermanandPolishversionsagreeonthis).Moreworryingly,Tarski’sargumentfor(α)ofTh.1isdonewithdis-playedthesis(2)(asGómez-Torrentenotes[Gómez-Torrente,2004,34]).Buthegoesontoarriveat(3)andthensays“inthiswaywehaveprovedthefirstpartofthetheorem”.Butwhat,then,istheargumentfor(3)theretoestablish?Likewise,healsosaysthat“fromthisthesecondpartofthetheoremimmediatelyfollows”.Butthesecondpartofthetheo-rem,(β),saysthatifwedoaddadefinitionthatimpliestheT-sentencestothetheory,thetheorywillberenderedinconsistent.Butthisdoesn’tjustimmediatelyfollow,giventhat(3)isalreadyonboard:ithasalreadybeenshown—if,thatis,thetextthatprecedes(3)isanargumentthatifweaddanintuitivelyadequatedefinitionoftruththeT-sentencewillbeatheorem.Furtherproblemsarisewhenwetrytoaddresstheseissues.Supposeweholdthat(3)andthesentencesleadinguptoitaretheremerelytoestablishthat(2)isthenegationofaT-sentence.ThiswouldmakesenseofTarski’ssayingthat(β)remainstobeshown,butfollowsimmediately:(3)hasn’tbeenclaimedtofollowfromthemetatheoryyet;itwasjustmentionedbywayofestablishingthat(2)isthenegationofaT-sentence.However,itmakeslesssenseofwhatstatus(1)through(3)aresupposedtohave.Tarskisaysinanoteon249that“insteadofsayingthatagivensentenceisprovableinthemetatheory,weshallsimplyassertthesen-tenceitself”intheproof-sketch.Thiswouldseemtomeanthatwe’retotaketheclaimtobethat(1)through(3)followfromthemetathe-ory.Thismakessensefor(1)and(2),butitsticksuswiththeviewthat(3)isclaimedtofollowfromthemetatheoryasare(1)and(2).Moregenerally,thereisnoassumptionintheproofthat“Tr”hasadefinitionadequateaccordingtoConventionT;thetheoremissupposedtocon-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01cernapredicateformallydefined“inwhateverway”,and,again,at251we’retoldthatitremains(easily)tobeshownthatifweaddadefinitionadequateaccordingtoConventionTthemetatheoryisrenderedincon-sistent.Hencethesuggestedwayofmakingsenseof(3)’sroleandtheclaimthat(β)islefttobeshownisn’tcompatiblewith[Tarski,1983a,249note1].10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE151—#8\n152AlfredTarski:PhilosophyofLanguageandLogicFinally,thereisafurtherissueofhistoricalaccuracyhereconcerninginexactlyhowdiagonalizationplaysaroleintheargument.Boththereadingsin[Gómez-Torrente,2004]and[Vaught,1986,871]attributeanargumentinvolvingaageneralformofthediagonallemmatoTarski’stext.NowTarskidoesspeakof“applyingthediagonalprocedurefromthetheoryofsets”intheshorterproof-sketchat[Tarski,1983a,248],buttheargumentofthelongerproof-sketchfailstoinvolveanygeneralclaimamountingto(DIAG).Tarskicites[Fraenkel,1928,48],whichdiscusses,asTarskisays,theuseofdiagonalconstructionsinsettheory,butthereisnostatementofthediagonallemmathere—notsurprisingly,sinceFraenkelisconcernedwithsettheoryratherthanproof-theory.[Vaught,1986,871]notesthesimilarityto[Gödel,1967b],butGödel,likeTarski,concernshimselfonlywithparticularinstancesof(DIAG).GödelhimselflaterattributedthegeneraldiagonallemmaneithertohimselfnortoTarski,buttoCarnap’sLogicalSyntaxinafootnoteaddedlatertothewrite-upofhis1934Princetonlectures[Gödel,1986,363note23].Thus,noreadingthatattributessomeappealtoageneraldiagonallemmatothepassagecanbeaccurate.3Thesearetheproblemsfacedbythesyntacticreading.Asmentioned,I’vecomearoundtothinkingthatthesyntacticreadingmustcapturewhatTarskihadinmind.Beforesayingwhy,letuslookbrieflyathowthesemanticreadingwouldinterpretthepassage.Theideaisthattheargumentlookslikethis:SupposethatTristhesetoftruthsoftheobjectlanguage.Nowbydiagonalizationthereisasentencessuchthats↔¬Tr
sisatheoremofourtheory.(Sofar,thisisthereadingof(1)and(2).)Now,sinceTristhesetoftruthsoftheobjectlanguage,ifsisatheorem,soisTr
sandviceversa,sincewe’reassumingthatourtheoryistrueandassumingthatTrdefinesthesetoftruths.Sos↔Tr
sisalsoatheorem.(That’s(3).)Hence,ifinfactTrsemanticallydefinesthesetoftruths,ourtheoryisinconsistent.Soifourtheoryisconsistent,nothinginitsemanticallydefinesthesetoftruths.Themeritofthisreadingisthatitmakestheprogressionfrom(1)to(3)onpage250asensibleargumentforaconclusionthatTarskicouldCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01havebeenprovingin1931inviewofGödel’swork:itmakesuseoftheassumptionthatthesetTroftruthsisdefinedsemantically,itgives(3)anobviousjobtodo,itmakessenseofTarski’sclaiminthenoteto249thathe’llstatethesesmeaningbythemthattheyaretheoremsofthemetatheory,anditmakessenseofthepassagebetween(2)and(3)asanargumentthat(3)isatheoremwithoutimportingtheassumptionthat10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE152—#9\nIndefinabilityandInconsistency153itissobecause“Tr”hasaformaldefinitionthatisintuitivelyadequateaccordingtoConventionT,therebyleavingthepointaboutConventionTyettobeshown,justasitisclaimedtobeon251.(WhyisConventionTevenmentionedthen?SimplytoremindthereaderoftheT-sentences:noticethatTarskijustsays“applycondition(α)ofConventionT”notsomethinglike“applytheassumptionthatTrhasanexplicitdefinitionthatisintuitivelyadequateaccordingtoConventionT”andthathesim-ilarlyreferstotheT-sentencesthiswayinthestatementsofTheoremsIIandIII.)Theweaknessofthisreadingisthattheargumentitattributestothepassagedoesmorethantoprove(α)andsetupforanimmediateproofof(β):itprovessomethingthatTheoremIsaysnothingabout,namelythatnoone-placeopensentenceofthemetatheorycansemanticallydefinethesetoftruths.TheoremI,asstated,simplyisn’taboutthis.ThereasonthatIhavecometofavorthesyntacticreadingisthatIhavecomearoundtothestoryI’mtellinginthisbook:itwasnotuntilwellafterthe1931publicationoftheworkthatTarskireallycametoseesemanticsashehadintroduceditashiscentralconcern.In1931Tarskiwasconcernedwithgivingformaldefinitionsofsemantictermsthatwereintuitivelyadequatebyhisstandards.Toshowthatnosuchdefinitionispossibleinthecaseathandin§5,Tarskionlyneedsthesyntactictheorem,andsincehissemanticswasmerelyatoolforcraftingformaldefinitionsofsemanticnotions,hesimplywouldn’thaveseensemanticdefinitionofthesetoftruthsasthecentralissue.4SothereadingofthepassageIwouldnowfavoristhatTarskiatleastwantsthe“syntactic”result,sincethatistheresultstatedbyTh.I,andsincethatiswhat,onmygeneralstoryhere,hewouldhavebeeninter-estedinin1931–3.Thetextwehaveon250,however,isfarfromaclearattempttodothat.Noticealsothatin1931–3Tarskiisnotyetworkingcomfortablywithadistinctionbetweensyntacticandsemanticconse-quence.Sinceitwasonlyseveralyearslaterthathewastoformulatethelatternotionandclearlytodistinguishitfromtheformer,itisn’tterriblysurprisingtofindhimfailingtobeclearaboutitinthepassageathand.Inlightofthistheapparentshiftfromasyntactictheoremtoaseman-ticproofseemsexplicable,sincetheargumenton250isclearlyliftedCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01from§1of[Gödel,1967b].Gödel’sinformalpresentationoftheresultthathegoesontoproveindetailrunsasfollows:considertheclassKofnumbersnsuchthatitisn’tprovableinP(AversionofSTTplusPeanoArithmetic)thatthenthformulainsomeenumerationofformu-lasappliestothem.SomeformulaofP,saytheqth,definesK;callitR(q),andlet[R(q),n]betheformulainwhichthefreevariableinR(q)is10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE153—#10\n154AlfredTarski:PhilosophyofLanguageandLogicreplacedbyanumeralforn.Nowsupposethat[R(q),q]isprovableinP.Thenq∈K.Butbyhypothesis,ifq∈Kthen[R(q),q]isnotprovableinP.Contradiction.SoSupposethat¬[R(q),q]isprovableinP.Then,byhypothesisq∈/Kandthus¬[R(q),q]isprovableafterall.Contradiction.Soneither[R(q),q]nor¬[R(q),q]isprovableinP.Gödelgoesontonotethatsince[R(q),q]isnotprovable,itistrue[Gödel,1967b,598].Thisargumentinvolvestheback-and-forthbetweenaformulaandthesetitsemanticallydefinesthatischaracteristicofthemoderntakeontheresults,andthatisattributedtopage250ofTarski’sworkwithsomejus-ticebythesemanticreading.SinceTarskibyhisownadmission[Tarski,1983a,275]wasspurredtoaddTheoremIandtheproofsketchesby[Gödel,1967b],wecanimaginehimworkingbynotingthat,ashesays,“theresultsobtainedforthesystemPcaneasilybecarriedovertothepresentdiscussion”[Tarski,1983a,248].Withthegoalofshowingthatnointuitivelyadequatedefinitionoftruthcouldbesuppliedforthelan-guageofthegeneraltheoryofclasses,TarskineededtorecastGödel’sargumenttoconcerntruthratherthanprovabilityandtoinvolvethederivationofthenegationofaT-sentence;whatresultsissomethingthatbeganlifeinGödel’shandsasconcerningsemanticdefinitionofasetbutbecomes,inTarski’shandsagarbledcombinationofthatwithapointaboutformaldefinition.Ifwerememberthatthesethingswerenewthen,theunevencharacterofthepassage—which,onbalance,seemstobeaproofofthesemanticresultinsertedintoadiscussionofthesyntac-ticresult—isunderstandable:Tarskiwastryingtocarryoveranargumentforthesemanticresultandputittosyntacticpurposesanddidn’tdoaparticularlythoroughjobofit.HeseesthatthefirstpartofGödel’sargu-mentcaneasilyberecastasaproofthatthenegationofaT-sentenceisprovablewithsuitableadjustments,butleavesinfromGödel’spresenta-tiontheassumptionofthedefinitionofasetandsomeargumentthataninconsistencyresultsgiventhedefinitionofthesetandtheresultofthefirstpartoftheproof.Indeed,afurtheraspectof247–251mightexplainwhyanapparentproofofamoreGödelianresultappearsastheproofofTheoremI.Afterintroducingthetheorem,onpages247–8Tarskigivesaverygeneralsketchofthebasicapplicationofdiagonalizationtoproducethenega-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01tionofaT-sentence.Hethen,ratheroddly,saysatthetopof249“Weshallsketchtheproofalittlemoreexactly”andstartsoverwiththeproof-sketchwehavebeendiscussing.5Thisstutteringpresentationinvitesthefollowinghypothesis:TarskifirstwrotetheadditionstoCTFLthatfollowpage246usingonlythefirst,shortproofsketch,verysoonafterlearningofthebasicthrustofGödel’sresults.Perhapsheevensenttheextension10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE154—#11\nIndefinabilityandInconsistency155of§5totheprinterinthatform.AftercomposingthismaterialTarskicontinuedtostudyGödel’sresults,tryingoutvariousthingshimselfandexploringvariouslines;sinceGödel’sdiscussionreallyisasemanticone,andsinceTarskiwaspreparedtounderstanditthatway,suchconsidera-tionswouldhavebeentovaryingextentsalongGödel’ssemanticlines.AtsomepointitoccurredtoTarskithatitwouldbegoodtohavesome-thingalittlemoreconcreteasaproof-sketchthanwhatwenowhaveonpage248,soheinsertedthesecond,longerproof-sketch,whichwecanthinkofasperhapssomethingtakenfromworkingnotes.Somehowhewasn’tcarefultomakethelateradditionmatchthesurroundingtext.Whateverthereason,thiswouldexplainthewayinwhichthemate-rialon250ismoreorlessanindependentproofofaresultdifferentfrom,butcloselyrelatedto,TheoremI,anditspoorfitwithwhathesayshehasdoneandhaslefttobedoneatthetopof251.Infavorofthisreading,notethatifoneskipsfromthelastsentenceof248tothesentencethatbeginsonthelastlineof250,oneproducesadiscussionofTheoremIandits“proof”thatmakesperfectlygoodsense:onpage248we’retoldthatdiagonalizationproducesthenegationofaT-sentence,therebyestablishing(α),andthen251makesthepointthat(β)followseasily,sinceaddinganintuitivelyadequatedefinitionwillbringthecor-respondingT-sentenceinitswake.Whatliesbetweenthosetwopointsisapassagethatstandsaloneastheproofofasemanticindefinabilitytheoremmuchmorelikethecontemporarytakeon“Tarski’sTheorem”.Thisisspeculative,ofcourse.Mybestsourcestellmethattheyknowofnoextantproofsheetsormanuscriptsthatwouldanswerquestionsabouttheorderofcompositionofthepassages.Butthestorymakesenoughsenseoftheproblematicfeaturesofthepassagefrom247–251thatIaminclinedtoacceptituntilevidenceagainstitcomesin.5.1.3TheoremIandIntuitionisticFormalismAsIsaidabove,IhavecometothinkthatthetheoremaccordingtothesyntacticreadingiswhatTarskiintendedtoprove.ThisisbecauseitmatchesmuchmorecloselytheanimatingconcernsofCTFLthandoesthesemanticresult.BywhatIhavepositedistheoriginalendofthework—thetopofpage246inLogic,Semantics,Metamathematics—TarskiCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01isinthefollowingposition.Hisgoalisexplicitlytodefine“∈Tr”sothattheT-sentencesinvolvingitaretheorems;therequirementisthatthedef-initionshouldbelogicallyinorder(“formallycorrect”)andthatwhenaddedtothebackgroundtheorytheT-sentencesshouldbecometheo-rems(“intuitivelyadequate”).GiventheembeddingoftheconstructioninSTTandTarski’scommitmenttothisembeddingasconsonantwith10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE155—#12\n156AlfredTarski:PhilosophyofLanguageandLogictherequirementsofthetheoryofsemanticalcategoriesand,thereby,thebasicstructureofintuitivemeaning,itturnsoutthatonlyforlanguagesoffiniteordercansatisfactionbedefinedonthewaytodefiningtruthassatisfactionbyallsequences.Tarski’sviewrequiresthattheT-sentencesbetheorems;therequire-mentthat“istrue”explicitlybedefined,andmoreoversodefinedagainstasyntactic(ultimately,arithmetic)backgroundtheorycomesnotfromthedesirenottoexpresstheintuitiveconceptionoftruth,buttoexpressitconsistently—thatis,towardoffsemanticparadox.ThetheoryofsemanticcategoriesasTarskiappliesit—onlyasapplyingtoexpressionswith“independentmeaning”and,forthepurposesofthework,onlytovariablesandpredicates—doesn’tpassanyparticularverdictontheT-sentencesthemselvestotheeffectthatthemetalanguagecancontainthemonlyifitisofhigherorderthantheobjectlanguage.SowhatTarskihaswiththepassagetoalanguageof“infinite”orderissimplythatiftheT-sentencesaretobeintroduced,itcan’tbebywayofanyrelationofsatisfaction,whichaccordingtothetheoryofsemanticalcategoriesdoesneedtobeofhigherorderthanallthevariablesoftheobjectlanguage.Heisthusleftwithaquestion:isthereanywaytogettheT-sentencesconsistentlyintothemetalanguagewhentheobjectlanguageisofinfi-niteorder?Isthere,inparticular,anywaytointroduceadefinitionthatimpliestheT-sentencesconsistentlyinsuchacase?TheoremIandtheargumentforitfunctiontoshowthatitisn’tjustsomeweaknessoftheapproachofdefiningtruthbyrecursiononsatisfactionthatleadstheprojecttogrindtoahaltatinfiniteorder;diagonalizationshowsthattheT-sentencescannotbetheoremsforanyarithmeticallydefinablepredicateinfullSTT.Intheabstract,thisleavesachoiceofviews:eithertruthisn’tarithmeticallydefinable,ortheT-sentencesaren’ttheorems.ThepointofinteresttousisthatTarskiacceptsthefirstratherthanthesecondview,therebysettinghimselfatoddswithnearlyallcontemporarylogicaltreatmentsofsemanticpara-dox.AsTarskialreadybeginstorecognizemoreclearlyinthePostscript,therealupshotofGödel’sresultsisthatthemetatheorycanhaveapred-icatethatsemanticallydefinesthesetoftruthsoftheobjectlanguageonlyifthemetalanguagecansemanticallydefinemoresetsthantheCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01objectlanguagecan,sincetheresultisthatnopredicateoftheobjectlanguagecandoso.Now,supposewe’reinterestedinthecaseinwhichthemetatheorycan’tdefinemoresets,saybecausewe’reinterestedinthecaseinwhichalanguageis,inTarski’s1944terms,“semanticallyclosed”—perhapsbecausewethinkthatnaturallanguagesaresemanti-callyclosedandwewanttomodeltheminthisrespect.Herewehavethe10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE156—#13\nIndefinabilityandInconsistency157twooptionsthatconfrontTarskiininterpretinghisresultin1931–3:wecansaythatthereisnosetoftruths,orwecansaythatthereisone,butthatthatatleastsomeoftheT-sentencesareuntrue.Inthecontempo-raryliterature,thesecondoptionisfavored.Onthetreatmentin[Kripke,1975],forinstance,theobjectlanguagecontainsapredicatethatseman-ticallydefinesthesetofitsowntruths,butsomeoftheT-sentencesintheobjectlanguageareuntrue(they’reuntrueforungroundedsentences,whichincludeparadoxicalsentences).Ofcourse,inKripke’smetalan-guagetheT-sentencesaretrue,butthemetalanguagecandefinemoresetsthantheobjectlanguagecan,inparticularthesetofuntruths.Mostcon-temporaryworkfollowsKripkeingivingupsomeT-sentencesinordertogetapredicatethatsemanticallydefinesthesetoftruthsintotheobjectlanguage.Tarskitakestheotherroute.Thewholepointoftheexercise,forTarski,istoexpresstheconceptoftruthinaccordwiththesemanticaldefinitioninadeductivetheory.ThisrequiresthattheT-sentencesbetheorems.IfwearetoproceedwiththisprojectinthefaceofTheoremI,theonlyoptionistogiveuponthearithmeticdefinabilityoftruth.ThisisjustwhatTarskidoes.Heexpresseshimselfasfollows:TheresultreachedinTh.Iseemsperhapsatfirstsightuncommonlyparadoxical.Thisimpressionwilldoubtlessbeweakenedassoonaswerecallthefundamentaldistinctionbetweenthecontentofthecon-cepttobedefinedandthenatureofthoseconceptswhichareatourdisposalfortheconstructionofthedefinition…Onlythankstothespecialmethodsofconstructionwhichwedevelopedin§§3and4havewesucceededincarryingouttherequiredreductionofsemanticalconcepts,andthenonlyforaspec-ifiedgroupoflanguageswhicharepooringrammaticalformsandhavearestrictedequipmentofsemanticalcategories…TheanalysisoftheproofofTh.Isketchedaboveshowsthatthiscircumstanceisnotanaccidentalone.Undercertaingeneralassumptions,itprovestobeimpossibletoconstructanadequate(richtige,trafnej)definitionoftruthifonlysuchcategoriesareusedwhichappearinthelanguageunderconsideration[Tarski,1983a,251–4](corrected).Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Theresultisapttostrikeustodayasunremarkable.Why,forTarskiatthetime,wasit“uncommonlyparadoxical”?Preciselybecausehewasperfectlyconfidentthattheintuitiveconceptoftruthcouldbeappliedtomeaningfulsentences,andthelanguageofthegeneraltheoryofclassesconsistsofmeaningfulsentences.Sowhatis“paradoxical”abouttheproofisitsapparentclashwiththesemanticaldefinitionoftruth,10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE157—#14\n158AlfredTarski:PhilosophyofLanguageandLogicaconceptionthatTarskitooktobeobviouslycorrect.Therestofthepassageexplainsawaythatappearanceofparadox:TheoremIdoesn’timpugnourintuitiveconceptionoftruth,itimpugnstheattempttoexpressitinatheorythatcontainsonly“structuraldescriptive”terms,thatis,itimpugnstheassumptionthatanexpressionthatexpressesthecontentoftheconceptoftruth(asexpressedinthesemanticaldefini-tion)canbeeliminativelydefinedinamathematicalbackgroundtheorywhentheobjectlanguagecandefineallthesetsthatthemetalanguagecan.Thevalidityofthesemanticaldefinitionoftruthandtherequire-mentofthetheoremhoodoftheT-sentencesgoesunquestionedinallofthis.Tarskicontinues:AninterpretationofTh.Iwhichwentbeyondthelimitsgivenwouldnotbejustified.Inparticularitwouldbeincorrecttoinfertheimpos-sibilityofoperatingconsistentlyandinagreementwithintuition(mitderIntuitionübereinstimmenden,zgodnegozintuicjka˛)withsemanticalconceptsandespeciallywiththeconceptoftruth[Tarski,1983a,255].Notetherecurrenceofthephrasethatendedtheoriginalmanuscript[Tarski,1983a,246],somethingobscuredbyWoodgerwho,followingBlaustien,variesthetranslationof“zgodnegozintuicjka˛”betweenthetwopassages.5.1.4AxiomaticsemanticsExplicitsyntactic(arithmetic)definitionbeingoutofthequestion,Tarskinotesthatthatitispossibletodefinetruthforsublanguagesoffiniteorderandthencontemplatesamovetoaxiomatictruth-theoryforthelanguageofthegeneraltheoryofclassesasawhole:TheoremIIIIftheclassofallprovablesentencesofthemetatheoryiscon-sistentandifweaddtothemetatheorythesymbol‘Tr’asanewprimitivesign,andalltheoremswhicharedescribedinconditions(α)and(β)oftheConventionTasnewaxioms,thentheclassofprovablesentencesinthemetatheoryenlargedinthiswaywillalsobeconsistent[Tarski,1983a,256].Sincethemetatheoryisonlycontradictoryifsomefinitesubsetofitis,Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01andsinceanyfinitesetofT-sentencescanbemadetheconsequencesofanexplicitdefinition,thesentencestheseT-sentencesconcernbeingofonlyfiniteorder,addingdenumerablymanyT-sentenceswillnotrenderthemetatheoryinconsistent.However,asTarskigoesontonote,theresultisbyhisstandardslogi-callyandconceptuallyunsatisfactory.Generaltheseswhoseprovability10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE158—#15\nIndefinabilityandInconsistency159Tarskidemandsaren’tavailable—onecannot,forinstance,provethatnosentenceoftheobjectlanguageisbothtrueanduntrue,thoughonecanproveofeachthatitisn’tbothtrueanduntrue.6Simplyaddingwhatwewantintheformofadditionalaxiomswon’taddresstherealissue,whichforTarskiisthattheaxiomsystemisn’tprovablycategoricalinthesenseof§2ofDC:Itseemsnaturaltorequirethattheaxiomsofthetheoryoftruth,togetherwiththeoriginalaxiomsofthemetatheory,shouldconsti-tuteacategoricalsystem.Itcanbeshownthatthispostulatecoincidesinthepresentcasewithanotherpostulate,accordingtowhichtheaxiomsystemofthetheoryoftruthshouldunambiguouslydeter-minetheextensionofthesymbol‘Tr’whichoccursinit,andinthefollowingsense:ifweintroduceintothemetatheory,alongsidethissymbol,anotherprimitivesign,e.g.thesymbol‘Tr’andsetupanalo-gousaxiomsforit,thenthestatement‘Tr=Tr’mustbeprovable.Butthispostulatecannotbesatisfied.Foritisnotdifficulttoprovethatinthecontrarycasetheconceptoftruthcouldbedefinedexclusivelybymeansoftermsbelongingtothemorphologyoflanguage,whichwouldbeinpalpablecontradictiontoTh.I[Tarski,1983a,258].SincethetheorythatresultsfromaddingtheT-sentencesandanyothersentenceswepleaseiseitherinconsistentornotprovablycategorical,wecanonlyincompletelyexpresstheconceptoftruthaxiomatically.Soinmovingtoaxiomatictruththeory,wegiveupallofTarski’slogicalandexpressivegoals:ourtheoryisn’tfinitelyaxiomatizable,wedon’texpressthecontentoftheconceptoftruthinatheorythathasunprob-lematicallymathematicalprimitiveexpressions,andevenatthatcostwedon’tcompletelyexpressit.Tarskiistherebyunenthusiasticabouttheprospectsforaxiomatictruththeory.Nevertheless,axiomatictruththeoryisadequate(trafny),aswelearninthesisCat[Tarski,1983a,266],[Tarski,1986a,184],[Tarski,1933,114].AnotheradditiontoaxiomatictruththeoryTarskiconsidersistheω-ruleor“ruleofinfiniteinduction”[Tarski,1983a,259]thatallowsonetoinferanumericalgeneralizationfromthe(denumerablyinfinite)setofitsinstances.Therulewasofsignificanceduetoitsrolein[Gödel,Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-011967b]and,aswewillsee,itplayedamajorroleinCarnapandTarski’sassimilationofGödel’sresults;Tarskihimselfwasconsideringitssignif-icancefromatleast1926onward[Tarski,1983a,260,note1].Tarskicontemplatesaddingtheruletoaxiomatictruththeoryassofardis-cussed,butnotesthatduetoits“non-finitistnature…itmaywellbedoubtedwhetherthereisanyplacefortheuseofsucharulewithin10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE159—#16\n160AlfredTarski:PhilosophyofLanguageandLogicthelimitsoftheexistingconceptionofthedeductivemethod”[Tarski,1983a,260].Hethennotesthattheruleisintuitivelyvalidinthesensethat“italwaysleadsfromtruesentencestotruesentences”(afactthatwillplayarolelater;§7.4.2),andthataddingittoaxiomatictruth-theoryremediesmanyofitsweaknesses,e.g.thelackofcategoricity[Tarski,1983a,261].But,hecautions,itisnotknownwhetheraxiomatictruththeorysupplementedbytheω-ruleisconsistent;whatisknown,fromTh.I,isthattruthremainsindefinableinsyntaxinthecontemplated“deductive”(inanextended,non-finitarysense)theory.(Thisquestiongetssettled,inasense,inthePostscript,butseeourremarksonthedif-ferencebetweentheGeneralTheoryofClassesandCarnap’stheoryoflevelsin§6.1.)5.2InconsistencyineverydaylanguageTarski’sviewsontheinconsistencyof“colloquiallanguage”havebeenthesubjectofincomprehensionandderisionfordecades.Wearenowinapositiontounderstandthem.In§4.1.3wediscussedtheargumentof§1,focusingonthewayinwhichstraightforwardemploymentofthe“semanticaldefinition”oftruthwouldnothavesuitedTarski’spur-poses,sincethesemanticaldefinitionisn’tformallycorrect.Wecannowlookintothetreatmentof“colloquiallanguage”inthesection.Hav-inggonethroughthedifficultiesinvolvedinworkingoutthesemanticaldefinition,Tarskiseguesintoaremarkableconclusion:Acharacteristicfeatureofcolloquiallanguage(incontrasttovariousscientificlanguages)isitsuniversality.Itwouldnotbeinharmonywiththespiritofthislanguageifinsomeotherlanguageawordoccurredwhichcouldnotbetranslatedintoit;itcouldbeclaimedthat‘ifwecanspeakmeaningfullyaboutanythingatall,wecanalsospeakaboutitincolloquiallanguage’.Ifwearetomaintainthisuniversalityofeverydaylanguageinconnexionwithsemanticalinvestigations,wemust,tobeconsistent,admitintothelanguage,inadditiontoitssen-tencesandotherexpressions,alsothenamesofthesesentencesandexpressions,andsentencescontainingthesenames,aswellassuchCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01semanticexpressionsas‘truesentence’,‘name’,‘denote’,etc.Butitispresumablyjustthisuniversalityofeverydaylanguagewhichistheprimarysourceofallsemanticalantinomies,liketheantinomiesoftheliarorofheterologicalwords.Theseantinomiesseemtoprovideaproofthateverylanguagewhichisuniversalintheabovesense,andforwhichthenormallawsoflogichold,mustbeinconsistent.This10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE160—#17\nIndefinabilityandInconsistency161appliesespeciallytotheformulationoftheantinomyoftheliarwhichIhavegivenonpages157and158,andwhichcontainsnoquotation-functionwithvariableargument.Ifweanalysethisantinomyintheaboveformulationwereachtheconvictionthatnoconsistentlan-guagecanexistforwhichtheusuallawsoflogicholdandwhichatthesametimesatisfiesthefollowingconditions:(I)foranysen-tencewhichoccursinthelanguageadefinitenameofthissentencealsobelongstothelanguage;(II)everyexpressionformedfrom(2)byreplacingthesymbol‘p’byanysentenceofthelanguageandthesym-bol‘x’byanameofthissentenceistoberegardedasatruesentenceofthislanguage;(III)inthelanguageinquestionanempiricallyestab-lishedpremisshavingthesamemeaningas(α)canbeformulatedandacceptedasatruesentence.Iftheseobservationsarecorrect,thentheverypossibilityofaconsis-tentuseoftheexpression‘truesentence’whichisinharmonywiththelawsoflogicandthespiritofeverydaylanguageseemstobeveryquestionable,andconsequentlythesamedoubtattachestothepossibilityofconstructingacorrectdefinitionofthisexpression[Tarski,1983a,164–5].Thepassageprovokesanumberoffamiliarcomplaints:1.Naturallanguagescan’tbeinconsistentbecauseonlytheoriescanbeinconsistent,andlanguagesaren’ttheories[Putnam,1975,73],[Burge,1984,83–4],[Soames,1999,63].2.Naturallanguagesobviouslyaren’tinconsistentsinceiftheywerecommunicationinthemwouldbeimpossible,buttheveryfactthatTarski’sviewcanbestatedshowsthatcommunicationinnaturallanguageispossible[Sloman,1971,133–4].3.Naturallanguagescan’tbeinconsistentbecauseaninconsistentlan-guagewouldbeoneinwhichcontradictionsweretrue,butifcontra-dictionsaretrueeverythingisthecase,and,clearly,itisnotthecasethateverythingisthecase[Soames,1999,55,151].Infact,asIwillargue,Tarski’sviewsonnaturallanguagearequiteCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01workableandperfectlysensibleinviewofhisconceptionofmeaning.Tarskiseemstotakeitasobviouswhat“everyday”or“colloquial”lan-guageis;presumablyhehasfamiliar“natural”languageslikePolishandEnglishinmind,butinfacthisremarksfailtosettleeventhis.Asisstandard,Iwilltakeitasunderstoodthatwecaninterprethisremarksintermsofsuchlanguages.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE161—#18\n162AlfredTarski:PhilosophyofLanguageandLogic5.2.1InconsistentKotarbinskianconventions´Despitetheextenttowhichtheyhavebeenpilloried,Tarski’sviewsoneverydaylanguageareperfectlysensibleinlightoftheconceptionofmeaningandcommunicationwithwhichheworkedaround1930.RecallthefollowingpointsaboutKotarbi´nski’sconceptionofmeaning:•Theutteranceofaparticularsentencedirectlyexpresseswhateverthespeakerintendsittoexpress,alongwhatwewouldnowthinkofasbroadlyGriceanlines.•Asentenceindirectlyexpresseswhatevertheconventionsofeverydaylanguageimplyitwouldbeusedcorrectlydirectlytoexpress.•Theconceptintendedtobeexpressedbyatermisitsconnotation,thesetofpropertiesthatwouldbegraspedbysomeonewhouseditcorrectlyinaccordwiththeconventionsdirectlytoexpressathought.•Therecanthereforebeamismatchbetweentheformalroledeter-minedbytheconventionsforatermandtheconceptitisintendedtoexpress.ThiscombinationofviewseasilygivesrisetoTarski’sclaimsabouttheinconsistencyofcolloquiallanguage.First,itisentirelypossiblefortheconventionsgoverningalanguagetobeeitherinconsistentinthem-selvesorinconsistentgivencontingentfact.Thiswillhappenpreciselywhentheconventionsinvolveexpressionsthatarethemselvesboundupwiththoughtsaboutwhattheconventionsrequire—thatis,preciselywhensyntacticandsemanticexpressionsareintroduced.WecanseehowthisplaysoutentirelywithinKotarbi´nski’sconcep-tionbylookingatasimpleconstructionoftheGrellingalonghislines.Considerthefollowingsetofconventions:1.Someonecorrectlyaccepts‘Xy’iffshethinksthat‘X’appliestoyandcorrectlyaccepts‘¬Xy’iffshethinksthat‘X’doesnotapplytoy.(Correctlyacceptinganatomicsentencerequiresthinkingthatthepredicateappliestotheobjectsdenoted,whilecorrectlyacceptinganegatedatomicsentencerequiresthinkingthatthepredicatedoesnotapplytotheobjectsdenoted.)Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-012.Someonecorrectlyaccepts‘Axy’iffshethinksthatxappliestoy.(‘A’isanapplicationpredicate.)3.Someonecorrectlyaccepts‘Hx’iffshecorrectlyaccepts‘¬Axx’(‘H’isaheterologicalitypredicate.)4.Someonecorrectlyaccepts‘Xh’iffshethinksthat‘X’appliesto‘H’.(‘h’namestheheterologicalitypredicate.)10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE162—#19\nIndefinabilityandInconsistency1635.Either‘¬p’or‘p’iscorrectlyaccepted.(Bivalence:eitheraclaimoritsnegationisalwayscorrectlyaccepted.)Theseconventionsdeterminethatsomeonecorrectlyaccepts‘Hh’ifandonlyifshethinksthat‘H’appliestoh,by(1),andhence,by(4)ifandonlyifshethinksthat‘H’appliesto‘H’.By(2)and(3)theyalsodeterminethatsomeonecorrectlyaccepts‘Hh’ifandonlyifshethinksthathdoesnotapplytoh,andhence,by(4),ifandonlyifshethinksthat‘H’doesnotapplyto‘H’.Sosomeonewillcorrectlyaccept‘Hh’ifandonlyifshethinksboththat‘H’appliestoitselfandthat‘H’doesnotapplytoitself.Sofarweseemonlyhavetheunproblematicconclusionthatnobodycancorrectlyaccept‘Hh’withouthavingbeliefsthatimplyacontradic-tion;allthatwouldappeartofollowfromthatisthatnobodycanaccept‘Hh’withouthavingbeliefsthataretrueonlyifsomethingbothisandisnotthecase.However,parallelreasoningestablishesthatnobodycanaccept‘¬Hh’withouthavingbeliefsthatimplyacontradiction,either.By(1),someonecorrectlyaccepts‘¬Hh’ifandonlyifshebelievesthat‘H’doesnotapplytoh,andhence,by(4),ifandonlyifshebelievesthat‘H’doesnotapplyto‘H’.By(2)and(3),someonecorrectlyaccepts‘¬Hh’ifandonlyifshethinksthatitisnotthecasethathdoesnotapplytohandtherefore,eliminatingthedoublenegation,ifandonlyifshethinksthathappliestoh.By(4),suchapersonalsobelievesthat‘H’appliesto‘H’.Thus,‘¬Hh’iscorrectlyacceptedonlybysomeonewhohasbeliefsthatcanonlybetrueifsomethingbothisandisnotthecase.However,by(5),oneofeither‘Hh’or‘¬Hh’iscorrectlyaccepted.Ifeitheris,though,aconventionforconjunctionintroductionwillgetusanexplicitcontra-dictionbeingcorrectlyacceptedandafewmoreconventionsforclassicallogicwillgetusthateverythingiscorrectlyaccepted.SinceKotarbi´nskiandTarskiassumethattheconventionsgoverningtheobjectlanguageinvolveclassicalconnectives,theaboveconventionsspecifyalanguageinwhicheverysentenceiscorrectlyaccepted.Thus,ifonebelieveswhattheseconventionssayonehasbeliefsthataretrueonlyifsomethingbothisandisnotthecase.Furthermore,theconventionsaresuchthatCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01ifoneacceptsthemthenoneoughttoconcludethatallsentencesarealwayscorrectlyaccepted.Clearly,onemighttakethisaccountofconventionstobeoverlysimple;obviousemendationswouldmaketheconventionsthemselvesinconsistent,andotherobviousemendations,e.g.addinganotionofrejection,7wouldchangethelandscapeinotherpertinentways.Our10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE163—#20\n164AlfredTarski:PhilosophyofLanguageandLogicpointhereisthatKotarbi´nskiandTarskiareworkingwithasimplenotionofcorrectacceptancethatmatchestheconceptionoftheoremhoodintheirunderstandingofadeductivetheory(§1.4.3).Thelanguagegov-ernedbytheseconventionsis“inconsistent”inthesensethatanyonewhoagreestotheconventionsiscommittedtologicalfalsehoodsand,ifweaddclassicallogic,toeverysentence’sbeingcorrectlyaccepted.WehavenowreconstructedTarski’sdiscussionofeverydaylanguageusingonlyKotarbi´nski’snotionsofaconventionandindirectstatement,andtheaccountofcorrectassertiontheseconceptionsentail.Alanguageisinconsistentjustincaseitsconventionsjointlydeterminethatbothsomesentenceanditsnegationarecorrectlyaccepted;conventionsthatmakesometermssemanticandothertermssyntactic,inthesensethattheyrefertoexpressions,aresufficientforthis.Letusnownoteanumberofthingsthatdonotfollowfromtheexis-tenceofinconsistentconventionsonKotarbi´nski’sview.First,itdoesn’tfollowthattheintuitivemeaningsofexpressionsgovernedbytheseconventions,thatis,theconceptsexpressedbythem,arethemselvessomehow“inconsistent”.Theconceptexpressedbyatermisitscon-notation,asetofproperties.‘A’connotesthepropertythattwothingshavewhenthefirstappliestothesecond.Thethoughtthatonehasifandonlyifonecorrectlyaccepts‘Axy’isthattheorderedpair(x,y)hastheapplicationproperty.Theremaywellbesuchapropertyeveniftheconventionsgoverningsomelanguageinwhich‘A’appearsarejointlyinconsistent.Theexistenceofinconsistentconventionsgovern-ing‘A’andrelatedexpressionsareperfectlycompatiblewiththeclaimthatspeakersgrasptheconceptthat‘A’isintendedtoconnote,andevenwiththeclaimthatthisconceptissomehowsimpleandobvious.Second,itdoesn’tfollowthatanyindividualconventiononitsownissubjecttocriticism.Theapplicationpropertymaywellbesuchthatanexpressionthatconnotesitandwhichhasaroleinalanguagethatcapturesthisconnotationmustbegovernedbyconvention(2)above;likewise,thepropertyofheterologicalitymaybesuchthatanyexpres-sionthatconnotesitmustbegovernedby(3).Noneoftheseconventionsneedbeflawedonitsown,and,indeed,eachmayberequiredforade-quateexpressionofintuitivemeaning.YetitmaystillbethecasethatCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01theconventionscannotcoherentlybecombined—whichistosaythattheremaybenosuchthingasalogicallynon-triviallanguageinwhicheveryconceptcanbeexpressed.Takencorrectly,thisissimplyacorollaryofthestandardlimitativeresultsoftheperiod[Patterson,2009,414].Third,itdoesn’tfollowthatthosewhousetheseconventionsdonotexpressdeterminatethoughtsinparticularassertions.Sincespeakers’10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE164—#21\nIndefinabilityandInconsistency165acceptanceoftheconventionsisn’tclosedunderlogicalconsequence,theymightacceptasetofconventionsthatinfactimplythateverysentenceisassertibleineverycircumstancewithoutalsoacceptingthateverysentenceisassertibleineverycircumstance.Whatspeakersacceptissomefinitesetofclaimsaboutwhichsentencesexpresswhichthought-constituents,andthesespeakerswillingeneralapplytheirbeliefsaboutthisonlytothepointoftheirmostimmediateimplicationofsomeclaimaboutwhatthoughtagivensentenceiscorrectlyusedtoexpress.Speak-ersasagroup,then,willconvergeonbeliefstotheeffectthatvarioussentencesarecorrectlyassertedonlybythosewhohavecertainbeliefs,andsuchspeakerswill,inusingthosesentences,directlyexpress(touseKotarbi´nski’sterminology)preciselythosebeliefsintheuseofthosesentences—eventhough,unbeknownsttothem,theconventionstheyacceptdon’tactuallydeterminethatparticularthoughtsareindirectlyexpressedbyparticularsentences.Fourth,speakersdonotneedtobeespeciallydim-wittedtoendupspeakinganinconsistentlanguage.Speakerswhoagreetoconventionsthatareinfactinconsistentinthissensewithoutnoticingitcangoabouttheirbusinessprettymuchunhindered.Assertionofaparticularsentencetakentobecorrectbyfellowlogicallyignorantadherentstotheconven-tionswillleadthemtoattributeadeterminatethought,andifoneisinfactusingthesentenceasintended(thatis,directlyexpressingwithitthatthoughtwhichtheconventionssaythatitindirectlyexpresses),itwillbeexactlythethoughtonedirectlyexpressedintheassertion.Socommunication,asKotarbi´nskiconstruesit,canhappenunimpededbythefactthattheconventionsofalanguagejointlyreduceittologicaltriviality.8Finally,itisn’tanypartofKotarbi´nski’sconceptiontomaintainthatcontradictionsaretrue.Theviewisonlythatcontradictionsarecorrectlyassertedaccordingtotheconventionsoflanguagethatspeakersactuallyaccept.Thereisnomoremysteryinthisthanthereisintheideathatanyothersystemofrulesorconventionsthatpeopleacceptmightbesuchthatitsstatementissuchthatonecouldactuallyabidebytherulesonlyifeverythingwerethecase.Nowsuchspeakers,iftheygettotryingsystematicallytoworkoutlinesCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01ofthoughtbytracingoutwhathastobecorrectlyassertiblegiventhatotherthingsare,giventheconventions,maygetthemselvesintotrouble,andtheymaynoticethatthereisaproblem.Butthebusinessofexpress-ingthoughtsinparticularsituationsmayproceedunimpededatnocost.Furthermore,acceptinginconsistentconventionsdoesn’trequireany-onetobelievecontradictions;theconventionsofcoursedeterminethat10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE165—#22\n166AlfredTarski:PhilosophyofLanguageandLogiccertainsetencesbothareandarenotcorrectlyasserted,butitisn’tacon-ditionofacceptingtheconventionsthemselvesthatonemustrecognizethisconsequence.5.2.2TarskiafterKotarbinski´NoneoftheusualcomplaintsaboutTarski’snotionofaninconsistentlanguageappliestotheKotarbi´nskianview.WecannowmakeshortworkofunderstandingTarski’snotoriousremarks.Aninconsistentlanguageisonetheconventionsgoverningwhichimplythatallofitssentencesarecorrectlyasserted.9Aswesawearlier,thebasicnotionisnottruthbutcorrectassertability;hence,Tarskidoesnotmeanbyaninconsistentlanguagealanguageinwhichcontradictionsorallsentencesaretrue—contraryto[Soames,1999,55]andinaccordwith[Ray,2003,71]and[Patterson,2006,157].WhenTarskiwritesthat:Iftheseobservationsarecorrect,thentheverypossibilityofaconsistentuseoftheexpression‘truesentence’whichisinharmonywiththelawsoflogicandthespiritofeverydaylanguageseemstobeveryquestionable,andconsequentlythesamedoubtattachestothepossibilityofconstructingacorrectdefinitionofthisexpression[Tarski,1983a,164–5].hemeansthatgiventhecontent“truesentence”isintendedtoexpress,theexpressionofwhichrequirestheT-sentencestobeassertiblebytheconventionsgoverningthelanguage,introducingsuchconventionsintoalanguageinwhich(a)theconnectivesareclassical,and(b)“thespiritofeverydaylanguage”ismaintainedinthesensethatitismaintainedthateveryintuitiveconcept(andso,forinstance,syntacticconceptsdenot-ingexpressionsofthelanguagecanalsobeexpressed)willrenderthatlanguageinconsistentinthesenseadumbratedabove:theconventionsjointlywilldeterminethateverysentenceisassertible.Wecanalsounderstandwhyitdoesn’tfollow,onTarski’sview,thatcommunicationisimpossibleineverydaylanguage:givenKotarbi´nski’saccountsofconvention,indirectexpressionandcommunication,com-municationispossibleinaninconsistentlanguage.Itlikewisedoesn’tfollow,forreasonsstatedabove,thatonlyirrationalpeoplecanspeakinconsistentlanguages.Finally,then,itdoesn’tfollowthattheveryfactCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01thatTarskispoke,wroteandwasunderstoodineverydaylanguageisarefutationofhisview.Quitethecontrary:theoverallviewexplainspefectlywellhowitsowntruthiscompatiblewithTarski’sbeingabletodoso.Asnotedinplacesabove,Tarski’sviewisnotthattheconceptoftruthitselfis“inconsistent”-or,moreaccurately,thatthecontentof10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE166—#23\nIndefinabilityandInconsistency167theconceptoftruthis“inconsistent”-([García-Carpintero,1996,125–6][Soames,1999,51]andmanyothers).Theconcepthasaperfectlyclearcontent,statedinthe“semanticaldefinition”.Languages,governedbyconventions,canbeinconsistentinthesensethattheconventionsrequiretheassertionofeverysentence,andtheparadoxesshowthat“col-loquial”languagesarelikethis.NotethattheviewthatTarskiholdsthattherecanbeinconsistentconceptsisverynaturallyalliedtothecommonmisconceptionthateachdefinedTarskiantruth-predicateexpressesadif-ferentconcept(§4.3.2):if“true”inEnglishexpressesan“inconsistent”concept,butadefined“true”forsomeformalizedlanguageexpressesaconsistentconcept,thendifferenttruthpredicatesexpressdifferentcon-cepts.WeseeherehowvariousaspectsofthestandardreadingofTarskireinforceoneanother.WehavehereareadingonwhichTarski’snotionofaninconsistentlanguagedoesnotinvolvetheclaimthataninconsistentlanguageisoneinwhichcontradictionsaretrue.Aninconsistentlanguageissim-plyonegovernedbyconventionsthatimplythatcontradictionsmaycorrectlybeasserted.AswenotedwithKotarbi´nski,nothinguntowardfollowsfromthis:attemptstolaydownrulesinanydomaincanmisfire,andTarski’sclaimissimplythattherulesofusethatspeakersofnaturallanguageattempttoimplementhavethisflaw.Soames[Soames,1999,64]respondstothiswiththeobjections(1)that“itishardtoimaginethatitshouldbeaconditionofmyspeakingEnglishthatIbewillingtoassertthingsthatarenottrue”,and(2)that“Tarskinowhereexplicitlystatesthatalanguagecandictatetheassertabilityofasentencethatisnottrue”.Inresponseto(2)wecannote,first,thatTarskinowheresaystheopposite,either,andthattheconceptionofdeductivesystemsinarticleslike[Tarski,1983g]and[Tarski,1983c]obviouslyallowsforthepossibilityofinconsistentrules.Asfor(1),weneedtobecarefulwiththenotionof“speakingEnglish”.IftospeakEnglishistodoeverythingtherulesrequire,thenTarski’sviewimpliesthatitisimpossibletospeakEnglish—butonthisdefinitionof“speakEnglish”theviewisquiterea-sonable,anditdoesn’timplythatitisaconditionofspeakingEnglishexceptinthetrivialsensethat(givenclassicallogic)everyconditionisanecessaryconditionofdoingtheimpossible.Ifwetakethemorereason-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01ableviewthatspeakingEnglishisamatterofintendingtoabidebyitsconventions,thenpeopledospeakEnglish,butbytheaboveconsidera-tionsonhowrationalpeoplecanspeakinconsistentlanguages,itdoesn’tfollowthatitisanecessaryconditionofintendingtoabidebythecon-ventionsofEnglishthatonebewillingtoacceptexplicitcontradictionsorotherobviousfalsehoods;itfollowsonlythatonecanonlydowhatoneintendstodoifonedoesso.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE167—#24\n168AlfredTarski:PhilosophyofLanguageandLogicIn§5of[Patterson,2006]IextendedsimilarremarkstotheviewthatTarskiholdsthatnolanguagesaresemanticallyclosed,thoughsomeappeartotheirspeakerstobe.Liketherestofthatpaper,thoughthepointstillseemstometobeessentiallyright,itisvitiatedbymyadher-enceatthetimeofwritingtomorestandardviewsofTarski’sphilosophyoflanguageonwhichTarskiwasreallyonlyconcernedwithintroducinganexpressionthathadasitsextensionthesetoftruthsofalanguage.Itiscorrect,asIclaimedthere,thatnolanguageisuniversalinthesenseofhavingexpressionsthatdefineabsolutelyallsets,orevenuniversalinthesenseofhavingexpressionsthatdefineallofthesubsetsofthesetoftheirownsentences.10Wecanhoweveralsosay,aswasnotsaidthere,thatnaturallanguagesare“universal”inthesenseofintuitivemean-ing:foreveryintuitiveconcept,naturallanguagecanbeextendedtoincludeanexpressiongovernedbyaconventionsufficientforsentencesinvolvingitindirectlytoexpressthoughtsinvolvingthatconcept(con-trarytothetypicalresponsetothemodalobjectiondiscussedabove,aswenoted,Tarskiallowsthatlanguagescanbeextended[Tarski,1983a,164]).Inprinciple,everythingcanbeindirectlyexpressedaccordingtotheconventionsofnaturallanguage,and,moreobviously,everythingcanbedirectlyexpressedwithasentenceofnaturallanguage,giventheGriceanstoryinthebackgroundofTarski’sKotarbi´nskianviews.Inthatsense,naturallanguagesareuniversal;theproblemisthattheybuythisuniversalityatthepriceofformalincoherenceandtotalsemanticinde-terminacy:strictlyspeaking,takingtherulesseriously,everysentenceofanaturallanguageisassertible:justconstructasemanticparadoxandreasonbyexfalsofromthere.Moreover,preciselybecauseeverythingisassertible,nopredicatecanbeassignedanyparticularextensionasopposedtoanyother.11Istandbyanothercontentionofthatarticle,asshouldbeclearfromtheabove:Tarskidoesn’tinanyserioussenseproposeaccountingforthesemanticsofnaturallanguagebyimposinga“Tarskihierarchy”onit.Ashewrites,thiswouldinvolve“thethanklesstaskofareformofthislanguage”[Tarski,1983a,267].Sincedoingsoisrequiredtopursuethesemanticsofcolloquiallanguageby“exact”methods,wecancon-cludethatTarskidoesn’tholdtheinformalaccountofsuchlanguageinCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01termsofconventioninKotarbi´nski’sstyletobeexact,butinthebodyofCTFLheclearlyfollowsKotarbi´nskiinholdingthatthecorrectaccountofmeaninginnaturallanguageisintermsofconventionandintention,notextensionalsemantics.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP05”—2011/11/29—14:25—PAGE168—#25\n6Transitions:1933–1935Wellpast1931semanticswasforTarskisimplyatoolforproducingexplicitdefinitionsofcertainconceptsthatappearedbothsuspectanddifficulttoavoid:truth,denotation,satisfactionand(semantic)defini-tion.Someofthetoolstherebydevelopedwereevenmathematicallyfruitful,asshowninthelaterpagesofODSanditssequel,“LogicalOper-ationsandProjectiveSets”,co-authoredwithKuratowski.ButitseemsnottohaveoccurredtoTarskitothinkofmeaningitselfintermsofsemanticword–worldrelations;hewastooweddedtotheconceptionofmeaningintermsoftheexpressionofthoughtandtheconstraintsplacedonthisbythedeductivestructureofanaxiomsystem.Hiswillingnesstopresent“SomeMethodologicalInvestigations”in1934andpublishitin1935showshowfarhewas,eventhatlate,fromthinkingthatsemanticscouldstandonitsownortaketheplaceoftheoldermeaning-theoreticnotionshehadacquiredfromKotarbi´nskiandLe´sniewski.Padoa’smethodclearlyshouldbeconstruedinsemanticterms,yetin1934Tarskiiswillingtore-usematerialfrom1926inwhichthenotionofaninterpretationofatermreceivesthesamepsychologistictreatmentitgetsfromPadoahimself.Giventhetimingandthenatureofthechange,especiallyasitwillbediscussedinthenextchapterwithrespecttoTarski’sconceptionoflogicalconsequence,oneimpetusfortheshiftinTarski’sthinkinghastobehiscarefulreadingofCarnap’sLogicalSyntaxofLanguageandhisCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01interactionwithCarnaphimself.Inbrief:seeingwhatgoeswronginCarnap’sdefinitionof“analytic”forLanguageIIledTarskitoseethatwhatwasmissingwashissemantics.This,inturn,inthespaceofaboutayearledTarskitoshedtheolderconceptionofmeaningwithwhich16910.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP06”—2011/11/28—13:02—PAGE169—#1\n170AlfredTarski:PhilosophyofLanguageandLogichisprojectbeganandtomovetoamorecontemporaryconceptionoftruth-theoreticsemantics.IbeginwithonetextthatshowstheinfluenceofLogicalSyntaxtosomeextent,butalsoshowsTarskimovingawayfromhisPolishroots,especiallywithrespecttoLe´sniewski,inrepudiatingtheallegianceofthemaintextof“TheConceptofTruth”tothetheoryofsemanticalcategoriesandhencetoSTTasthetruesystemoflogic:the1935PostscripttotheGermantranslation.6.1The1935postscriptInthebodyofCTFLTarskisubscribedtoaninterpretationofSTTintermsofLe´sniewski’stheoryof“semanticalcategories”.Forourpurposeshereitsufficestonotethatthesystemassumesthatanygivenfunctionalcategorydeterminesthecategoriesofitsarguments—so,inparticular,thereisnosuchthingasafunctionalcategorythatcantakeeithernamesorpredicatesasargumentsandformasentence.Atthattime,Tarskiwrote:thetheoryofsemanticalcategoriespenetratessodeeplyintoourfundamentalintuitionsregardingthemeaningfulnessofexpressions,thatitisscarcelypossibletoimagineascientificlanguageinwhichthesentenceshaveaclearintuitivemeaningbutthestructureofwhichcannotbebroughtintoharmonywiththeabovetheory[Tarski,1983a,215].Thiscommitmentshapeshisdiscussionofthepossibilityofadefini-tionoftruthforalanguageofinfiniteorder,implyingasitdoesthatforsuchlanguagestruthcannotbeexplicitlydefined,cannotthereforereductivelybeeliminatedinfavorofunproblematicexpressionsfromthe“morphologyoflanguage”,and,concomitantly,thatnoexpressionofthemetalanguagecansemanticallydefinethesetoftruthsoftheobjectlanguage,doomingconsistentaxiomatictruth-theorytoafailureofprov-ablecategoricity.Theupshotofthewholediscussionisthatasappliedtothemostinterestingandpowerfullanguages,theintuitiveconceptoftruthcanbeonlyimperfectlyexpressed.TarskirelaxeshisadherencetothetheoryofsemanticalcategoriesinCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01thepostscript:TodayIcannolongerdefenddecisivelytheviewIthentookofthisquestion.Inconnexionwiththisitnowseemstomeinterestingandimportanttoinquirewhattheconsequenceswouldbeforthebasicproblemsofthepresentworkifweincludedinthefieldunder10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP06”—2011/11/28—13:02—PAGE170—#2\nTransitions:1933–1935171considerationformalizedlanguagesforwhichthefundamentalprin-ciplesofthetheoryofsemanticalcategoriesnolongerhold[Tarski,1983a,268].Notethehesitationhere:Tarskicannotdefendtheview“decisively”—hardlyawholesalerejection.Relatedly,noticethatTarskidoesn’tquiterepudiatetheviewthatnolanguageofwhichthefundamentalprinci-plesofthetheoryofsemanticalcategoriesfailtoholdcanhavesentenceswithclearintuitivemeanings.Rather,thebasicproblemsoftheworkwillbeconsideredwithrespecttolanguagesofwhichtheseprinciplesdonothold.Itisoftenclaimed(e.g.[Frost-Arnold,2004,272],[Feferman,2008,86],[Creath,1999,69])thatTarskichangeshisviewinthePostscriptandcomestoholdthattruthcanbedefinedfortheGeneralTheoryofClassesusingexpressionsofinfiniteorder,asthoughtheshiftinthePostscriptweremerelyamatterofaddingsometransfinitelevelsatopthehierarchyofSTT.Thisisn’taccurate:Tarskidoesn’tchangehisviewsaboutGTCformulatedinSTT.Rather,hewidensthescopeofhiscon-siderationstolanguagesthatdon’tadheretothetheoryofsemanticalcategoriesandthenpointsoutthatforthoselanguagesatruthdefinitionisavailablebecauseexpressionsthatare,bytheirstandards,“ofinfiniteorder”areavailable.Itremainsthecasethattruthcan’tbedefinedforGTCinalanguagethatadherestothestricturesofthetheoryofseman-ticalcategories.1(Admittedly,Tarskihimselfisn’thelpfulonthispoint.ThesesAandBof§6[Tarski,1983a,265]aren’tactuallyincontradictionwithThesesAandBofthePostscript,sincetheshiftfromthetheoryofsemanticalcategoriestoCarnap’stheoryoflevelsinducesanambiguityin“order”.)Now,onemightthinkthatsincethebasicproblemsoftheworkhadeverythingtodowithintuitivemeaning,thefocusisstillsquarelyonthat.ThePostscript,however,issilentonthetopicofintuitivemeaning,makingsimplytheformalpointthatifoneallowsexpressionsofinfiniteorder,thenforeverylanguagethereisalanguageofhigherorderandhenceforeverylanguagethereisametalanguageinwhichanexplicittruth-definitioncanbegiven.WhatisintheoffingisnotsomuchanextensionofpreviousquestionsaboutintuitivemeaningasawaningofCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01interestinthem.IfTarskihadremainedinterestedintheLe´sniewskianviewsaboutmeaningthathadanimatedhiminthefirstplace,hewouldhaveremarkedonhowtheabandonmentofthetheoryofsemanticalcategoriesaffectedtheconceptionofmeaning.TheshiftthatmatterstotheresultshighlightedinthePostcriptisTarski’sallowingthatthenotionofanexpressionoftransfiniteorder10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP06”—2011/11/28—13:02—PAGE171—#3\n172AlfredTarski:PhilosophyofLanguageandLogiccanbecountenanced.Thelargershiftisallowingexpressionsthatdon’tfixtheordersoftheirarguments:Sincetheprinciplesofthetheoryofsemanticalcategoriesnolongerhold,itmayhappenthatoneandthesamesignplaysthepartofafunctorintwoormoresententialfunctionsinwhichargumentsoccu-pyingrespectivelythesameplacesneverthelessbelongtodifferentorders[Tarski,1983a,269].SinceTarskiwantstoadheretotheprinciplethattheorderofanexpres-sionistheleastordinalgreaterthananythatspecifiestheorderofanyargumentittakes,butthereisnofiniteordinalαsuchthatωistheleastordinalgreaterthanα,theonlywaytogetexpressionsoftransfiniteorderistohaveexpressionsthattakeargumentsofallfiniteordersandhencetoallowvariabilityintheorderoftheargumentsthatafunctionalexpressiontakes(cf.[Sundholm,2003,118]).ItisinthisshiftthatTarskiabandonsthetheoryofsemanticalcategoriesandhencesimpletypethe-oryashehadconstruedit.Aswehavenotedin§4.2.2theconstraintsofSTThadlongbeenanirritantthatTarskisalvedwith“systematicallyambiguous”formulationsanyway.Whatismotivatingtheshiftawayfromthetheoryofsemanticalcat-egoriesofwhichTarskihadbeensocertainonlyafewyearsearlier?Theanswercomesatfootnote2to[Tarski,1983a,270]:TarskihasthetheoryoflevelsfromCarnap’sLogicalSyntaxinmind.SothePostscriptiswrit-ten,really,torelaxthetheoryofsemanticalcategoriesdowntothemoreliberalconstraintsofthetheoryoflevelsof§53ofLogicalSyntax,andthistheoryallowsbothexpressionsofinfiniteorderandpredicatesandfunc-torsthattakeargumentsofvariableorder[Carnap,2002,188–9]—e.g.theidentitypredicategetsassignedorderωbecauseittakesargumentsofeveryfiniteorder.2TheabandonmentofthetheoryofsemanticalcategoriesthenallowsthattruthcanexplicitlybedefinedforalanguageofanyCarnapianorder,finiteortransfinite,andTarskicorrespondinglymodifiestheconclusionshedrewintheoriginaltext.TheoriginalthesesAandBstatedthattruthcanbedefinedforaformalizedlanguageoffiniteorder,butnotoneofinfiniteorder[Tarski,1983a,265],whiletherevisedthesesstatethatCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01truthcanbedefinedforeveryformalizedlanguageinametalanguageofhigherorder,beitfiniteortransfinite[Tarski,1983a,273].TheoriginalthesisCstatedthatforalanguageofinfiniteorder“theconsistentandadequate”(konsequenterundrichtiger,konsekwentnieitrafnie)useoftheconceptoftruthwaspossibleonthebasisofanaxiomatictruththeory.AsTarskinotes,itlosesitsimportancesinceaxiomatictruth-theorydone10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP06”—2011/11/28—13:02—PAGE172—#4\nTransitions:1933–1935173inalanguagenotofhigherorderthantheobjectlanguagebecomesamerecuriositywhentruthforthesameobjectlanguagecanbedefinedinametalanguageofhigherorder.Concomitantly,theopenquestionsabouttheconsistencyofaxiomatictruth-theoryaugmentedwiththeω-rule,aresettled:sincetruthcanconsistentlybedefined,andsincetheaxiomatictheorywiththeω-ruleisasub-theoryofthetheoryinclud-ingthedefinition,theextendedaxiomatictheoryisconsistent[Tarski,1983a,273].3WeshouldnotethatinadditiontorelaxingthetheoryofsemanticalcategoriesdowntoCarnap’stheoryoflevels,thePostscriptconsidersashifttofirst-orderaxiomaticsettheoryaswell[Tarski,1983a,271note1].Themainpointsinthepostscriptcarryover:onecandefinetruthforafirst-ordersettheoryonlywithamorepowerfulsettheory.InthisformthePostscript’streatmentoftheissuesmorecloselyresemblesacontem-porarytakeon“Tarski’sTheorem”:arithmetictruthisnotarithmeticallydefinable,though,asnoted,theexplicitfocusinthePostscriptisstillonexplicitdefinitionandnotsemanticdefinition.4WhatdoweseeinthePostscriptthatisofmostinteresttoourstudyhere?ThemostimportantshiftistheweakeningoftheholdonTarskioftheviewsaboutmeaningheinheritedfromLe´sniewski,andadiminutionofinterestinthequestionthatgothimstartedinthefirstplaceofhowintuitivesemanticconceptsaretobeformallyexpressed.ThePostscript,forinstance,isalmostsilentonwhathappenstointuitivemeaningwhenthetheoryofsemanticalcategoriesisscrappedinfavorofCarnap’stheoryoflevels.AllTarskigivesusistheremarkthathenolonger“defendsdeci-sively”thetheoryofsemanticalcategories.SinceatthisdateitisunlikelythatTarskihasceasedtoholdthattheproblemofdefiningtruthrequiressentencestohave“concrete,intelligiblemeanings”[Tarski,1983a,167],allwecanconcludeisthatCarnaphasconvincedTarskithatitisn’tobviousthatexpressionswithinthetheoryoflevelsdon’thavesuchmeanings,contrarytoTarski’semphaticendorsementofthetheoryofsemanticalcategoriesat[Tarski,1983a,215].NowgiventhetightconnectioninLe´sniewskiandthereforeTarski’sthoughtbetweenthecapacityoflanguagetoexpressthoughtsandthatlanguage’sbeingstructuredinaccordwiththetheoryofsemanticalcat-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01egories,itfollowsfromthePostscript’sapostasyfromthattheorythatthePostscripthaslittletosayabouttheeffectontheintuitivethoughtexpressedbylanguageoftherelaxationofthetheoryofsemanticalcat-egoriesdowntoCarnap’stheoryoflevels.Withincreasingtolerancecomesadecreasedemphasisontheideathatlogicexpressesthought(§1.4.2).Whenlanguageisviewedasexpressingthoughtsthatstandin10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP06”—2011/11/28—13:02—PAGE173—#5\n174AlfredTarski:PhilosophyofLanguageandLogicadeterminatesetofdeductivelogicalrelationships,logicistheattempttocaptureexactlytheserelationshipsandnoothers.Inorder,bycon-trast,tofindroominone’sviewformultiplelogicalsystemsthatareallequally“right”,thispictureneedstobereplacedbyoneofvariousotherfamiliarones—e.g.thatoftherulesofvariousgamesthatcanbeplayed,onamoreconventionalistview,orathoroughlyempiricistviewofthesorttowhichTarskiwasultimatelyattracted[Frost-Arnold,2004],[Mancosu,2005].ButinthePostscriptweseethatthebasicexpressiveconceptionoflanguagefromwithinwhichhisprojectbeganwaslos-ingitsgriponTarski,andthathewasmovingtoamoreopen-minded,Carnapian,“tolerant”conceptionofalternativelogicalsystems.Inotherrespects,thePostscriptremainssquarelywithintheearlierperiod.ExplicitdefinitionthatimpliestheT-sentencesremainsthecen-tralconcern,withthedefinabilityofsetsonlyinthebackgroundandlargelyrelegatedtoasketchyfootnoteonfirst-orderaxiomaticsetthe-ory.Theconceptionoflogicalconsequenceisstillfullyderivational:consequenceistobedefinedastheclosureofsomeaxiomsundersometransformationrules[Tarski,1983a,269].ThoughtheolderconceptionwaslosingitsgripabitatthetimeofthewritingofthePostscript,themostsignificantchangeswereyettocome.6.2CarnaponanalyticityandtruthAsisoftennoted,Tarskistudied[Carnap,2002]carefullyandisthankedforsuggestionsandcorrectionsinmanyplacesthroughoutthe1937expandedEnglishedition.WhenhestudiedLogicalSyntaxTarskiwouldhavefoundtwopassagesthatwereclosetohisinterestswithrespecttothetopicsthatconcernushere:theargumentof§60bthattruthcannotbedefinedinsyntax,andthedefinitionsof‘analytic’and‘contradictory’in§34.5ByTarski’sstandardstheformerisinerrorandthelatterinvolvesconvolutionsthatTarski’sconceptionoftruthwouldstripaway.Theargumentof[Carnap,2002,§60b]purportstoshow,first,thattruthcannotbedefinedforalanguagesufficienttoexpressitsownsyntaxinthatlanguageitselfand,second,thateveninalanguageofgreaterexpressivepower,truthcannotbedefined.TheargumentforthefirstCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01claimrunsviaafamiliardemonstrationthatintroducingapredicatesuchthattheT-sentencesaretheoremsintoalanguageSthatcanformulateitsownsyntaxwillgeneratesentencesthatattributeuntruthtothemselves[Carnap,2002,214–6].Carnap’sofficialformulationinvolvesaversionofthe“visitingcardparadox”involvingtwosentences,thefirstofwhichsaysthatthesecondisfalsewhilethesecondsaysthatthefirstistruein10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP06”—2011/11/28—13:02—PAGE174—#6\nTransitions:1933–1935175order,asCarnapexplains,toavoidtheimpressionthatself-referenceissomehowtheproblem.Sofarthisisperfectlystandard.Theargumentforthesecondclaimiswherethingscomeofftherails.Carnapwrites:Thiscontradictiononlyariseswhenthepredicates‘true’and‘false’referringtosentencesinalanguageSareusedinSitself.Ontheotherhand,itispossibletoproceedwithoutincurringanycontradictionbyemployingpredicates‘true(inS1)’and‘false(inS1)’inasyntaxofS1whichisnotformulatedinS1itselfbutinanotherlanguageS2.S2can,forinstance,beobtainedfromS1bytheadditionofthesetwopredicatesasnewprimitivesymbolsandtheerectionofsuitableprimitivesentencesrelatingtothem,inthefollowingway:1.EverysentenceofS1iseithertrueorfalse.2.NosentenceofS1isatthesametimebothtrueandfalse.3.If,inS1,S2isaconsequenceofK1,andifallsentencesofK1aretrue,thenS2islikewisetrue.Atheoryofthiskindformulatedinthemannerofasyntaxwouldneverthelessnotbeagenuinesyntax.Fortruthandfalsehoodarenotpropersyntacticalproperties;whetherasentenceistrueorfalsecannotgenerallybeseenbyitsdesign,thatistosay,bythekindsandserialorderofitssymbols[Carnap,2002,216].Carnapgoesontoproposethatthisdoesn’tmatterbecauseinthesyntaxlanguageforS1alltheworkonewouldhavewantedtodowith‘SentenceAistrue’canbedoneeitherbyAitselforby‘Aisanalytic’.TheproblemhereisCarnap’sitalicizedsentenceandtheexplanationthatfollowsit.For‘true’tobedefinableinsyntaxandhence,ultimately,inarithmeticitneedn’tbethecasethatitis‘syntactic’inthesensethatitisapropertyofasentencedeterminedbythekindsandserialorderofitssymbols;itneedonlybethecasethatitcanbedefinedintermsofexpressionsthatapplytosentencesinvirtueofthekindsandorderoftheirsymbols.Coffa,at[Coffa,1987,566–7]notestheproblem,butgivesonlyapartialdiagnosis:Ingeneral,whetherasentenceistrueorfalsecannot“begathered”throughthetechniquesavailableinsyntax.NoticethatwhetheraCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01sentenceisanalyticornot(inCarnap’ssense)cannotbegatheredfromthetechniquesavailableinCarnap’ssyntax;and,moretothepoint,whetherasentenceistrueorfalsecannotbegatheredbythetechniquesavailableinsemanticseither.WhatcouldhaveledCarnaptothinkthatwhetheraconceptCisdefinableonthebasisofcertaintechniquesdependsonwhetherthosetechniquesallowustoidentify10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP06”—2011/11/28—13:02—PAGE175—#7\n176AlfredTarski:PhilosophyofLanguageandLogictheinstancesofC?Theobviousansweris:averificationistprejudice[Coffa,1987,567].Itmaywellberightthatthatverificationismencouragesthethoughtthatiftruthcanbedefinedintermsofsyntaxthensyntaxmustdecidewhetheranygivensentenceistrue,theideabeingthatthemeaningwithwhichanypredicateisendowedmustbeassociatedwithaverificationprocedure.TheproblemCoffaalludestoisthat“analytic”itselfdoesn’tapplytoasentencemerelyinvirtueofthe“kindsandserialorderofitssymbols”,sinceanalyticitydependsonwhethervarioussubstitution-variantsofasentenceareconsequencesofeverything.Carnaplaterdescribedhismentalblockatthetimethisway:WhenTarskitoldmeforthefirsttimethathehadconstructedadefi-nitionoftruth,Iassumedthathehadinmindasyntacticaldefinitionoflogicaltruthorprovability.Iwassurprisedwhenhesaidthathemeanttruthinthecustomarysense,includingcontingentfactualtruth.SinceIwasonlythinkingintermsofasyntacticalmetalan-guage,Iwonderedhowitwaspossibletostatethetruth-conditionforasimplesentencelike“thistableisblack”.Tarskireplied:“Thisissimple:thesentence‘thistableisblack’istrueifandonlyifthistableisblack”[Carnap,1963,60].Oncetheobjectlanguagewasincludedinortranslatedintothemet-alanguage,onenolongerhadtocastaboutwithinthesyntacticandproof-theoreticresourcesofthemetalanguagetofindthetermsinwhichtoconstructthedefinitionoftruth.Aslongastheobjectlanguageitselfwastakentobescientificallylegitimate,therewas,asTarskisaw,noreasonnottomakeuseofitsexpressiveresourcesinthemetalanguageaswell.Turningtooursecondtopic,Tarski,alreadyarmedwithhisproceduresforexplicitlydefiningtruth,wouldimmediatelyhaveappreciatedhowunnecessarilycomplicatedCarnap’sdefinitionofanalyticityforLan-guageIIwas,whileitneverthelesswas,initsessentialstructure,quiteclosetoTarski’sownwayofdoingthings.ThispointabouttherelationbetweenCarnapandTarskiisfamiliarfromtheliterature;seee.g.[Coffa,1987]and[deRouilhan,2009].Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01AnalyticityisdefinedforLanguageIIintermsofvaluations.Carnapfirstpointsoutthatalthoughwecan,infinesyntacticform,takethevaluationofafirst-ordervariabletobeadenumerablyinfinitesetofnames(weseeherethesameunderstandingofthevaluesofavariableaswefound(§3.1.1)inthe1929Abriss)andsounderstandasentenceopenonafirst-ordervariableasanalyticwheneverysentencethatresultsfrom10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP06”—2011/11/28—13:02—PAGE176—#8\nTransitions:1933–1935177substitutinganameforthevariableis,whenitcomestohigher-orderquantificationthismethodwon’tgivetheresultwewant,forthe(now)obviousreasonthat“asaresultofGödel’sresearchesitiscertain,forinstance,thatforeveryarithmeticalsystemtherearenumericalpropertieswhicharenotdefinable”[Carnap,2002,106].WithonlydenumerablymanysymbolstoworkwithCarnapmakesthestrikinganddesperatemoveofassociatinghigher-ordervariableswithset-theoreticconstructsoutofnamesthemselves,lettingsecond-order(one-place)predicatevariablesrangeoverthepowersetofthesetofnames—thusbuyinghimselfmorethandenumerablymanyitemsforasecond-ordervariabletorangeover[Carnap,2002,107].DespitetheascenttothetransfinitesomehowCarnapconvincedhimselfthatthiswasstillproperly“syntactic”.Withsufficientnumbersofinterpre-tantsforhigher-ordervariablessecuredinthismanner,Carnapcanthenproceedinwhatis(followingTarski)theobviousway—auniversalgen-eralizationisanalyticjustincaseallofitsinstancesare,etc.[Carnap,2002,107ff].Theissue,however,ishow“analyticity”isdefinedfortheresultofsubstitutingnappropriateconstructsoutofthesetofnamestonopenpositionsinsomeopensentence.HereiswhatCarnapwriteswithrespecttotheclosureofasecond-orderone-placeopensentence:NowifB1isaparticularvaluationfor‘F’ofthiskind,andifatanyplaceinS1‘F’occurswithSt1asitsargument(forexample,inthepartialsentence‘F(0

)’),thenthispartialsentenceis—sotospeak—trueonaccountofB1,ifSt1isanelementofB1,andotherwisefalse.Now,bytheevaluationofS1onthebasisofB1,weunderstandatransformationofS1inwhichthepartialsentencementionedisreplacedbyNifSt1isanelementofB1,andotherwiseby∼N.Thedefinitionof‘analytic’willbesoframedthatS1willbecalledanalyticifandonlyifeverysentenceisanalyticwhichresultsfromS1bymeansofevaluationonthebasisofanyvaluationfor‘F’.AndS1willbecalledcontradictorywhenatleastoneoftheresultingsentencesisacontradictorysentence.Weshalllaydownanalogousrulesforotherp-types[Carnap,2002,107].AsTarskiwaswellinapositiontoappreciate,thewholeproceduremakesCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01nosenseatallatthecrucialpointunlessonekeepstheintuitivenotionoftruthwhichissupposedlybeingeliminatedinmind.Thisisn’ttoaccuseCarnap’sdefinitionofcircularity—thiswouldbeanalogoustothemis-takencriticismofTarskiwerejectedabovein§4.3.1.Thepoint,rather,isthatsimplyadmittingthatthenotionoftruthisinplaymakesallofthecomplicationsunnecessary.Whyshouldthefactthataisinasetof10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP06”—2011/11/28—13:02—PAGE177—#9\n178AlfredTarski:PhilosophyofLanguageandLogicnamesassociatedwithFallowusto“transform”Fainto‘0=0’ratherthan‘0=0’?Afterall,inthemselves,wejusthaveabunchofnameshere.Theonlyreasonfortheprocedureisthatwetacitlyallow—asCar-napcan’trestrainhimselffrommentioning—thataname’sbeinginasetofnamesstandsinfortheideathatwhatisnamedisinthesetofthingsthatsatisfythepredicateandthatthesentenceistherebytrue(cf.[Creath,1999,69]).SowhatTarskifoundin§§60and34bwasthat,ontheonehand,Carnaphadnogoodargumentagainstthepossibilitythatsemanticexpressionsbedefinedinsyntax,andthat,ontheother,Carnap’sownprocedurewithrespecttoLanguageIIappealedtoreference,satisfactionandtruthineverythingbutname.TheonlythingpreventingCarnapfromseeingthefruitfulnessofthetruth-definitionshehadnearlyarrivedathimselfwasalingeringprejudiceagainstsemanticnotionsbackedupbytheoversightof§60onwhichaproperlyscientificmetalinguistictreatmentofalanguagecouldn’tinvolvethetermsofthatlanguageinadditiontoitsownpropersyntacticterms.Beforelookingattheactualdevelopmentsinthenextchapter,weshouldpausenowtounderstandtheshiftintheoffingonceonerealizesthatCarnap’sdefinitionofanalyticityisbestreplacedbyaconceptionbasedonTarski’ssemantics.Aslongasconsequenceisconceivedofintermsofprimitivelyvalidtransformationrules,syntacticandproof-theoreticnotionsarebasictoone’sconceptionoflogic.Evenwithsemanticsaddedinlaterinthestyleof[Tarski,1983f],proof-theoreticconsequenceisneededbeforeextensionscanbeassignedtoexpressions,sincethesetofprovabletheoremsasawholeiswhatconstrainstheirsemanticinterpretation.Bycontrast,ifwetaketheextensionalsemanticvaluesassignedtoprimitiveterms(betheyobjects,sets,n-tuples,orvariousfunctionstherefromandthereto)asbasicandgrantourselvesmerelyformationrules,anotionofconsequencedropsoutoftheaccountforfree:asen-tenceisaconsequenceofothersrelativetoacertainfixedvocabularyifandonlyifeverysemanticassignmenttotheremainingexpressionseithermakestheformertrueoroneofthelatterfalse.Ofcourse,inplaceofwranglingoverwhichtransformationrulesarevalidhereoneCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01getsinsteadthenow-familiarquestionofthelogicalconstants:whichselectionoffixedvocabulary,ifany,producesanaccountoflogicalcon-sequence?Theprimaryquestiontobeaddressedwhenitcomestotheaccountoflogicalconsequencedevelopedinthiswayiswhetheritcon-stitutesanyadvanceovertheaccountintermsoftransformationrules.Butthepointnowisthatonceonehastheideaofdefininganalyticity10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP06”—2011/11/28—13:02—PAGE178—#10\nTransitions:1933–1935179andmoregenerallyconsequenceintermsoftruth,oneseesthatonecanjudgethetransformationrulespreviouslytreatedasprimitivelyvalidbythestandardsoftheresultantaccountofconsequence.Overall,then,Tarski’sencounterwithLogicalSyntaxallowedhimtoseethatCarnap’streatmentofanalyticityreallyappealstointuitivesemanticnotionsanyway,thatCarnaphasnoargumentagainsttheinclusionofsemanticnotionsinserioustheory,andthatonceCarnap’streatmentofanalyticityisreplacedwithTarskiansemantics,everythingcanbedonewithoutthetransformationrules.6TarskibeganwithaderivabilityconceptionofconsequenceassociatedwiththeIntuitionisticFormalistideathatadeductivetheorywaspri-marilyavehiclefortheexpressionofthoughtandreasoning.Butthiscouldonlybetakensofar;theconvolutionsofCarnap’sdefinitionofanalyticityforLanguageIIshowswhyitultimatelycan’tbeworkedout.Andthenthesemanticshedevelopedalongthewayasasubsidiarytoolturnedouttobejustwhatwasrequiredtofixtheproblem.Theresultwastruth-theoreticsemanticsandthedefinitionofconsequenceasweknowit.6.3TheestablishmentofscientificsemanticsHavingsetouttoestablishthathissemanticsshouldreplacesyntaxinCarnap’ssystem,Tarskiwasledtoseewhatsemanticscouldhavebeendoingthatsofarhewasn’tdoingwithit.Wehaveseenthatin1934TarskistillunderstoodPadoa’smethodinIntuitionisticFormalisttermsandultimatelypsychologistically,andthataslateasthePostscripttoCTFLTarskishowednoinclinationtodefinelogicalconsequencesemanticallyratherthanintermsofalistofintuitivelyvalidtransformationrules.ButonceTarskisethimselftothinkingaboutwhathecouldaccomplishwithhistechniquesfordefiningsemanticterms,withinafewmonthshehadcastasidetheoldwayofunderstandingconsequenceandwithit,thoughlessexplicitly,theconceptionofmeaningfromwhichhehadbegun.Wewilllookatthefirstpointinthenextchapterandthesecondinthechapterafterthat.TarskicouldonlyhavewantedtoletCarnapandanyonewhofollowedCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01himknowhowmuchmoreeffectivelysemanticswouldservethepur-posestowhichCarnapwasputtingtheconvolutionsofhis“syntax”.ThefoginthedefinitionofanalyticityforLanguageIIcouldbeclearedbyastraightforwarddefinitionoftruthinacceptabletermsofjustthesortthatTarskiwasinapositiontoprovide.TarskigothischanceindiscussionswithCarnap,whowasquicklyconvinced,andwhoencouragedTarski10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP06”—2011/11/28—13:02—PAGE179—#11\n180AlfredTarski:PhilosophyofLanguageandLogictopresenthisideasattheplannedSeptember1935UnityofSciencecongressinParis[Carnap,1963,61],[FefermanandFeferman,2004,95].TarskiarrivedattheInternationalConferenceofScientificPhiloso-phyinPariswithtwopaperstopresent:“TheEstablishmentofScienficSemantics”and“OntheConceptofLogicalConsequence”.ThefirstismoreorlessanadvertisementforTarski’smethodofdefiningtruth,whilethesecondrepresentsanessentialextensionofsemanticstoadomainpreviouslyrelegatedtotherealmofsyntaxandintuition,theconse-quencerelation.Wewillcoverthefirstbrieflyhere.Thesecondwillbeourtopicinthenextchapter.In[Tarski,1983b]TarskireviewsthecontentofCTFLand,fortheaudi-enceoftheCongress,triestomakethingsasexcitingaspossible.Thefirstfewparagraphsreadliketheintroductionto“OntheConceptofTruth”:semanticconceptsareintuitivelyclearbutattemptstomakerig-oroussenseofthembogdowninparadox.Thesolutionisrigorouslytodistinguishobjectandmetalanguageandtoaccepttherelativizationofthedefinitionstocometooneobjectlanguageatatime.Wesetupaformalsyntaxinthemetalanguage,includetranslationsoftheobjectlanguage’sdistinguishedvocabulary,andthendefinetruthinthestyleof§3ofCTFL.ThearticlecloseswithabriefrecapitulationoftheremarksonGödelfromthePostscript.Missinginallofthisismuchphilosophizingabouttheconceptoftruthofthesortthathadanimatedtheprojectjustfouryearsearlier.Asinglesentenceintheconcludingparagraphgivesanodtothephilosophi-calambitions,buttheoverallpitchofthearticleistothe“scientific”philosophersintheaudience.Interestingly,Tarskidoesn’tanticipatethedefinitionofconsequencesetoutintheotherlecture.Tothisweturninthenextchapter.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-0110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP06”—2011/11/28—13:02—PAGE180—#12\n7LogicalConsequenceAswithhisaccountoftruth,Tarski’saccountoflogicalconsequencehascomeinforagreatdealofabuse.CharacteristichereisEtchemendy’sfamiliarassessmentthatTarski’s“accountoflogicaltruthandlogicalcon-sequencedoesnotcapture,orevencomeclosetocapturing,anypretheo-reticconceptionofthelogicalproperties”[Etchemendy,1990,6].Tarski’sconceptionofconsequencehaslikewisebeensubjectto“defenses”thateitherattempttomakevirtuesoutofitsleastattractivefeaturesorareimplausibleasreadingsofwhatTarskiactuallysaid.Manyofthesedefensesread“OntheConceptofFollowingLogically”throughthelensofthelecture“WhatareLogicalNotions?”,alecturedeliveredthirty-oneyearsaftertheoriginalpresentationoftheconceptionofconsequence.ThoughthereissomebasisforthisinworkcontemporarywithCLC,e.g.[Tarski,1983i],theastonishingfactabouttheliteratureisthatprettymuchnobodyhasbotheredtofollowupthearticle’smanyreferencestoCarnap’sLogicalSyntaxinanydetail—thisdespitethefactthattheonlyconceptionofconsequenceTarskimentionsisCarnap’s.IwillarguethatthearticleintendsnothingmorethananimprovementonCarnap’scon-ceptionthatconsistsinreplacingCarnap’srelianceonthenotionofsub-stitutionwithTarski’streatmentofsemantics,andtransformationrulestakenasprimitivewiththenotionoftruth-preservationleavingevery-thingelseuntouched,andthatalloftheimportantclaimsandargumentsinthearticlearedirectedatCarnap.Inparticular,Tarski’sconceptionofCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01formalityisnottheoneextractablefromthelaterarticle,onwhichlogicis“formal”inthesenseofbeingconcernedwithmathematicalstructure,butisrathernothingotherthanCarnap’snotionofsyntacticform.Read-ingTarskiintermsofCarnapwillallowusbetterunderstandingofmanypassagesofthearticlethathavecauseddifficultyanditwillalsoallowustohighlightjustwheretheimplausibilityoftheconceptionlies.18110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE181—#1\n182AlfredTarski:PhilosophyofLanguageandLogic7.1Tarski’sdefinition7.1.1SynopsisWecanbeginbybrieflysummarizingthetext.Tarskiannouncestheaimofcapturingthecontentoftheconceptoflogicalconsequence(or“followinglogically”,as[Tarski,2002]hasit),withcaveatsaboutthe“everyday,‘pre-existing’”usageofthetermexpressingitbeingunclear,theintuitionsaboutitbeing“murky”,andsoforth—compareheretheopeningremarksinODSandCTFL.HeretheaimseemstobeinaccordwithIntuitionisticFormalism,inthattheaimistocapturethecontentofanordinaryconcept—butthatwearenowatsomeremovefromIntu-itionisticFormalismisindicatedamongotherthingsbythefactthatTarskimakesnoheavyweatheroverexplicitlycharacterizingtheobjectlanguageandthemetalanguage.Inpartthisisduetothegeneralityinherentinthecase:Tarskiwillshow,givenadefinitionofsatisfaction,howtodefineamodel,andintermsofthat,consequence.Butitisalsoduetothefactthatassignmentsofsemanticvaluewillbemadetodotheworkpreviouslydonebytransformationrules.Hethensurveysthefamiliarsyntacticconceptionofconsequencetowhichhehimself,aswehaveseen,subscribed,withreservations[Tarski,1983a,252],aslateasthePostscripttoCTFL,andnotesthatGödel’sresultsshowthatitdoesn’tcapturethe“everydayintuitions”[Tarski,2002,178]aboutwhatfollowsfromwhat.1Inparticular,asGödelhadrecentlyshown,syntacticconceptionsofconsequenceareω-incomplete[Tarski,2002,177–8].Theω-rule,whichallowsthederivationofauni-versalgeneralizationovernumbersfromallofitsnumericalinstancesfixesthisbutisn’tfinitary[Tarski,2002,178].UsingGödeliantechniquesallowstheformationoffinitaryrulesthatallowthederivationofuni-versalgeneralizationsfromcodedclaimstotheeffectthatalloftheirinstancesareprovable[Tarski,2002,180],butbythesameGödelianresultstheresultisstillω-incompleteness[Tarski,2002,181].Moregen-erally,Gödelsentencesforatheoryfollowlogically,accordingtoTarski,fromthistheory“intheeverydaysense”,butforanyproposedsetofrules,includingfinitarycodedreplacementsfortheω-rule,thereissomesuchsentencethatisn’tprovable.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01TarskinextnotesCarnap’sattemptatcapturingthe“properconcept”oflogicalconsequence,faultingCarnap’sdefinitionforLanguageIIas“toospecialandcomplicated”[Tarski,2002,182],hisdefinitionforLan-guageIasbeingunsuitedforextensionto“lesselementary”languages[Tarski,2002,192],andhisdefinitioninGeneralSyntaxasmakingthe“denotation”ofthedefinedconcept“dependentinanessentialwayon10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE182—#2\nLogicalConsequence183therichnessofthelanguagewhichistheobjectofconsideration”(thisistheissuewithCondition(F)andTarski’smodificationoftheaccount;seebelow),andasdependingonabasicconceptofconsequenceintermsofstipulatedtransformationrulesthatCarnapleavesasprimitive[Tarski,2002,193].Thelastcriticismdoesn’toccurintheGermanor,therefore,Woodger’stranslation;wecanprobablyacceptHitchcock’sconjecture[Stroi´nskaandHitchcock,2002,158]thatTarskiaddedittothePolishinresponsetodiscussionatthePariscongress.However,ratherthansofteningthecriticismofCarnap,asHitchcocksuggests,thesec-ondpointemphasizesaperhapsmorefundamentaladvantageofTarski’saccount,whichistheeliminationofanydependenceontransformationrulestakenasprimitivelyvalid.Thisisthereallyrevolutionaryaspectof[Tarski,2002].Tarskithenstatesacriterionofintuitiveadequacy2forthedefinitionofconsequence:Thepointofdepartureforuswillbecertainconsiderationsofanintu-itivenature.LetusconsideranarbitraryclassofsentencesKandanarbitrarysentenceXwhichfollowsfromthesentencesofthisclass.Fromthepointofviewofeverydayintuitionsitisclearthatitcan-nothappenthatallthesentencesoftheclassKwouldbetruebutatthesametimethesentenceXwouldbefalse.Sincemoreoveritisaquestionhereoftherelationoffollowinglogically,i.e.formally,andthereforeofarelationwhichhastobecompletelydeterminedbytheformofthesentencesamongwhichitobtains,thusfollowingcannotdependonoutknowledgeoftheexternalworld,inparticularonourknowledgeoftheobjectswhicharespokenaboutinthesentencesoftheclassKorinthesentenceX,cannotbelostasaresultofourreplacingthenamesoftheseobjectsinthesentencesunderconsid-erationbynamesofotherobjects.Boththesecircumstances,whichseemhighlycharacteristicandessentialfortheconceptoffollowing,findjointlytheirexpressioninthefollowingcondition:(F)IfinthesentencesoftheclassKandinthesentenceXwereplacetheconstanttermswhicharenotgeneral–logicaltermscorrespondinglybyarbi-traryotherconstantterms(wherewereplaceequiformconstantseverywhereCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01byequiformconstants)andinthiswayweobtainanewclassofsentencesK
andanewsentenceX
,thenthesentenceX
mustbetrueifonlyallsentencesoftheclassK
aretrue[Tarski,2002,183–4].Tarskithennotesthattheconditionisn’tsufficient,sinceitmaybesatis-fiedonlybecausesomeobjectslackdenotingconstantsinthelanguageunderconsideration.Onewouldn’twanttheinferencefromFatoGato10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE183—#3\n184AlfredTarski:PhilosophyofLanguageandLogiccomeoutlogicalmerelybecausesomeobjectthatisFbutnotGlacksaname.This,too,lookstobepartofIntuitionisticFormalism:wehaveacon-sequencethatissupposedtofollowfromadefinition,bywhichthedefinitionistobejudged.Butcondition(F)playsadifferentrolefromtheT-sentences.Thelatteraresuchthatimplicationofthembyadeduc-tivetheoryissufficientforintuitiveadequacy,whilecondition(F)isexplicitlymarkedasnecessarybutinsufficient.Thisispartofashiftofemphasisinthearticle:thedefinitionofconsequence,unlikedefini-tionsinearlierpapers,appearstocarrytheweightofconceptualanalysisitself.Itisn’tthataddingthedefinitionbringsintuitiveconsequences;rather,intuitiveconsequencesareevidencethatadefinitionthatisitselfintendedtobeanalyticoftheordinaryconceptisinfactsuch.Thesolutiontotheproblemthatinferencesmaybeaccounted“logi-cal”bycondition(F)onlybecauseoftheimpoverishmentoftheobjectlanguageistoreplacesubstitutionwithsemantics.Tarskiexplainsthenotionofsatisfactiontothereaderanddefinesamodelofasetofsen-tencesasasequencethatsatisfiesallofthem.Hethendefineslogicalconsequenceintermsofmodels:WesaythatthesentenceXfollowslogicallyfromthesentencesoftheclassKifandonlyifeverymodeloftheclassKisatthesametimeamodelofthesentenceX[Tarski,2002,186].Tarskithenargues,inapassagesubjecttomuchdispute,thatfollowingaccordingtoconditionFisanecessaryconditionoffollowingaccordingtoTarski’sdefinition:theconsequencesoftruesentencesmustbetrue,andargumentsdeclaredvalidareallsuchthateveryoneoftheirsub-stitutioninstanceshaseitherafalsepremiseoratrueconclusion.3HethennotesthathisdefinitionisessentiallyequivalenttoCarnap’s.Tarskiendsthearticlebynotingthatthedefinitionofconsequencedependsonthedistinctionbetweenlogicalandnon-logicalconstantsandraisesthepossibilitythatthisdistinctioncannotbedrawninaprincipledway,somethinghetakestobesignificantforTractarianandViennaCircledoc-trinesaboutthewayinwhichlogic“saysnothingabouttherealworld”[Tarski,2002,189].Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01ThoughIemphasizeTarski’sattentiontoCarnapbecauseitisCarnapthatTarskimentions,weshouldnotethatadefinitionofconsequenceresemblingTarski’swasknownintheLvov–WarsawSchool.Łukasiewiczgivesthisdefinition:Therelationofimplication,ortherelationbetweenreasonandconsequence,holdsbetweentwoindefinitepropositionsaandbifforeverypairofvaluesof10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE184—#4\nLogicalConsequence185thevariablesoccurringinaandbeitherthereasonayieldsafalsejudgmentortheconsequencebyieldsatruejudgement[Łukasiewicz,1970a,17].Thisonlyconcernsone-premisedinferencesandclearlytheappealtoformalityislacking,butthesimilaritytoCondition(F)ismanifest.InsomethingofananticipationofnotoriousaspectsofTarski’sview,Łukasiewiczhappilynotesthatgiventhisconception“xisgreaterthan4”has“xisgreaterthan3”asaconsequence.InanotherarticleŁukasiewiczallowsmultiplepremisesandspeaksoflogicasinvolving“formal”conse-quence,butonlyinadegeneratesenseof“formal”onwhichitamountstocontainingvariables:Whenwespeakaboutconsequenceinlogic,weusuallymeanformalconsequence.Theexpression:apropositionis“aformalconsequence”ofanotherproposition,isusedtodenoteacertainrelationwhichholdsbetweenindefinitepropositionsorpropositionalfunctions.Thesetwotermsareusedwithreferencetopropositionswhichcon-tainsomevariables…Henceifwewanttoexaminewhethercertaindefinitepropositionsare,orarenot,formalconsequencesofotherpropositions,wemustfirsttransformallthesepropositionsintoindef-initepropositions,thatis,wemustreplacedefinitetermsbyvariablesrepresentedbycertainletters.TheindefinitepropositionZisaconsequenceoftheindefinitepropositionsP,R,S,…,ifthereexistnovaluesofthevariablescon-tainedinthepropositionsP,R,S,…,ZwhichverifythepropositionsP,R,S,…,butdonotverifythepropositionZ.[Łukasiewicz,1970b,66–7]NoticethatafeatureofTarski’sownprocedure,namely,replacingcon-stantswithvariablesratherthansimplyreinterpretingtheconstantsthemselvesinthedefinitionoflogicalconsequence,ispresenthere.ThismightbepartoftheexplanationofTarski’sprocedure,thoughofcoursegiventhatIntuitionisticFormalismdemandedthataconstantgetoneandonlyonedefiniteinterpretation,Tarskiwouldhavefoundthemovenecessaryanyway.7.1.2ObjectionstoTarski’saccountCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01ThemajorknowncriticismsofTarski’sdefinitionconcernthreebasic,inter-relatedpointsaboutitsadequacyandonequestionaboutTarski’sintentions.First,theeffectofTarski’sdefinitionistotreatthelogicaltruthofasentenceastheplaintruthofanassociateduniversalgeneralization[Etchemendy,1990,95].Etchemendyreferstothisclaimthatlogicaltruthreducestotheplaintruthoftheassociateduniversalgeneralization10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE185—#5\n186AlfredTarski:PhilosophyofLanguageandLogiconnon-logicalvocabularyasthe“ReductionPrinciple”[Etchemendy,1990,98].EspeciallygiventhefactthatTarskiappearsnottoconsidervaryingthedomainsofhismodels,theresultisthatahostoftruthsandfalsehoodsthatareattheverybestcontroversially“logical”—claimsabouthowmanyobjectsthereare,thecontinuumhypothesis,andsoon—turnouttobe“logicaltruths”or“logicalfalsehoods”onTarski’sdef-inition.Similarpointshold,mutatismutandis,forlogicalconsequenceingeneral:logicalconsequenceissimplymaterialconsequenceamongtheassociatedgeneralizations;onesentenceisalogicalconsequenceofothersiffeitheroneofthegeneralizationsoftheseothersisfalseoritsgeneralizationistrue.Argumentsformulatedwithonlyvocabularysingledoutaslogicalarethereforedeclaredvalidaslongastheyhaveeitherafalsepremiseoratrueconclusion.Accordingtotheintuitionsofmany,however,“thereisonething,thereforetherearetwothings”isaparadigmofinvalidity,yetonastandardselectionoflogicalconstantsitcomesoutasvalidonTarski’saccountgrantedonlythattheuniversedoesn’tcontainexactlyonething.Callthis,followingEtchemendy’sterminology,theovergenerationproblem.4Second,bytreatinglogicaltruthandconsequenceintermsoftheplaintruthofassociatedgeneralizations,Tarski’sdefinitionseemsentirelytolosetheintuitiveideathatalogicalconsequenceofasetofsentences“mustbe”[Tarski,2002,187]trueifthemembersofthesetare,that“itcannothappen”[Tarski,2002,183]thatthepremisesofalogicallyvalidargumentaretruebutitsconclusionisfalse.Ifitjustturnsoutthatthegeneralizationassociatedwithasentenceistrue,thesentencewillbecountedaslogicallytruebyTarski’sdefinition;asEtchemendywrites,“thishappenswheneverthelanguage,strippedofthemeaningsofitsnon-logicalconstants,remainsrelativelyexpressive,oriftheworldisrelativelyhomogenous,orboth”[Etchemendy,2008,271–2].Callthisthemodalityproblem.Third,Tarski’sconceptionofconsequenceissupposedtorespecttheintuitionthatlogicalconsequenceistiedtothe“forms”ofsentences[Tarski,2002,183],but,asTarskinotesatsomelength,theassumedaccountof“form”restsonadivisionofexpressionsofthelanguageintologicalandnon-logical,adivisionthatTarskihimselfallowsmayCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01admitofnoprincipledcharacterization[Tarski,2002,188–90].Thislastproblemonitsownmightappearnomoresignificantthananyothernoteworthybutmanageableindeterminacyinananalyticcategory,buttheissuebecomessignificantbecauseitisheldbymanycommentatorsthatthefirsttwocriticismsmightbeansweredbyaproperandprinci-pledaccountofwhatmakesanexpressionlogical.ThisiswhereTarski’s10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE186—#6\nLogicalConsequence1871966“WhatareLogicalNotions?”becomessignificant,andwewillturntoitinduecourse.Callthistheformalityproblem.Finally,turningtothequestionaboutTarski’sintentions,giventheapparentdifficultiesofdefendingTarski’sdefinitionoflogicalconse-quenceassomethinglikeananalysisofwhatisoftenmeantwhenonespeaksinanordinarywayofvalidarguments,thereisasignificantques-tionintheliteratureastowhoseconceptionofconsequenceTarskiwastryingtorepresent,andtowhatstandardsofaccuracyheheldhimself.SomecommentatorstakeittobeobviousthatTarskiheldhimselfrespon-sibletothe“ordinary”meaningsoftheseterms,whileotherstakeittobeequallyobviousthatTarskiwasattemptingmerelytodefinethenotionofconsequenceasitappearedinmathematics,mathematicallogicor“formalaxiomatics”[Jané,2006].Callthistheanalyticproblem.Wewilldiscussitinthelastsectionofthischapter,wherewewillevaluateTarski’saccount.7.2ConsequenceinLogicalSyntaxAlthoughalloftheseproblemsarewellworkedoverintheliterature,surprisinglylittleismadeofTarski’smanyreferencestoCarnap’sLogi-calSyntax.Etchemendy,forinstance,doesn’tmention[Carnap,2002]asaprecursortoTarski’sanalysisat[Etchemendy,1990,7],andotherworkhaslargelyfollowedEtchemendyintheomission,[Coffa,1987]and[deRouilhan,2009]beingtwonoteworthyexceptions.5Thisisespe-ciallystrangegiventhattheonlyexampleofadefinitionofconsequenceTarskimentionsinthearticleisCarnap’s,andthatinthelastparagraphsofthearticleTarskiaddressesthesignificanceofthedistinctionbetweenlogicalandextra-logicaltermsforsometypicalViennaCircleviews.7.2.1L-consequenceandconditionFCarnap’saccountofconsequenceisanimprovementoverastraightfor-warddefinitionintermsoftransformationrulesofthesortthatTarskihimselffavoredthrough1934,andTarskisawfromLogicalSyntaxthatitwasheadedintherightdirection.Hewasalsoabletosee,though,asweCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01discussedinthepreviouschapter,thatitbreaksdownatjustthepointwheresemanticscanbeusedtofixit.TheadvanceinCarnap’saccountwasthatbyaddingthe“indefinite”rulesoftransformation,inparticulartheω-rule,CarnaphadawayofrespondingtoGödel’sincompletenesstheorem,onwhichderivabilitywasinsufficienttocaptureallsentencesthataretrueinthelanguageofatheoryifthetheoryistrue.SinceCarnap10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE187—#7\n188AlfredTarski:PhilosophyofLanguageandLogiceschewed“true”infavorof“analytic”,heneededawaytostateGödel’sresultwithouttheforbiddenterm,andintroducinganinfinitarytrans-formationruleallowedhimtostateGödel’sresultastheclaimthatthereareconsequencesoftheoriesextendingarithmeticthataren’tderivable.Thestudyofthe“formalproperties”ofexpressionsproceedsintermsofadivisionofexpressionsinto“syntactic”categories,whichgroupsymbolsintotypes.InthepreliminarydiscussionsofLanguagesIandII,Carnapsimplyspecifiesthecategories;forLanguageIwehavevari-ables,numerals,predicates,functorsand,importantlyforourpurposes,eleven“individualsymbols”:theparenthesesandcomma,thesucces-sorstroke(which,seebelow,isnottreatedasafunctor),foursententialconnectives,theidentitysign,andthesigns“∃”and“K”,whichformpartsoftheexistentialquantifierandtheK-operator—basicallyakindofdescriptionoperatorwecanglossas“theleastxlessthanysuchthat”.In“GeneralSyntax”,bycontrast,thesyntacticcategoriesaredefinedintermsofsubstitutionsthatpreservesentencehood:twoexpressionsbelongtothesame“genus”justincasesubstitutingonefortheotherinanysentencealwaysproducesasentence,syntacticrulesofforma-tionforsentencesbeingassumed[Carnap,2002,169].Anexpressionisgenerically“isolated”ifithasagenustoitself.InthetreatmentofLanguagesIandIIthedistinctionbetweenlogi-calanddescriptivevocabularyismadeenumeratively:forLanguageI,anyundefinedpredicateorfunctor,oranydefinedpredicateorfunc-torwhichisdefinedultimatelyintermsofanundefinedpredicateorfunctor,isdescriptive;allothersymbols(includingthesuccessorstrokeandthenumerals)arelogical.6Anexpressionisdescriptiveifitcon-tainsadescriptivesymbol,andlogicalotherwise.Ingeneralsyntax,thedefinitioncomesintermsofconsequence(whichwewilladdressinamoment):anexpressionislogicalifandonlyifitbelongstothesmall-estsetofexpressionssuchthateverysentenceformedoutofthemisaconsequenceoftheemptysetofsentencesorhaseverysentenceasaconsequence[Carnap,2002,177–178].Sinceanysentencecontain-ingaprimitivepredicateorfunctorofLanguagesIandII(orsomethingdefinedintermsofone)is“indeterminate”—neitheraconsequenceoftheemptysetofsentencesnorhavingeverysentenceasaconsequence—Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01thecharacterizationmatchesthetreatmentofdescriptiveexpressionsinthetreatmentoflanguagesIandII(thesuccessorstroke,whichofficiallywasn’ta“functor”inIandIIendsupbeinglogicalbecause,giventhetransformationrulesstatedforIandIIitbelongstothesetofexpres-sionssuchthateverysentenceformedonlyfromthemisdeterminate,despitethefactthatsentencehoodispreservedbysubstitutingvarious10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE188—#8\nLogicalConsequence189functorsforit).Alanguageislogicaljustincaseallofitssentencesaredeterminate.Logicality,then,ingeneralsyntax,assumesaprimitivenotionofcon-sequence.Twodistinctionsarerelevanthere;Carnaphasdirectandindirectconsequenceandderivability.Asentenceisdirectlyderivablefromothers,orisadirectconsequenceofthem,ifandonlyifthereisadefinintetransformationrulethatstatesasmuch[Carnap,2002,168];repeatedapplicationsoftheserulesgiverisetoindirectderivation[Carnap,2002,171–2].Thedistinctionbetweenconsequenceandderiv-abilityhastodowithtransformationrulesthatare“definite”asopposedto“indefinite”.Thisdistinction,ingeneral,concernswhetherornotunrestrictedquantificationoverinfinitetotalitiesisinvolved;Carnap’sLanguageIisdefiniteinthatitsquantifiersarealwaysfinitelyrestricted.Whenitcomestotransformationrules,theseareindefinitewhentheystatethatacertainsentenceisatransformationofsomeinfinitesetofsentences.Theexampleinthetextistheω-rule,whichwehaveseenTarskicall“theruleofinfiniteinduction”[Tarski,1983l,259],whichappearsasCarnap’sDC2inthediscussionofLanguageI[Carnap,2002,38].AsCarnapnotes,althoughinallbutthesimplestlanguagesitisnotthecasethateverysentenceformedwithonlylogicalexpressionsiseitherderivableorrefutable,withtheω-ruleadded,everysuchsentenceoritsnegationiseitherconsequenceoftheemptysetofsentencesorhaseverysentenceasaconsequence[Carnap,2002,40].IfonlyCarnapallowedhimselftomakeserioususeofthenotionoftruth,hewouldbeabletosaythatanalyticsentencesarethosethataretrueinvirtueoftheirsyntacticform,butasitishehastorelegatethepointtoaninformalgloss:Bymeansoftheconcept“analytic”,anexactunderstandingofwhatisusuallydesignatedas“logicallyvalid”or“trueonlogicalgrounds”isachieved.Hithertoithasforthemostpartbeenthoughtthatlogicalvaliditywasrepresentablebytheterm“demonstrable”—thatistosay,byaprocessofderivation.Butalthough,forthemajorityofpracticalcases,theterm“demonstrable”constitutesanadequateapproxima-tion,itdoesnotexhausttheconceptoflogicalvalidity.ThesameCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01thingholdsforthepairs“demonstrable”–“analytic”and“refutable”–“contradictory”,asforthepairs“derivable”“consequence”.Inmaterialinterpretation,ananalyticsentenceisabsolutelytruewhatevertheempiricalfactsmaybe.Hence,itdoesnotstateanythingaboutfacts…Asyntheticsentenceissometimestrue—namely,whencertainfactsexist—andsometimesfalse;henceitsayssomethingas10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE189—#9\n190AlfredTarski:PhilosophyofLanguageandLogictowhatfactsexist.Syntheticsentencesarethegenuinestatementsaboutreality[Carnap,2002,41].Thisis,however,simply“materialinterpretation”;theofficialpositionissimplythatananalyticsentenceisaconsequenceoftheemptyset,andconsequenceoutstripsderivability;theinformalnotionoflogicaltruthisformallytakenupbybeingaconsequenceofeverything.NotetheworkthatthenotionofanalyticityheredoesinupholdingCarnap’sTractatus-inspiredconceptionoflogicandthecorrelativetreatmentofmathematics.Wewillreturntothisinamoment.WecomefinallytoCarnap’sunderstandingof“L-consequence”,whichTarskitakesasthenotionoflogicalconsequenceanddefinesinwhathebelievestobeamorecompleteandsuitablemanner.Carnapinformallydistinguishes“L-rules”from“P-rules”intermsoftheformer’sbeing“logico-mathematical”[Carnap,2002,180]andwrites:NowhowisthedifferencebetweenL-rulesandP-rules—whichwehavehereonlyindicatedinaninformalway—tobeformallydefined?Thisdifference,whenrelatedtoprimitivesentences,doesnotcoin-cidewiththedifferencebetweenlogicalanddescriptivesentences.AnSlasaprimitivesentenceisalwaysanL-rule;butanSdasaprimitivesentenceneednotbeaP-rule.[Example:Let‘Q’beaprdofLanguageI.Then,forexample,‘Q(3)⊃(∼Q(3)⊃Q(5))’(S1)isadescriptiveprimitivesentenceofthekindPSIi.ButS1isobvi-ouslytrueinapurelylogicalway,andwemustarrangethefurtherdefinitionssothatS1iscountedamongsttheL-rulesandiscalled,notP-valid,butanalytic(L-valid).ThatS1islogicallytrueisshownformallybythefactthateverysentencewhichresultsfromS1when‘Q’isreplacedbyanyotherprislikewiseaprimitivesentenceofthekindPSIi.]TheexamplemakesitclearthatwemusttakethegeneralreplacabilityoftheAdasthedefinitivecharacteristicoftheL-rules[Carnap,2002,180–1].Carnaphereexpresseshisviewaboutwhatittakestobe“trueinapurelylogicalway”:asentenceisanalyticifitfollowsfromtheemptysetCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01andcontinuestodosounderalluniformsubstitutionsofnon-logicalvocabulary.WorkingthisideaoutgivesuswhatwecanrecognizeasthebasicideaTarskiworkswithincondition(F):asentenceisanL-consequenceofasetofsentencesjustincaseeveryuniformsubstitutionofnon-logicalexpressionsproducesasentenceandasetsuchthatthesentenceisaconsequenceoftheset[Carnap,2002,181].HereisCarnap’s10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE190—#10\nLogicalConsequence191definition;rememberthatconsequenceitselfissimplyamatterofstip-ulatedtransformationrulesandthatthedefinitionsortsconsequencesintoL-consequencesandtherest:LetS2beaconsequenceofK1inS.Herethreecasesaretobedistin-guished.1.K1andS2arelogical.2.DescriptiveexpressionsoccurinK1andS2,butonlyasundefinedsymbols;heretwofurthercasesaretobedistinguished:2a.foranyK3andS4whichareformedfromK1(orS2)bythereplacementofeverydescriptivesymbolofK1(orS2respectively)byanexpressionofthesamegenus,andspecificallyofequalsymbolsbyequalexpressions,thefollowingistrue:S4isaconsequenceofK3…Thiswouldbecompleteifitweren’tforthefactthattheuseofdefinedsymbolscandisguisetheunderlyingformofthesentence,andinpartic-ularwhetheradefinedpredicateisultimatelydefinedintermsoflogicalornon-logicalprimitives[Bays,2001,1724];henceweneedtoconsiderthethirdcase:3.InK1andS2defineddescriptivesymbolsalsooccur;letK¯1andS¯2beconstructedfromK1(orS2respectively)bytheeliminationofeverydefineddescriptivesymbol(includingthosewhicharenewlyintroducedbytheresultofanelimination);3a.theconditiongivenin2aforK1andS2isfulfilledforK¯1andS¯2…Incases1,2a,3a,wecallS2anL-consequenceofK1…ThustheformaldistinctionbetweenL-andP-rulesisachieved[Carnap,2002,181].Tarski’scondition(F)issimplyamodificationthatreplacestheappealtoconsequencedefinedintermsofprimitivetransformationruleswiththeideathatifthepremise-setistrue,theconclusionmustbetrue.7(Tarskileavesoutthethirdconditiononly“forthesakeofsimplifyingourcon-siderations”[Tarski,2002,184]).Tarski’saimissimplytorestoretheintuitiveconnection,brokenbyCarnap’sskepticismaboutsemantics,betweenlogicalconsequenceandtruth,andtomakeupforthedefini-tion’srelianceonsubstitution—thetwoaspectsofCarnap’sskepticismaboutsemanticswediscussedin§3.1.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-017.2.2TractarianismintheViennacircleThat,then,isthebasicidea:TarskiissimplyouttoadjustCarnap’sdefini-tionofL-consequencefromGeneralSyntaxbyreplacingasubstitutionalinterpretationofL-validitywithanobjectualone,andtransformationruleswithasemanticaccountofconsequencethatimpliesthatthecon-sequencesoftruesentencesaretrue.However,Tarskialsohadanaxe10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE191—#11\n192AlfredTarski:PhilosophyofLanguageandLogictogrindwithCarnap.Carnap,alongwiththemajorityoftheViennaCircle,heldthattheTractariandoctrinethatlogicsaysnothingabouttheworldiscrucialtoascientificworldview,since,ontheirview,itwastheonlysatisfactoryconceptionofmathematicscompatiblewithempiricism:TheconceptionofthenatureofmathematicswhichwedevelopedinthediscussionsoftheViennaCirclecamechieflyfromthefollowingsources.IhadlearnedfromFregethatallmathematicalconceptscanbedefinedonthebasisoftheconceptsoflogicandthatthetheoremsofmathematicscanbededucedfromtheprinciplesoflogic.Thusthetruthsofmathematicsareanalyticinthegeneralsenseoftruthbasedonlogicalone.ThemathematicianHansHahn,oneoftheleadingmembersoftheCircle,hadacceptedthesameconceptionundertheinfluenceofWhiteheadandRussell’swork,PrincipiaMathematica.Fur-thermore,Schlick,inhisbookAllgemeineErkenntnislehre(1918),hadclarifiedandemphasizedtheviewthatlogicaldeductioncannotleadtonewknowledgebutonlytoanexplicationortransformationoftheknowledgecontainedinthepremises.Wittgensteinformulatedthisviewinthemoreradicalformthatalllogicaltruthsaretautological,thatis,thattheyholdnecessarilyineverypossiblecase,thereforedonotexcludeanycase,anddonotsayanythingaboutthefactsoftheworld.Wittgensteindemonstratedthisthesisformolecularsentences(i.e.thosewithoutvariables)andforthosewithindividualvariables.Itwasnotclearwhetherhethoughtthatthelogicallyvalidsentenceswithvariablesofhigherlevels,e.g.,variablesforclasses,forclassesofclasses,etc.,havethesametautologicalcharacter.Atanyratehedidnotcountthetheoremsofarithmetic,algebra,etc.,amongthetautologies.ButtothemembersoftheCircletheredidnotseemtobeafundamentaldifferencebetweenelementarylogicandhigherlogic,includingmathematics.Thuswearrivedattheconceptionthatallvalidstatementsofmathematicsareanalyticinthespecificsensethattheyholdinallpossiblecasesandthereforedonothaveanyfactualcontent.WhatwasimportantinthisconceptionfromourpointofviewwasCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01thefactthatitbecamepossibleforthefirsttimetocombinethebasictenetofempiricismwithasatisfactoryexplanationofthenatureoflogicandmathematics.Previously,philosophershadonlyseentwoalternativepositions:eitheranon-empiricistconception,accordingtowhichknowledgeinmathematicsisbasedonpureintuitionorpurereason,ortheviewheld,e.g.,byJohnStuartMill,thatthe10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE192—#12\nLogicalConsequence193theoremsoflogicandofmathematicsarejustasmuchofanempiricalnatureasknowledgeaboutobservedevents,aviewwhich,althoughitpreservedempiricism,wascertainlyunsatisfactory[Carnap,1963,46–7].CarnapsawthetreatmentoflogicandmathematicsinLogicalSyntax,onwhich“Inmaterialinterpretation,ananalyticsentenceisabsolutelytruewhatevertheempiricalfactsmaybe.Hence,itdoesnotstateany-thingaboutfacts”[Carnap,2002,41]asawayofcontinuingtoworkoutthebasiclogicistreconciliationofempiricismwiththestatusofmathematics:Theideaoccurredtomethat…thereseemedtobeapossibilityofreconcilingtheconflictbetweenintuitionismandformalism.Sup-posethatmathematicsisfirstconstructedasapurelyformalsysteminHilbert’sway,andthatrulesarethenaddedfortheapplicationofthemathematicalsymbolsandsentencesinphysics,andfortheuseofmathematicaltheoremsfordeductionswithinthelanguageofphysics.Then,itseemedtome,theselatterrulesmustimplicitlygiveaninterpretationofmathematics.Iwasconvincedthatthisinterpre-tationwouldessentiallyagreewiththelogicistinterpretationofFregeandRussell[Carnap,1963,48].Since,asLogicalSyntaxheld,“Inlogic,therearenomorals”[Carnap,2002,52],Carnapfeltfreetostipulateformationandtransformationrulesthathadtheconsequencethatmathematicalstatementswere“analytic”inthesenseofbeingderivablefromeverything.Suchlanguageswereintendedtobeappliedinscienceinaccordwithcoordinatingdefini-tions[Carnap,2002,78],butthequestionofthestatusofmathematics,anditsreducibilitytologic,wereavoidedviathetolerantconceptionoflogic.(OnthedevelopmentfromFrege’slogicismandtheroleofsuchcoordination,see[Ricketts,2004]and[Ricketts,2009].)Thecornerstoneoftheapproach,however,wasstilltherigiddistinctionbetweenlogicandfactualstatementwhich,combinedwithlogicismandtheTractarianideathatsincelogicaltruthsexcludenothingtheyaren’treally“aboutCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01reality”,obviatedtheperennialquestionsaboutthestatusofmathemat-icalobjectsandourknowledgeofthem.AsIwillarguebelow,Tarski’sremarksonthelogicalconstants,andhisclaimthatlogicalconsequencedegeneratestomaterialconsequenceifallexpressionsofthelanguagearetakentobelogical,areintendedtoexpresshisdisagreementwiththeseviews.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE193—#13\n194AlfredTarski:PhilosophyofLanguageandLogic7.3TheovergenerationproblemanddomainvariationTheinterpretivehypothesisisthatTarski,inhistreatmentofconse-quenceitself,intendsin[Tarski,2002]onlytomodifyCarnap’sdefinitionofL-consequencebysupplying,viasemantics,objectualquantificationinfavorofCarnap’ssubstitutionalquantification,andtruth-preservationinfavorofprimitiverulesof“directconsequence”.Theevidenceforthehypothesisisofseveralsorts.First,thatTarskipresentedthepapertoaroomfullofCarnapandhisfriendsandcolleaguesandtheParisUnityofSciencecongress.Tarskiwouldhavewantedtoshowcasewhathehadtoofferwhilearousingasfewextraneousanddistractingdisputesaspos-sible.HethereforeassumedallelementsofCarnap’sconceptionsavethosehespecificallywantedtoreplacewithsemantics.Second,Tarski’sadequacyconditionisanobviousmodificationofCarnap’sdefinitionofL-consequence.Finally,theremainingfeaturesofthearticle—namely,theskepticismaboutthedistinctionbetweenlogicalandnon-logicalconstantsthedefinitionitselfuses—aretobeexplainedbyTarski’sdis-agreementwithCarnapovertheTractarianconceptionoflogic.Thebesttestofthehypothesis,however,isinthetreatmentofthestandardinter-pretivepuzzles.Letusnowseehowtheproposedreadingfareswiththeovergeneration,modality,formality,andanalyticproblems,startingwiththefirst.87.3.1DomainvariationTheessenceoftheovergenerationproblemisthat,bytreatinglogicaltruthastheplaintruthoftheuniversalgeneralizationofasentencewithrespecttoitsnon-logicalvocabulary,Tarski’sdefinitiondeclaresanysentenceformulatedonlyinlogicalvocabularyaslogicallytrueorlogicallyfalse.Yetmanysuchsentences,suchas(∃x)(∃y)(x=y),seemneitherlogicallytruenorlogicallyfalse.Thestandardmodel-theoreticaccountoflaterdecadesavoidsthisconsequencebyvaryingthedomainofquantificationacrossinterpretationssothatthissentenceisneitherlogicallytruenorlogicallyfalse,asthereareinterpretationswithdomainsoffewerthantwoelements.Sympatheticinterpretersherefallintotwocategories,maintainingeitherthatTarski’saccountinfactdoesn’thaveCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01theconsequencesdeemedobjectionablebecauseTarskiactuallyassumedsuchvariationasread,orthattheseconsequenceswerenotinfactobjec-tionabletoTarskibecausehedidn’tthinkthatlogicshouldlacksuchmathematicalcontent.ThefirstaccounttakesitsinspirationfromTarski’sdiscussionoftheconceptsof“satisfactionandcorrectsentence…withrespecttoagiven10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE194—#14\nLogicalConsequence195individualdomaina”at,e.g.,[Tarski,1983a,239–40](§4.2.3).SinceinthispassageTarskiexplicitlyrecognizesthepossibilityofrelativizingthedomainofquantificationandinparticularofthepossibilityofstudyingwhichsentencesremain“correct”ineveryrelativizeddomain,andsincehenotesthat“thepointofviewisbetterestablished”thatlogicisonlyconcernedwithsentencescorrectineveryindividualdomain,thereisplausibilityinthesuggestionthatTarskiassumesin[Tarski,2002]thatthedomainvariesacrossinterpretations,andthissuggestionisthebasisofthefamilyofinterpretationsonwhichitisarguedthatTarski’saccountisn’tsubjecttotheovergenerationproblembecauseTarskiassumeswith-outmentioningitthatthedomainofquantificationissupposedtovaryacrossinterpretations,e.g.[Ray,1996,628–30],[Sher,1991,45ff],[Gómez-Torrente,1996,143–5].9Thewell-knownweaknessoftheread-ingisthatTarskinowherein[Tarski,2002]saysthatthedomainvariesand,moreover,thatTarskiholdsthatlogicalconsequencereducestomaterialconsequencewhenallexpressionsofthelanguagearetreatedas“logical”,whichwouldbewrongifthedomainvaried,since“thereisonething,thereforetherearetwothings”wouldhaveatruepremiseandafalseconclusioninaone-elementdomain.Attemptstoreaddomainvariationinto[Tarski,2002]taketwoforms.Oneeither(i)arguesthatTarskijustisassumingthatthesequencesthatformhismodelsareofvariablelengthwithoutmentioningit([Sher,1996],[Ray,1996](endorsedby[Hanson,1999,610]andStroinskaandHitchcock’sintroductionto[Tarski,2002,170]),or(ii)arguesthatthoughTarski’ssequencesareallinfinite,suitableuseofimplicitrestrictionoftherangeofthevariablesgivesTarskitheeffectofvariabledomainswithintheconfinesoftheconceptionmoreclearlyonthesurfaceof[Tarski,2002]([Gómez-Torrente,1996]).Ourwaywith(i)canbebrief:asEtchemendynotes[Etchemendy,2008,280],thereissimplynoevi-dencethatTarskiassumesthatthedomainvariesin[Tarski,2002]andthereadingisincompatiblewithTarski’sclaimthatlogicalconsequencedegeneratestomaterialconsequenceifalltermsaretreatedaslogical.RayholdsthatthisissimplyamistakeonTarski’spart,butbyhisownadmissionthisdependsontheassumptionthatitisamarkofgreaterstupiditytothinkthatthat“therearetwothings”isalogicaltruthCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01thanitistothinkthat“therearetwothings”istrueinaoneelementdomain.Likemostinterpreters[Bays,2001,1709],[Etchemendy,2008,280],[Gómez-Torrente,2009,268],Ifindthisimplausible.Thestrongerapproachis(ii).Gómez-TorrentepushesthelinethatTarskioftenformulatedtheorieswithapredicateDcharacter-izingthedomainofobjectswithwhichthetheorywasconcerned10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE195—#15\n196AlfredTarski:PhilosophyofLanguageandLogic[Gómez-Torrente,1996,143].Theideahereisthatsincethispredicateisnon-logical,itwillreceivedifferentinterpretationsindifferentmodels,andsinceitspecifieswhatthetheoryisabout,thevariableswithwhichothernon-logicalconstantsarereplacedinTarski’stestforconsequencewillrangeoveritandhenceoveravariabledomain.Thus(∃x)(∃y)(x=y)isreallyshorthandfor(∃x)(∃y)((Dx&Dy)&(x=y))and,theideais,thisisfalseonanyTarskianmodelwherethevariablethatreplacesDisassignedanextensionof<2elementsandthesentenceistherebynotclassifiedaslogicallytrue.However,asMancosupointsout[Mancosu,2006,220–6]thediscus-sionofcategoricityin[Tarski,1983k]andotherpassagesmakeclearthatTarski,followingLangford,allowedanindeedoftendemandedthatthe-oriesquantifyoverobjectsnotwithintheextensionofD.SinceTarskiisworkingwithinSTT,thismeansthatforthetheoriesheconsidersingen-eralD=Vandthusthatthevariabilityoftheextensionof“D”impliesnothingaboutthevariabilityofVand,correlatively,thatthequantifiersaren’ttobereadasimplicitlyrelativizedtoD.NotealsothatMancosudistinguishes“strong”from“weak”fixeddomainviews,whereastrongviewholdsthatVisfixedonceandforallineveryapplicationofSTT(presumablyto“allobjects”)whiletheweakviewissimplythatthoughthedomainofthequantifiersisfixedtobeVinanygivendefinitionofconsequenceforSTT,Vitselfistakentohavedifferentcardinalitiesindifferentapplications.AsMancosualsopointsout,itwascommonforotherauthors,inparticularGödel,toworkinSTTwithVfixedtobesomethingotherthaneverythingthatexists[Mancosu,2006,232–6].AllthatisshownbyTarski’svariationofVacrossexamplesisthatVwasn’tconceivedofbyhimorotherstoberequiredactuallytobetheentiretyoftheuniverseinaparticularapplication.10Inordertosortthingsouthereweneedtointroduceapreviouslyneglecteddistinction.OnequestioniswhetherTarskiinthe1920sand1930shadtheideaofasetofsentencesbeingsatisfiedbydifferentsetsofobjectsandrelationsuponthem,where,inparticular,thesesetscouldvaryincardinality.AsecondquestioniswhetherinthesameperiodTarskimadesomeuseoftheideathatmodelsofvariouscardinalitieswerequantifiedoverinthedefinitionofconsequenceitself.ItisbeyondCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01questionthatTarskihadthefirstideafromveryearlyon[Feferman,2008,78–9].Whatisn’tclearatallisthatsuchdomainvariationplayedanyroleinhisconceptionoflogicalconsequenceandlogicaltruth.ThetextualevidencemarshalledbyGómez-Torrenteestablishesthefirst,butonlytheill-supportedsuggestionthat,inatleastsomeapplicationsofthedefinitionofconsequenceTarskitiestherangeofthevariablestothe10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE196—#16\nLogicalConsequence197extensionofthedomainpredicate,makesitlookasthoughtheanswertothesecondisintheaffirmative.ThemerefactthatTarskihadtheideathatthatsometheories,orevenlogicitself,consistedofsentencesthataretrueindomainsofvariablecardinalitydoesn’tofitselfestablishthatheconceivedofsemanticconsequenceasheintroduceditintermsofmodelswithdomainsofvariablecardinality.Thisdistinctioninhand,wecanconsiderthefactthat[Gómez-Torrente,2009,255ff]respondstoMancosuwithmanyexamples,mostlytakenfromthetextbook,ofTarskitakingitthatachangeintheexten-sionofthedomainpredicateofatheorycorrelateswithachangeintheintendedrangeofthevariablesinitssentences.TheproblemisthattheexamplesarealleasilyaccountedforbyMancosu’sidea[Mancosu,2006]thatTarskiallowsVtovaryacrossapplicationsbutdoesn’tholdthatD=V,aslongasweassume,asisreasonable,thatifthereisanyplacethatTarskiisn’tbeingfullyexplicitaboutthelogicalbackdropofthetheoriesheconsiders,itisinthetoyexamplesofthetextbook.ThepassagesGómez-Torrentecitesneversaythattherangeofthevariableschangesbecausethedomainpredicatehasadifferentextension;theysimplyratherinarticulatelymakethenovicereaderfamiliarwiththeideaofvaryingtheinterpretationofatheoryinanintuitiveway.Ican’trespondtoallpointsofdetailhere,butforthesortofthingIhaveinmind,considerGómez-Torrente’sargumentthatonthefixeddomainconceptionoftherangeofthevariableswegettheresultthatcontradic-tionsareprovableinthetheoryiftheextensionofthedomainpredicateismovedoutsideofthefixedrangeofthevariables[Gómez-Torrente,2009,258].This,however,assumesthe“strong”fixeddomainconcep-tion;the“weak”conception,bycontrast,handlesitjustfine:whenTarskiconsidersreinterpretinghistoytheoriesinthetextbook,hechangestheinterpretationsofboththerangeofthevariablesandthedomainpred-icate,withoutassumingthattherangeofthevariablesissettledbytheextensionofthepredicate.AtmostestablishingthatTarskicontemplatedtheoriesinwhich,e.g.(∃x)(∃y)(x=y)isreallyshorthandfor(∃x)(∃y)((Dx&Dy)&(x=y))wouldestablishthatTarski’saccountoflogicalconsequencegivesreasonableverdictsonwhatfollowslogicallyfromnon-logicaltheoriesformulatedCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01inthismanner.Itwouldn’taddresswhatEtchemendytakestobethepressingproblem,namely,counting(∃x)(∃y)(x=y)asaplainlogicaltruth.ItistruethatTarskicangettheeffectofvariabledomainsforatheorystatedwithinSTTthroughrestrictingeverythingbyadomainpredicate,butthisdoesnotshowthatTarskiallowsthatthenotionoflogicalconsequenceforSTTitselfisavariable-domainnotion.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE197—#17\n198AlfredTarski:PhilosophyofLanguageandLogicApotentialflyintheointmenthereisTarski’sdiscoveryoftheupwardLöwenheim–Skolemtheoreminthelate1920s,asannouncedinanoteinFundamentaMathematicae[Mancosuetal.,2009,433–4].Afterall,iftheresultisthatanyfirst-ordertheorywithacountablyinfinitemodelhasamodelineveryinfinitecardinality,isn’tTarskithinkingofmod-elsasvaryingincardinality?However,thisproblemiseasilysolvedoneitherGómez-Torrente’sorMancosu’sapproaches,sincetheresultisthatanytheoryexpressedfirst-orderthathasacountablemodelhasmodelsinfurtherinfinitecardinalities.WehavealreadyseenthatvariabledomainsforatheorycaneasilybeaccommodatedbyTarski’svariousapproachesatthetime,evenifVisheldfixedinSTTandtherebyinthedefinitionofconsequence.Forthepresentapplication,simplyletVbe“everything”oratleastsomeproperclass,andletDbeanon-logicaldomainpred-icate;thegeneralLöwenheim–Skolemresultisthatafirst-ordertheorycanatmostforcetheextensionofDtobeinfinite,butcannotsettleitscardinalityanyfurtherthanthat.Noneofthisgoestoshowthattherangeofthevariablesinthelanguageofthetheoryitselfchangesacrossinterpretations,andhencenoneofitgoestoshowthatinthetestforconsequenceitselfTarskiisallowingtherangeofthevariablestovary.Thepositionthatemergesiscoherent.Logicitselfassumesafixeddomain:allindividuals.11Whendefininglogicalconsequenceandlogi-caltruththisistheinvariantdomain(cf.[Milne,1999,158–9]).However,amatterofinterestistoconsiderwhatfollowslogicallyfromatheorystatedwithinabackgroundlogicwhenitsdomain,specifiedbyanon-logicalpredicate,variesinaccordwithaperfectlystandardapplicationofthe(domaininvariant)definitionofconsequence.Whenthingsaredonethisway,ofcourse,a“variabledomain”conceptionofwhatfollowsfromtheaxiomsofthattheoryarisesautomatically:thelogicalconsequencesoftheaxiomsarejustwhatfollowsfromthembythefixed-domainaccountofconsequence,inthattheinclusionofthedomainpredicateforceswhatfollowslogicallyfromtheaxiomstobewhatfollowsnomatterwhatthecardinalityofD’sextensionis.Whatfollowslogicallyfromthetheoryisthenautomaticallywhatfollowsindependentlyofthecardinalityofitsdomain.Seeninthislight,Tarski’sinvestigationoftheorieswithdomainpredicatesisperfectlycompatiblewithaconceptionoflogicproperthatCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01assumesafixeddomainofallindividuals.7.3.2ConsequenceinGödel’scompletenesstheoremTheprecedingpointinhand,wecanconfrontthesuggestionthatissometimesmadethatTarskiwasawareofvariabledomainconcep-tionsofconsequencebecausetheyplayedaroleinthestatementof10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE198—#18\nLogicalConsequence199thecompletenesstheoremforfirst-orderlogic.Gómez-Torrente[Gómez-Torrente,1996,138]takesthefactthatTarskiwasawarepassageslikethisonefromHilbertandAckermannasevidencethathewasfamiliarwithvariabledomainconceptionsofconsequence:Whetherouraxiomsystemiscomplete(vollständig),atleastinthesensethatalllogicalformulaethatarecorrect(richtig)ineverydomainofindividuals(Individuenbereich)canactuallybederived,isyetanunsolvedquestion[HilbertandAckermann,1928,68](mytranslation).Theweaknesshere,however,isasabove:Tarski’sfamiliaritywithGödel’scompletenesstheoremshowsonlythathehadthenotionoftruthindomainsofallcardinalities,andnotthathisconceptionofconsequenceitselfinvolvedvaryingthecardinalityofthedomain.However,thepassagesimplydoesnotexpressavariabledomainconceptionofconse-quence.WhatHilbertandAckermanndoisthis.Theyhavealanguage,andanaxiomatictheoryinthatlanguage.Thesetofsentencesofthelanguagehasacertainsubset,namely,thesetofthosesentencesofthelanguagethatare“correct”(thatis,true)ineveryindividualdomain.Nowthequestionarises:canallofthesentencesofthatsetbederivedfromtheaxiomsbytherulesofinferencegiven?Thereisnothinghereaboutthesesentencesbearingsomerelationotherthanderivabilitytotheaxioms,nothing,e.g.abouttherebeinganon-derivationalrelationof“following”thatsentencescanbeartotheaxiomsonlyinawaythatisindependentofthecardinalityofthedomain.Thequestionissimplywhetherthosesentencescorrectineveryindividualdomainarederiv-ablefromtheaxioms.Theonlyconceptionofconsequenceatworkinthequestionisderivability.Thepassagesimplyprovidesanexam-pleofwhatweknowTarskiwasawareof,thatthenotionoftruthinagiven,orevery,individualdomainwasofinterestandthatthecorrel-ativequestionsabouttherelationsbetweenprovabilityandcorrectnessineveryindividualdomainwereofinterest.Thepassagedoesn’tshowthatHilbertandAckermannhadanon-derivational,semantic,variabledomainconceptionoflogicalconsequencein1928(cf.[Etchemendy,1988,75]).Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Nordoes[Gödel,1967a]showthatGödelhadsuchaconception.Gödel’sconceptionofthequestionheanswersisexactlyHilbertandAckermann’s:iseverysentenceofHilbertandAckermann’s“restrictedfunctionalcalculus”thatis“validineveryindividualdomain”[Gödel,1967a,584]provablefromtheaxioms?ThereisinGödel’spapernomorethaninHilbertandAckermann’sbookanydefinitionorimplicit10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE199—#19\n200AlfredTarski:PhilosophyofLanguageandLogicconceptionof“followingfrom”intermsofmodels,betheyofvariablecardinalityornot.Thequestionissimplywhetherallsentencesthatarevalidineveryindividualdomainarederivablefromthegivenaxiomsbythegivenrules.12Thereis,inhindsight,aconceptualrestrictionhere:whybeinter-estedmerelyinwhethereverysentencethatistrueineveryindividualdomainisprovablefromtheaxiomsbytherules,andnotinthemoregeneralquestionofwhethereverysentencesandsetsuchthatineverydomaineitheramemberofisn’ttrueorsis,isalsosuchthatscanbederivedfromgiventheaxiomsandrulesofinference?Ifthesecondquestionhadn’treducedtothefirst(viathedeductiontheorem)inthecaseHilbertandAckermannandGödelconsidered,amodel-theoreticconceptionofofconsequencewouldhavehadtohavebeenarrivedatbyHilbertandAckermanninordertoformulatethequestionthatGödelthenanswered.13Giventhattheirinterestwaspreciselyin“correctness”ineverydomain,suchaconceptionofconsequencewouldhaveinvolvedvariabledomainsandhistoryintherelevantrespectsmighthavebeendifferent.Asitwas,however,thesecondquestiondidreducetothefirstandsonoconceptionofconsequenceotherthanderivabilitywasrequiredtoformulatethecompletenesstheoremasGödelprovedit(cf.[Etchemendy,1988,68]).Weshouldnotletourfamiliaritywiththenow-standardstatementofthetheoremthatif|=sthensmakeusoverlookthis(cf.[Etchemendy,1988,76–7].NowtherearehintsofatleastadistinctionbetweenderivabilityandconsequenceinsomeauthorsknowntoTarskiandpredatingGödel.InVeblen’swork,e.g.[Veblen,1904,346],assuggestedbyMancosu,ZachandBadesa[Mancosuetal.,2009,328]and[Veblen,1906,28],assug-gestedbyAwodeyandReck[AwodeyandReck,2002,18]onefindsatleastarudimentarydistinctionbetweensemanticconsequence(iftheaxiomsaretrue,thenagivensentenceistrue)andderivability.TheVeblenpassagesaretantalizingandifVeblenintendedtosaywhatheappearstosayhewaswellaheadofhistime,butthesuggestionsaresounderdevelopedthatitwouldbedifficulttoascribeanyhistoricalinfluencetothem.Moreproximally,Fraenkel[Fraenkel,1928,347–9]clearlydistinguishesderivabilityfromsemanticconsequenceanddoubtsCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01thatallsemanticconsequencesofatheorymustbederivable(accordingto[AwodeyandReck,2002,22],followingWeyl).However,inneitherFraenkelnorinVeblenisthereanythingaboutmodelsvaryingincardi-nalityortheinterestbeinginsemanticconsequenceinallcardinalities.Thenotionatworkisthesamefixeddomainsemanticnotionofconse-quencewithwhichwehaveseenTarskiwasfamiliarfromŁukasiewicz.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE200—#20\nLogicalConsequence201ButitisthevariablenotionthattheobjectionweareconsideringinthissubsectionclaimsmusthavebeenknowntoTarskifromearlierauthors.Contrarytotheobjection,however,thesituationisthesameinTarski’spredececessorsasitisinTarski’sworkitself,[Etchemendy,1988,70]:sometimespeopleworkedwithvariabledomains,sometimepeo-plevariedtheinterpretationofnon-logicalexpressions,butnobodywasputtingthetwoideastogetherintothemodel-theoreticconceptionofconsequenceaswehaveittoday(cf.[Milne,1999,159]).ImentionthispointbecauseitgoestotheheartofthematterwhenitcomestothevariabledomainreadingsofTarski.Thestrongestpointinfavorofthesereadingsis,supposedly,thatTarskiandothers,wellbefore[Tarski,2002],werethinkingabouttherelationoftruthor“correctness”thatheldbetweensentencesanddomainsofvariablecardinality.Onthisbasis,thesereadingsthensuggestthatsinceTarskiwasfamiliarwithvariabledomainconceptionsofmodelshemusthaveintendedtoofferavariabledomainconceptionoflogicalconsequencein[Tarski,2002],oreventhatavariabledomainconceptionofconsequenceis“presupposed”bythecompletenesstheorem,whichTarskiobviouslyknewandunder-stood.Theproblemwiththissuggestionisthatwhatwefindat[Tarski,1983a,239–40],[HilbertandAckermann,1928,68]or[Gödel,1967a]isn’tanythingthatmakesanyconnectionatallbetweentheideaofasetofsentenceshavingmodels(ofvariablecardinalityornot)andtheconceptoflogicalconsequenceitself.Wesimplyfindtheideathattherearesentencesthataretrueregardlessofthecardinalityofthedomainandthenthequestionofwhetherthesesentencesarederivableinthisorthatsystem.Ofcourse,especiallyinhindsight,themovefromthinkingofsentencesthataretrueineverydomaintothinkingofsentencesthatare,ineverydomain,trueifcertainothersaretrue,isnotabigstep.Butwhatwasabigstepwasthemovefromthinkingofconsequenceintermsoftherepeatedapplicationofagiven,simplesetofprimitiveformsofinferencetothinkingofitashavinganythingtodowiththerela-tionbetweensentencesandtheirmodels.OnlytherevelationofGödel’sincompletenesstheoremcouldunderminethestandingfaithintheideathatlogicalconsequencewassimplyamatterofderivabilityviarepeatedapplicationofcertainobviouslyvalidrules;thecompletenesstheoremCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01alonehadpreciselytheoppositeeffect.7.3.3Tarski’sfixeddomainAfurtherpointneedstobemade,inlightofthisdiscussionofGödel,aboutTarski’srelationshiptoCarnap.CarnaptookseriouslytheideathatwhatGödelshowedinthefirstincompletenesstheoremwas,atleast10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE201—#21\n202AlfredTarski:PhilosophyofLanguageandLogicinpart,thattheideaofonesentencefollowingfromotherscouldn’tbecashedoutintermsofderivation.Carnapdrewtheconclusionthatinfinitary“rulesofproof”couldclosethegap,andindeedtheycould,butrelianceonthemstretchedthemeaningof“proof”toofar.TarskirecognizedthatCarnap’streatmentofL-consequencetookanessentialstepawayfromaderivationalconceptionofconsequence,butalsothat,duetoCarnap’sTractarianhostilitytotalkofword–worldrelationsandhisconcomitantrelianceonsubstitutionalinterpretationsofgenerality,Carnap’snotionwasstilltootiedtotheactualexpressivepowerofthelanguagestowhichitwasapplied,andthatitwashobbledbyitsreferencetotheassumedtransformationrules.WorkingfromCarnap,hereplacedtherelianceonsubstitutioninCarnap’sconceptionofL-consequencewiththenotionofsatisfaction,andtheresult,tothatpoint,wasthedefinitionoflogicalconsequenceaswehaveitin[Tarski,2002].Con-ceivingofconsequenceintermsofmodelsatallwas,then,asignificantconceptualleap;asnotedabove,interpretivesuggestionstothecontrarynotwithstanding,theideasimplyisn’ttobefoundincommonlycitedpassagesofHilbertandAckermannorGödel.Tarskithenheldtothefixeddomainconceptionoflogicalconsequenceatleastthrough1940[Mancosu,2006].Nothinginthetreatmentoftruthinanindividualdomaininhiswritingsorthewritingsofthosehementionsbefore[Tarski,2002]speaksagainstthis.Withconsequencedefinedashedefinedit,Tarskiwasstillfreetocon-siderwhichsentencesofvarioustheoriesweretrueinvariousindividualdomains,andtoseewhichfollowedlogicallyfromthesetheorieswhentheywererestrictedbyanon-logicaldomainpredicatetheextensionofwhichthereforevaried,butallofthistookplaceagainstabackgroundconceptionoflogicalconsequenceitselfinwhichthedomainwasfixed.Now,sinceTarskiwasatmostbutastone’sthrowfromapplyingavari-abledomainconceptionofmodelsinthedefinitionofconsequencein1936,wehavetoaskwhyheoperatesresolutelywithafixeddomainofquantification.Moreover,wealsohavetoaddressaquestionaboutGödel’scompletenesstheoremasTarskiknewitandhisdefinitionofconsequence:asEtchemendypointsout[Etchemendy,1988,70],sinceGödelproperlyunderstooddidtreatlogicaltruthastruthindomainsofCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01allcardinalities,Tarskihastohaveknownthathisdefinitionin[Tarski,2002]impliedthatGödelhadn’tshownthatfirstorderlogicisactually“complete”,sinceTarski’smodelshaveafixedinfinitedomainandthedeductivesystemGödelconsideredcouldn’tprovethetruthsaboutthisdomainthatTarski’sdefinitiondeclareslogical.Thequestioninthebodyofthissectionwaswhetherthemodelsquantifiedoverinthedefinition10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE202—#22\nLogicalConsequence203ofconsequenceitselfareofvaryingcardinality.ThequestionhereisthesubsidiaryhistoricaloneofwhyTarskidoesn’tuntilsurprisinglylatecombinetwoideasthathehadhadinthe1930stoproducethedefini-tionofconsequencethatbecamestandard(andacceptedbyhim)bythe1950s.HereIthinktworelativelystandardsuggestionsareessentiallyright.ThefirstisthatsinceTarskitookSTTtobelogic,hesimplydidn’tthinktheaccount“overgenerated”inEtchemendy’ssense.14Tarskineverexplicitlytalksaboutlogicism(see[Murawski,2004]onthegeneralatti-tudetowardsuchquestionsinWarsaw),buthewascertainlyhappytocountenancehigher-orderlogicaslogic,andwiththiscame,viatheusualdefinitions,mathematics.Thesecondstandardsuggestion(e.g.[Milne,1999])isalsoimportant.Inthe1930sTarski’stargetnotionsarecon-ceivedofasabsolute:beingtrue,andbeingtrueifotherthingsare,intheuniverseasitis.Tarskitookthequestionofwhatfollowedlogicallyfromwhattobethequestionofwhatwastrueunderallinterpretations,holdinglogicalvocabularyfixed,period,notwhatwastrueinthisorthatsubregionoftheuniverse,orwhatwastrueineverysubregionoftheuni-verse.Logicwas,forTarskiatthetime,aboutwhatwastruegiventhatotherthingswere,nomatterhownon-logicalvocabularywasunder-stood.Truthinan,orevery,individualdomainwasaninterestingthingtoconsider,too,butithadnothingtodowiththerelationoflogicalconsequence.157.4Themodalityproblemand“Tarski’sFallacy”Avirtueoftheinterpretiveapproachbasedontying[Tarski,1983h]to[Carnap,2002]isthatitgivesusasimpleandnaturalreadingofapas-sageinthearticlethathasbeenthesubjectofagreatdealofcontention:Tarski’sclaim,immediatelyuponstatingthedefinitionoflogicalcon-sequence,that“itcanbeprovedonthisbasisthattheconsequencesoftruesentencesmustbetrue”[Tarski,1983h,417].TheliteraturehereconsistsofadebateoverhowTarskicouldpossiblyhavethoughtthatthedefinitionallowedhimto“prove”thattherelationdefinedhadanyCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01modalfeature,withEtchemendysuggestingthatTarskiissimplycaughtupinatransparentmodalfallacyandrivalcommentatorseithersuggest-ingthatTarskicouldestablishthathisconceptionofconsequencedoeshavemodalimportofsomekind,downplayingtheconceptionofthisimportsoastomakethesupposedproofeasier,orsomecombinationofthetwo.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE203—#23\n204AlfredTarski:PhilosophyofLanguageandLogic7.4.1ModalitiesOnthefirstsidewehaveEtchemendy,whoonmyviewquiterightlypointsoutthat:Themostimportantfeatureoflogicalconsequence,asweordinar-ilyunderstandit,isamodalrelationthatholdsbetweenimplyingsentencesandsentenceimplied.Thepremisesofalogicallyvalidargu-mentcannotbetrueiftheconclusionisfalse;suchconclusionsaresaidto“follownecessarily”fromtheirpremises.Thatthisisthesinglemostprominentfeatureoftheconsequencerelation,oratanyrateofourordinaryunderstandingofthatrelation,isclearfromeventhemostcursorysurveyoftextsonthesubject(1990,81).EtchemendyisalsorightthatTarski’sviewsupportsnothingofthesort.Asheputsitat[Etchemendy,2008,267],ifthevalidityofanargumentcomesdowntonothingmorethanthematerialvalidityofitsvariantsunderuniformsubstitutionofnon-logicalvocabulary(or,moregener-ally,Tarski’streatmentofthisintermsofassignmentstosubstitutedvariables),thenwehavenoalethicmodality,sinceitmaysimplybeacci-dentallytruethattherearenosuchvariantswithtruepremisesandafalseconclusion,andwehavenoepistemicmodalityeither,sincewhenpre-sentedwithanargumentwhosepremisesareknowntobetrue,allthatonecouldconcludeisthateitheritsconclusionistrueortheargumentis,byTarski’sdefinition,invalid[Etchemendy,2008,267].Asnotedabove,theresponsestoEtchemendyonthispointeitherinsistthatsomehowhisaccountcansupportsomemodality,orinsistthat“must”isintendedweaklyenoughthatTarski’sclaimtobeabletoestablishthattheconsequencesoftruesentences“mustbe”trueisacceptable.SinceitisamplyclearthatTarski’saccountinvolvesthe“ReductionPrinciple”onwhichlogicaltruthissimplytheplaintruthoftheuniversalgeneralizationofasentenceonitsnon-logicalvocabu-lary,theonlyrecourseforaninterpreterwhowantstoreadasignificantalethicmodalityintoTarski’saccountistheexpedientoftakingtheaccounttoquantifyovernon-existentobjects.Commentatorswhoargueforthrightlyforthisarehardtofindthesedays,thoughPopper,alwaysCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01anidiosyncraticinterpreterofTarski(cf.§4.3.4)in1959conflatesTarski’smodelswithpossibleworldsat[Popper,2002,432]whilearguingthatTarskididsuccessfullyreducethenotionof“logicalnecessity”togen-eralityonthewaytoproposinghisownparallelaccountof“naturalnecessity”.Anumberofcontemporaryinterpretersdomakeatleastten-tativesuggestionsinthisdirection,usuallyaspartofadefenseofTarski10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE204—#24\nLogicalConsequence205ontheovergenerationproblemonwhichitisclaimedthattheaccount’simplyingthattheexistenceofnthingsisalogicaltruthformanyvaluesof“n”isacceptable.Thestrategytempts,sinceifthequantifiersinthegeneralizationofasentencequantifyovermerepossibilia,overgenera-tionseemslessproblematic—itbeinglessofastretchtothinkthatlogictellsushowmanythingsmightexistthanhowmanythingsdoexist.Amodalaspectistherebyalsosecuredfortheconsequencerelation,sinceanargumentwillbedeclaredlogicallyvalidnotmerelyifallrelevantsetsofactualobjectsthatdosatisfythelogicalformsofthepremisesalsosatisfythatoftheconclusion,butifallrelevantsetsofpossibleobjectsdo.Priest,forinstance,suggeststhatTarski’saccountdoesn’tovergeneratebecausethequantifiersshouldbetakentorangeovermerepossibiliaandthatthereareasmanyoftheseasonepleases“isn’tasubstantivefactatall”[Priest,1995,291].Hefurthersuggeststhatthissolutiontotheproblemofovergenerationisacceptablebecause“themodalityinvolvedseemsjustwhatisrequiredifTarski’sreductionofamodalnotiontoaquantifieristoberight”[Priest,1995,291].Sher[Sher,1996,681–2]likewisearguesthatlogiconher“Tarskian”accountisconcernedwith“formalpossibility”andsuggeststhatmathematicsisthetheoryoffor-malpossibility,thoughitislessclearthanwithPriestthatsheattributestheviewtoTarskihimself.AswithPriest’saccount,takingmathe-maticstoconcernpossibiliaisalsosupposedtorendermoreplausiblethelogicalstatusgrantedtomathematicaltruthsbyTarski’saccount.16Etchemendy’sresponsehereissufficient:“I’mnotsurewhichismoredif-ficulttoaccept,theideathatwecanbuildstructuresoutofnon-existentobjectsortheideathatTarskihadthisinmind”[Etchemendy,2008,286].Inanycase,themoveisotioseaslongaswithinthesettheoryinwhichweexpressthemodeltheoryeverymodelconstructedfromtheurele-mentsofanypossibleworldisisomorphictosomemodelconstructedfrompuresets[McGee,1992,275–8],sincefromthisitfollowsthatforanystructurethatmightbeamodelsomesetofsentencesthereisastruc-turethatactuallyis.ThecomfortofthinkingthatonTarski’saccountlogiconlytellsushowmanythingsmightexist,ratherthanhowmanyCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01thingsactuallydo,isthereforelost.Ifonedoesn’tgoinforattributingamodalaspecttoTarski’scon-sequencerelationonthesegrounds,theotherpossiblestrategyistodownplaytheforceofTarski’s“must”.Infact,ofcourse,thetwostrategiesblendintooneanother;witnessSher’sequationof“formalpossibility”with“mathematicalexistence”atthepassagecitedabove.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE205—#25\n206AlfredTarski:PhilosophyofLanguageandLogicAnexampleofattemptingtodownplaytheforceofTarski’s“must”isRay’sreading:Ithinkthatineverycasethemodaltermsinquestioncanbecon-struedasafamiliarcolloquialuseofmodallocutions,orasindicatingnothingstrongerthandeductiveconsequence…IthinkthatTarskiissimplyusingthelogician’s“must”indicatingthelogicalnecessityoftheconsequence,nottheconsequent[Ray,1996,654].ThisbegsthequestionagainstEtchemendy:thewholeissueiswhetherTarski’sdefinitionsupportsanynotionof“logicalnecessity”.Ifdeduc-tiveconsequencehasmodalfeatures,eitheralethicorepistemic,andifthereissomequestionastowhetherTarski’sdefinitioncapturesthesefeatures,asthereis,thenonedoesn’tdefendTarskibysimplytakingitasgiventhattherelationhehasdefinedistherelationof“deduc-tiveconsequence”.Ray’snervous“logicalnecessity,ifyouwill”[Ray,1996,654]indicatesthatnotevenRayhimselfisparticularlyconvincedbythisslide.AsEtchemendyshowsquitewell,whateveritisTarskidefines,itlacksbasicfeaturesthatinformalglossesonconsequencelike“ifthepremisesaretrue,theconclusionmustbetrue”attributetotheconsequencerelation.17NeverthelessthequestionremainsofwhatTarskithinkscanbeproved,andalsoofwhathethinks“must”meansinbothcondition(F)andtheargumentwehavebeenconsideringthatcondition(F)holdsofaconsequencepairifTarski’sdefinitiondoes.7.4.2ConsequenceandtruthThe“fallacy”readingofthepassageandtheresponseswehavesurveyedassumethatwhatTarskiclaimscanbeprovedisthattheconsequencesoftruesentencesmustbetrue.Hereismysuggestion:Tarskiisn’tclaimingthatatall;thepointisratherthatitcanbeprovedthattheconse-quencesoftruesentencesmustbetrue.Theprofferedargument,then,isn’tintendedtowardoffthesuggestionthatTarski’saccountimpliesmerelythattheconsequencesaretrueandnot,further,thatthey“mustbe”true;itisintendedtoestablishthattheintuitiveconnectionbetweenCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01consequenceandtruthiscapturedbythedefinition.ThecontrasthereisCarnap’sconceptionofL-consequence,whichischaracterizedintermsof“analyticity”inthesenseoffollowingfromeverythingbyCarnap’stransformationrules.Aswehavealreadyseen,TarskiwantstoreplaceanalyticityinCarnap’sapproachbytruth,sincehethinksthatCar-nap’sobjectionstosemanticnotionsarebadandthathehimselfhas10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE206—#26\nLogicalConsequence207formulatedthemeanstomakeattributionsoftruthacceptablebyViennaCirclestandards.Insupportofthis,wecannotethatcondition(F)requiresexactlythatconsequencesofasetofsentencesbetrueifthemembersofthesetare.Condition(F),inturn,ismotivatedbytheconsiderationthat“fromthepointofeverydayintuitions,itisclearthatitcannothappenthatallthesentencesoftheclassKwouldbetruebutatthesametimethesen-tenceXwouldbefalse”[Tarski,2002,183].Moreover,thevalidityoftheω-ruleischaracterizedthesameway:thefailureofderivabilityconcep-tionsofconsequenceisthattheydon’tdeclareallω-inferencesvalid,yetoftheseinferences,withtheirpremisesA0,A1…anduniversallyquanti-fiedconclusionA,Tarskisaysthat,accordingtothe“everydayconcept”,“wheneverallofthesesentencesaretrue,thesentenceAmustbetrue”.AccordingtoTarski,then,the“everydayconcept”thatistobecapturedbyhisdefinitionischaracterizedintermsoftruth.ThuswhatTarskiclaimsforhisdefinitionisnotthatitcapturesthemodalfeaturesofthe“common”conceptofconsequence(moreonthislater),butthatitcor-rectsCarnap’sconceptionofL-consequencebyreinstatingtheordinaryconnectionbetweenconsequenceandtruth.Thisfollowseasily,sothe“proof”canbeomitted,butitbearsemphasisinthecontextofaresponsetoCarnap.18ThatthisaspectofhisdefinedconsequencerelationisofinteresttoTarskiaddressesthequestionsometimesraised(e.g.[Bays,2001,1718])astowhyTarskitakesitthatonagoodaccountofconsequencetheconclusionsofω-inferencesareconsequencesoftheirpremisesandthatGödelsentencesfortheoriesareconsequencesofthesetheories.We’llcomebacktotheω-rule,butinthelattercasetheconnectionisclear:thewholepointofaGödelsentenceisthatitistrueifthetheoryforwhichitisaGödelsentenceistrue,so,ifasentenceisaconsequenceofothersif,iftheyaretrue,itistrue,thenGödelsentencesareconsequencesofthetheoriesforwhichtheyareGödelsentences.Thisisn’t,ofcourse,thestandardtakeonGödel’sresults,butbothCarnapandTarskitake,intheimmediateaftermathofGödel’sresults,theinterestinglinethatwhatGödelhasshownisnotthattherearesentencesandtheoriessuchthatthesentencesaretrueifthetheoriesaretrueeventhoughtheseCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01sentencesaren’tconsequencesofthetheories,butthatconsequenceisn’taneffectivenotion—thisiswhyCarnapfeelsforcedinlightofGödel’sresultstointroduce“indefinite”transformationrules.Tarski,workingessentiallyfromCarnap’sapproach,acceptsthis(cf.[Etchemendy,1988,72],butwantstoreintroducetheintuitivelinkbetweenconsequenceandtruth.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE207—#27\n208AlfredTarski:PhilosophyofLanguageandLogicThusTarskiwasnotguiltyof“Tarski’sfallacy”—not,however,becausehegaveanargumentthatinaninferencedeclaredlogicallyvalidbyhisdefinition,necessarilyifthepremisesaretruesoistheconclusionthatwasinfactvalid,butbecauseheofferednoargumentforanyconclusionofthesort.Hedid,ofcourse,usecommonlocutionslike“mustbetrue”andwewillhavetolookintowhathecouldhavemeantbythis.7.4.3Tarski’s“must”BelowwewillconsidertheintuitiveadequacyofTarski’saccount.HerethequestioniswhatTarskihimselfmeantbythe“must”sincondition(F)andinhiscommentsonitandonhisdefinition.Thekeyhere,again,isthefactthatTarskiisattemptingsimplytomodifyCarnap’sdefinitionofL-consequence.AsTarskiwrites,whathewantsisatreatmentonwhichlogicalconsequenceis“arelationwhichistobeuniquelydeterminedbytheformofthesentencesbetweenwhichitholds”andonwhich:thisrelationcannotbeinfluencedinanywaybyempiricalknowledge,andinparticularbyknowledgeoftheobjectstowhichthesentenceXorthesentencesoftheclassKrefer[Tarski,1983h,414–5].19Whateverone’sopinionofthisasanattempttocapturetheintuitivemeaningof“followslogically”,thereissomethingmodalaboutthechar-acterization:logicaltruthsaretruthsindependentofthesemanticsofnon-logicalterms.“Mustbe”thenmeansthatlogicalconsequenceisindependentofempiricalknowledge,inthesensethatitisindepen-dentofthesemanticsofthelanguagetotheextentthatthisinvolvesobjectsthatareonlyknownempirically.Thenotionisatleast“modal”inthesensethatwhetherornotaconsequence-pair(K,X)instantiatestheconsequencerelationisn’tsettledbytheextensionalsemanticsofthesentencesinKandXalone,butdependsalsoonthesemanticval-uesofotherexpressionsthroughoutthelanguage.NoticeinparticularthatTarski’sconstructionherematchesCarnap’streatmentofanalyticityandL-consequence:logicaltruthsaresupposedtobeanalytic,followingfromtheemptyset,andhencenotdependingon“empirical”knowl-edge[Carnap,2002,180],andtheyaresupposedtobeindependentofthemeaningsofnon-logicalterms,inaccordwithCarnap’streatmentofCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01L-consequence[Carnap,2002,180–181].Itisonlytobeconcededthatthis“modality”hasnothingtodowithsupportingcounterfactualsandisnotepistemicinthatinferencesmightbelogicallyvalidorinvalidwith-outourbeingabletoknowthattheyaresincevaliditydependsonthedistributionoftruthvaluesacrosssentencesofthelanguagenotinvolvedintheargument.OnTarski’sCarnapianconception“ifthepremisesare10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE208—#28\nLogicalConsequence209true,theconclusionmustbetrue”means“eithertheconclusionistrueorapremiseisfalse,nomatterwhattheinterpretationofthenon-logicalvocabulary”.WewillreturntoTarski’sconceptionofformbelow;thepointhereisthatwhenTarskisaysthattheconsequencesoftruesentencesmustbetrue,hesimplyfollowsCarnapinmeaningthattheyaretruenomatterwhatarethesemanticvaluesofnon-logicalexpressionsinthelanguage:(K,X)isvalidifitismateriallyvalidoneveryuniforminterpretationofthenon-Fvocabularyitinvolves.Tarskidoeshaveasubstantial,inten-sionalconceptionbehindthe“must”incondition(F);itisjustthatwemustconcedethatitdoesn’tcaptureanyinterestingepistemicoralethicmodality.7.5TheformalityproblemandthelogicalconstantsSofarwehaveseenthatTarski’saccountcannotcapturethemodalfeaturesintuitivelyinvolvedinlogicalconsequence,whethertheseareepistemicoralethic,andthatTarski’saccountranksmanymathemati-caltruthsaslogicaltruths.20CommontobothproblemsisanapparentmismatchbetweenintuitivedefiniensandTarskiandefiniendum:Tarski’sdefinitionbothfailstocapturetheordinaryterm’smodalfeaturesandseemstogetitsextensionwrong.Inthecontextofconcernabouttheseproblems,itistemptingtointerpreterstoappealtotheotheraspectoflogicalconsequenceTarskimentionsjustbeforesettingouthisdefinition,its“formality”.7.5.1ConstantandconsequenceTheissueherecomesdowntoTarski’sappealtotheideathattruthsandinferencesareformalwhentheyinvolveacertaindistinguishedvocab-ulary.AsTarskihimselfnotes,“Underlyingourwholeconstructionisthedivisionofalltermsofthelanguageintologicalandextra-logical”[Tarski,1983h,188].Thisraisesthehopethattheotherwiseapparentfail-ingsofthedefinitioncouldbecorrectedbytheproperselectionoflogicalvocabulary.Etchemendy(asinotherplaces,substantiallyanticipatedbyKneale:[Kneale,1961,96–7])writes:Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01SinceTarski’sgeneralaccountcapturesnoneofthemodal,epistemicorsemanticcharacteristicsoflogicaltruthandlogicalconsequence,itseemsthatthesecharacteristicsmustsomehowemergefromthesought-aftersupplement,fromouraccountofwhatmakescertainexpressions“genuinelylogical”andothersnot.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE209—#29\n210AlfredTarski:PhilosophyofLanguageandLogicThisiswhythetaskofcharacterizingthelogicalconstantscomestoseematoncesoimportantandyetsodifficult.Indeed,mostoftheburdenofTarski’sanalysisseemstoshifttoexactlythisissue.Butbynowitshouldbeclearthattheissueisbasedonaconfusion—namely,theassumptionthatwhentheaccountworks,itworksduetosomepeculiarpropertyoftheexpressionsincludedinF.Butthisassumptionisfalse:thereisnopropertyofexpressionsthatguaranteestherightextensioninthesecases,nonewhatsoever.Afterall,anypropertythatdistinguishes,say,thetruthfunctionalconnectivesfromnamesandpredicateswouldstilldistinguishtheseexpressionsiftheuniversewerefinite.Butinthateventuality,Tarski’saccountwouldbeextensionallyincorrect.Thisobservationaloneisenoughtoshowthatitisnotanypropertyoftheexpressionsweholdfixed,theso-calledlogicalconstants,thataccountsfortheoccasionalsuccessofTarski’sdefinitions[Etchemendy,1990,128].Since,asTarskinotes,thewholeaccountdependsuponaprincipledidentificationofthelogicalterms,thetaskofinterpreterswhowanttoclaimthatTarski’sanalysisissubstantiallyrighthasbeentoclaimthatthereisadistinguishedsetoflogicalexpressions,thattheseexpressionsareindeedintuitively“formal”,andthatwhenFistakentobethissetofexpressions,theaccountatleastdoesn’tovergenerateandevenperhapscapturesthemodalityintuitivelyinvolvedinconsequenceafterall.SuchdefensesgenerallyalsorequireadefenseofTarski’sidentificationofatleastsomemathematicaltruthsaslogicaltruths,especiallygiventhat“=”isgenerallytakentobelogicaloncurrentviewsoflogic.Ourwaycanbeshortwiththesuggestionthataprincipledidenti-ficationoftheconstantswouldhelpwiththemodalityproblem.ItismanifestthatonanyselectionforFtheaccountcapturesnoappro-priatealethicmodality,sinceitgeneralizesonlyoveractualsemanticvalues.ItisequallymanifestthatonnoselectionforFdoesitcaptureanyappropriateepistemicmodality,sinceknowingwhetheraninfer-enceisvalidwouldrequire,bythedefinition,knowingthedistributionoftruth-valuesacrossahostofothersentencesinthelanguage.Whether,inturn,themovehelpswithovergenerationdependsnotsomuchonCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01thedetailsoftheconceptionofthelogicalconstantsoffered,butontheconcomitantdefenseofthelogicisminvolvedinthatselection.21Thatis,thoughofcoursetheplausibilityofthelogicisminvolveddependsontheselectionoftheconstants,therealweightoftheaccountrestsonthedefenseoftheclaimthatanymathematicaltruthsarelogical,andnotonthedetailsofwhichonescomeoutlogicalforthemootedselection10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE210—#30\nLogicalConsequence211oflogicalconstants.Sincethisissowewon’ttarrylonghereoverthequestionastowhetheraprincipledaccountoftheconstantswouldhelpwiththemodalityandovergenerationproblems,andwewillconfineourattentionsimplytoattemptstosupplysomecriterionoflogicalityonTarski’sbehalf.InterpreterswhowanttoclaimthatTarskiactuallyendorsesthesug-gestedapproachmustalsoreadhisskepticalremarksattheendofthearticleasmerelytheannouncementofaresearchprogram:Atthefoundationofourwholeconstructionliesthedivisionofalltermsofalanguageintologicalandextra-logical.Thisdivisioniscer-tainlynotentirelyarbitrary:ifwedidnotcountamongthelogicaltermse.g.theimplicationsignorthequantifiersthedefinitionpro-videdoffollowingcouldleadtoconsequencesmanifestlycontrarytoeverydayintuitions.OntheotherhandhoweverIknownoobjec-tivereasonswhichwouldallowonetodrawaprecisedividinglinebetweenthetwocategoriesofterms…Clearly,furtherinvestigationsmaythrowalotoflightontheques-tionwhichinterestsus;perhapsonewillsucceedwiththehelpofsomeweightyargumentsofanobjectivecharacterinjustifyingthedividinglinetracedbytraditionbetweenlogicalandextra-logicalterms.PersonallyIwouldnotbesurprisedhowevereveniftheresultoftheseinvestigationsweretobedecidedlynegativeandifhenceitwouldturnouttobenecessarytotreatsuchconceptsasfollowinglog-ically,analyticsentenceortautologyasrelativeconceptswhichmustberelatedtoadefinitebutmoreorlessarbitrarydivisionoftermsofalanguageintologicalandextra-logical;thearbitrarinessofthisdivi-sionwouldbeinsomemeasureanaturalreflectionofthatinstabilitywhichcanbeobservedintheusageoftheconceptoffollowingineverydayspeech[Tarski,2002,188–90].Ithinkitisfairlyclearfromthewordingthatthisexpressesskepticism,andIwillsubstantiatethisinwhatfollows,butmanyinterpreterstakethepassagetheotherway.TheprimarywaytosupplyTarskiwithanaccountoftheconstantsistoreadtheviewTarski’smuchlater“WhatareLogicalNotions?”of1966backintothe1935article,anditistoCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01readingsbasedonthismaneuverthatwenowturn.7.5.2AnachronisticreadingsTheconceptionoflogicinthe1966lectureisnowfamiliar.TarskireferstoKlein’sErlangerProgrammfortheclassificationofgeometriesintermsoftransformations,andnotesthatthattheconceptsofe.g.affinegeometry10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE211—#31\n212AlfredTarski:PhilosophyofLanguageandLogicandtopologymaybedistinguishedbythetransformationsunderwhichtheirextensionsareinvariant.Themoretransformationsoftheuni-verseofdiscourseontoitselfoneallows,thefewerarethepropertiesofobjectsthatareinvariantunderthetransformations,andhencethefewerandmoregeneralarethe“notions”involved.Tarskithenentershissuggestion:Nowsupposewecontinuethisidea,andconsiderstillwiderclassesoftransformations.Intheextremecase,wewouldconsidertheclassofallone-onetransformationsofthespace,oruniverseofdiscourse,or‘world’ontoitself.Whatwillbethesciencewhichdealswiththenotionsinvariantunderthewidestclassoftransformations?Herewewillhaveveryfewnotions,allofaverygeneralcharacter.Isuggestthattheyarethelogicalnotions,thatwecallanotion‘logical’ifitisinvariantunderallpossibleone-onetransformationsoftheworldontoitself[Tarski,1986j,149].SincehereTarskisimplyidentifies“notions”withextensions[Tarski,1986j,147]wearealreadyfarfromtheconcernsofthe1930s.TarskiclosesthearticlebynotingthateverythingdefinableinPrincipiaMathe-maticaislogicalinthedefinedsense,butthatthequestionofwhethermathematicsisabranchoflogiccanonlybedecidedifitisdecidedwhethersuchatype-theoreticconceptionoflogic,asopposedtofirst-orderlogicsupplementedbyasettheoryinwhichthemembershiprelationisnon-logical,iscorrect,aquestionTarskideclinestoanswer.ThereisasignificantliteratureonTarski’sinvariancecriterion,e.g.[Sher,1996],[Sher,2008],[Feferman,1999],[Feferman,2010],[McGee,1996],[Gómez-Torrente,2002],[Bellotti,2003],[Casanovas,2007],[Hanson,1997].Weneedn’tenterintoit,however,becausethedefi-nitionofaconstantofferedinthe1966lecturesimplyhasnothingtodowiththenotionofformalityatworkin[Tarski,2002].Admittedly,theinvariancecriterionof1966isrelatedtoTheoremIof[Tarski,1983i],buttheveryfactthatin1935:Everyrelationbetweenobjects(individuals,classes,relations,etc.)whichcanbeexpressedbypurelylogicalmeansisinvariantwithCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01respecttoeveryone–onemappingofthe‘world’(i.e.theclassofallindividuals)ontoitself[Tarski,1983i,385].isatheoremratherthanadefinitionoflogicalitygivesthegameaway.Rather,in[Tarski,2002]TarskiquiteforthrightlyretainsCarnap’scon-ceptionofalogicasformalinthesenseofbeingconcernedonlywiththeformsofsentences.The1966article,bycontrast,understandslogic10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE212—#32\nLogicalConsequence213intermsofthe“form”oftheuniverseofdiscourse—interms,thatis,ofmathematicalstructure.Whateverthevalueofsuchanunderstandingoflogic,itcan’thaveanythingtodowithwhatTarskiisinterestedinin[Tarski,2002],sinceintheearlierarticle,aswehavenowseeninsomedetail,Tarski’saimismerelytosupplementCarnap’sapproachwithhissemantics.Sincethisissoweneedn’tenterfurtherintothedetailsofsuchread-ings.ForemosthereareSher’sworks[Sher,1991],[Sher,1996],[Sher,2008],andwecannotethatSherdoesn’treallysuggestthatsupplement-ing[Tarski,2002]with[Tarski,1986j]produceshistoricalaccuracy.IfourgoalistounderstandwhatTarskithoughtaboutlogicalconstancyin1935,lookingforwardto1966isnottheanswer.7.5.3CarnaponformalityUnliketheanachronisticreadingsofTarski,areadingofthearticleintermsofLogicalSyntaxiswithoutproblems.ReturnagaintoTarski’simportantremarksontheconsequencerelationat[Tarski,2002,183–4]citedabove.Fourpropertiesareattributedtotheconsequencerelationbythispassage:1.itholdsbetweensentencesinawayuniquelydeterminedbytheirform2.itisindependentofempiricalknowledge3.itisindependentofthesemanticvaluesofexpressionsinthetype-theoretichierarchy4.itisinvariantundersubstitutionsofvocabularywithinthetype-theoretichierarchyAllofthesefeaturesarestraightoutofLogicalSyntax.ThefourthissimplyCarnap’scriterionofL-consequenceasstatedat[Carnap,2002,181],andthethirdisTarski’sgeneralizationofthesameidea.Thefirstisalsocon-sonantwithCarnap’s“syntactic”approach,andthesecondmakesthecontrastbetweenlogicandtheempiricalthatCarnapstressesrepeatedly:IfamaterialinterpretationisgivenforalanguageS,thenthesym-bols,expressions,andsentencesofSmaybedividedintologicalandCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01descriptive,i.e.thosewhichhaveapurelylogical,ormathematical,meaningandthosewhichdesignatesomethingextra-logical—suchasempiricalobjects,properties,andsoforth.Thisclassificationisnotonlyinexactbutalsonon-formal,andthusisnotapplicableinsyntax.Butifwereflectthatalltheconnectionsbetweenlogico-mathematicaltermsareindependentofextra-linguisticfactors,suchas,forinstance,10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE213—#33\n214AlfredTarski:PhilosophyofLanguageandLogicempiricalobservations,andthattheymustbesolelyandcompletelydeterminedbythetransformationrulesofthelanguage,wefindtheformallyexpressibledistinguishingpeculiarityoflogicalsymbolsandexpressionstoconsistinthefactthateachsentenceconstructedsolelyfromthemisdeterminate.[Carnap,2002,177].Tarski’spreferatoryremarkstocondition(F)simplyrecountthelastpartofthispassage.22Tarski’sconceptionoflogicalityin[Tarski,2002]issimplyCarnap’s.Notsurprisingly,then,itsharesitsweaknesses;onequestionherewaswhat,exactly,Tarski’sattitudewastowardtheseweaknesses.7.5.4Theω-ruleandGödelsentencesAcentraltopicinthereadingof[Tarski,2002]whichisofrelevancetoboththeovergenerationandformalityproblemsisthetreatmentoftheω-ruleat[Tarski,2002,177–81].Havingnotedtheweaknessesofconceptionsofconsequenceintermsoffinitaryproofrulesattheoutsetof[Tarski,2002],Tarskireturnstothetopicoftheω-rulefrom§5ofCTFLand[Tarski,1983l].Tarskinotesthathehasgivenexamplesofω-incompletetheoriesthatare“quiteelementary”.Oftheexistenceofsuchtheories,hewrites:Thisfactattestsbyitself,Ithink,thattheformalizedconceptoffollowing,whichuntilnowwasgenerallyusedintheconstructionofdeductivetheories,bynomeanscoincideswiththeeverydayconcept—afterall,fromthepointofviewofeverydayintuitionsitseemsindubitablethatthesentenceAfollowsfromthetotalityofsen-tencesA0…A1…An…:wheneverallthesesentencesaretrue,thenalsothesentenceAmustbetrue[Tarski,2002,178].Etchemendy,assumingthat—as[Tarski,2002,178]readmostflatlyindicates—Tarskiisthinkingoftheω-inferencehediscussesasformu-latedinafirst-ordertheorywithnumericalconstants,complainsthattheexamplemisfires:first-orderarithmeticisω-completeneitherproof-theoretically(that’swhatGödel’sincompletenesstheoremshows)normodel-theoretically(that’swhattheincompletenesstheoremplusthecompletenesstheoremforfirst-orderlogicshows)[Etchemendy,1988,Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-0173].Asaresult,Etchemendyargues,theonlywaythatTarskicangettheω-ruletocomeoutvalidistostipulatethatthenumeralsarelogicalconstants,but,thesuggestionis,thisisanadhocselectionofconstants.Thestandardresponseintheliteraturehereisrightasfarasitgoes,butsuffersasmuchastherestoftheliteraturefromignoringTarski’srelationshiptoLogicalSyntax.Asmanycommentatorshavepointedout10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE214—#34\nLogicalConsequence215(e.g.[Gómez-Torrente,1996],[Bays,2001])Tarski’sotherdiscussionsoftheω-ruleassumethattheinferencesinquestionareformulatedinSTTwithnumericaltermsdefinedinthelogicistmanner.Asthesecommen-tatorsnote,theω-ruleisvalidsoformulatedinSTTandtheselectionofconstantsinvolvedisn’tadhoc.(Therearedisputesofdetailovertheren-deringoftheω-rulethereadingassumes,e.g.[Edwards,2003],[Sagüillo,1997],butweneedn’tenterintothesehere.)This,again,isrightasfarasitgoesbutitneglectsthefactthatCarnapformulatestheω-ruleinLanguageIwiththesuccessor-strokeandthenumeralsstipulatedtobelogical[Carnap,2002,16,25].SinceTarskihasCarnapinmindelsewhereinthearticle,hisformulationoftheω-rulewithnumeralscouldwellsimplybeasEtchemendyhasit(contra[Bays,2001,1721])—withthedifferencethattreatingthenumeralsaslogicalisnotadhocbutrathersimplyderivedfromCarnap’spresentation.23AmoregeneralworryconcernsTarski’sequallyenthusiastictreatmentofGödelsentencesasconsequences,inhissense,ofthetheoriesforwhichtheyareGödelsentences.AsEtchemendysuggests[Etchemendy,1988,73],thisrequirestakingnearlyallofthevocabularyoftherelevanttheoriesaslogical.ThoughthepointstandsthatGödel’sresultsapplytoanytheorythatextendsaweaksystemofarithmeticandhencethat,inprinciple,aGödelsentencecouldinvolvejustaboutanyvocabularyonepleased,ifwetakeTarski’sattentiontohavebeenonGödelsentencesforplausiblymathematicaltheories(asthediscussionofproof-theoreticencodingsoftheω-rulethatprecedesthediscussionofGödelsentencesat[Tarski,2002,180–1]indicates)thentheresponseisthatTarskialreadycountedthisvocabularyaslogicalanyway,eitherbyfollowingCarnap’streatmentofLanguageIorbytakingthetheoriestobeformulatedinSTT.Formorefar-flungtheories(e.g.anextensionofarithmeticthatincludes“isacow”)Gödelsentencescouldinvolveintuitivelynon-logicalvocabulary,butthisvocabularywillhavefounditswayintotheGödelsentenceonlybybeingcaughtupintheconstructionoftheprovabilitypredicateforthetheoryand,plausibly,itshavingdonesowillreflectadecisiononthepartofthetheory-buildertotreatthevocabularyaslogical.Thusthereisn’treallyanyadditionalproblemhereforTarski.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-017.5.5AntitractarianismandthenatureoflogicAllthatsaid,Tarski’streatment,nolessthanCarnap’s,foundersontheassumedroleofthelogicalconstants—ifwetake[Tarski,2002]asseriouslyintendedtoofferaconceptionoflogicalconsequencethatissupposedtobesupplementedbysomedefinitiveaccountofthelogicalconstants.Onmyreading,thatnosuchaccountisavailableisactually10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE215—#35\n216AlfredTarski:PhilosophyofLanguageandLogicpartofthepoint.AsIhavesuggestedearlier,Tarski’smainaiminthearticle,beyondsupportingthesemanticsadvertisedinESSbyimprov-inguponCarnap’sworkistounderminetheTractarianconceptionoflogictowhichCarnapwasstillattracted.Forthispurpose,simplyrais-ingtheneedforacriterionoflogicalityandemphasizingtheunlikelinessthatonewouldbefoundwassufficient.SinceTarski’sviewwastheWarsawviewonwhichthereweren’tprincipleddistinctionsbetweenlogicandmathematics,orbetweenthoseandscience[Murawski,2004],[Wole´nski,1989],[Carnap,1963,30],Tarskiinthelectureisn’thopingforacriterionoflogicality,butisratherpointingoutthatnonewillbefound.Aswenotedabove,CarnapinLogicalSyntaxisthoroughlyconvincedthatthedoctrinethatlogic“saysnothingaboutfacts”plusatreatmentofmathematicsas“logical”istheonlywaytoasatisfactoryaccountoftheroleofmathematicsinscience.Iflogicalandmathematicalclaims“saynothingaboutthefacts”thentraditionalepistemologicalandmetaphys-icalquestionsaboutmathematicalfactsandobjectsandhowweknowthemcanbesetaside.Thewholeconstruction,however,dependsonaprincipleddelimitationoflogicandanargumentthatlogicsodelimitedhastheneededfeatures.Fromtheirearliestmeeting,Tarskihaddisputedtheseviews.HereareCarnap’snotesontheirmeetingofFebruary22,1930:8–11amwithTarskiinthecafé.Ofmonomorphism,oftautologies,hewillnotadmitthatthattheysaynothingabouttheworld:hesaysthatthereisonlyagradualandsubjectivedistinctionbetweentautologicalandempiricalsentences(quotedbyHallerat[Tarski,1992,5];mytranslation).Carnaprecountsthisdisagreementinhis“IntellectualAutobiography”aswell[Carnap,1963,30].InTarski’slettertoNeurathofSeptember7,1936,wehavethisrecollection:IrecallaconversationwithCarnapwhichIhadayearagoinPrague(duringwhichalsoMrsLutmannand,itseemstome,MrsCarnapwerepresent).Namely,atmyrequestCarnaptoldwhereinconsistedtheCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01influenceofWittgensteinontheViennaCircleandonhimpersonally.Hesaidthen,thatthisinfluencewasontheonehandchallenging,ontheother,hindering.Theinfluencewaschallenging,forW.focusedattentionontheimportanceofproblemswhichconcernthelanguage,forhebroughttolightthereductionof“philosophical”problemstolinguisticproblems;theinfluencewashoweveralsohindering,forjust10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE216—#36\nLogicalConsequence217W.contestedandrejectedthepossibilityofspeakingaboutalanguageinthelegitimateandfault-freemanner.AndnowCarnapwasoftheopinion,thatjusttheliberationfromthishinderinginfluenceofW.isduetotheVarsovians(andespeciallytomyViennalectures)[Tarski,1992,25].SoduringthePragueVorkonferenztotheParisUnityofSciencecongressTarskihadbeeninterestedtoknowwhatWittgenstein’sinfluencewas,andhadbeentoldbyCarnapthatthedoctrinethatonecouldn’tspeakaboutlanguagewasaretardinginfluenceofWittgenstein.TarskiwassurelypleasedtobecreditedbyCarnapwithhavingconvincedhimthatserioustheoryaboutlanguageisscientificallyacceptable.Noticehere,however,thatwhatCarnapdoesn’tsayin1934isthattalkaboutrelationsbetweenlanguageandtheworldisacceptable.Thatis,in1934CarnapcreditsTarskionlywithasmuchasisembodiedinLogicalSyntax,andattributesthistoTarski’slecturesof1930,whichconcernedTarski’sworkonconsequence—workwhichisaboutlanguagebutnotaboutlanguage–worldrelations.Havingdevelopedsemanticsinthemeantime,TarskiwouldhavewantedtodrawCarnapalongwithhimthatmuchfurther,anditistothisendthatESSandCFLarepitched.Tarski,then,wantstounderminethedistinctionbetwenlogicandsciencethatcontinuestobeenshrinedinCarnap’sdefinitionofL-consequence,andthedistinctionbetweenL-consequenceandconsequenceingeneral.24Tarski’sownwordingattheendof[Tarski,2002]hardlysuggestsendorsementofthepossibilityofaprincipleddelimitationofthelogicalconstants,sinceTarskisaysthathe“wouldhardlybesurprised”[Tarski,1992,189]ifnosuchdistinctioncouldbemadeinaprincipledwayand,moreover,TarskiexplicitlycharacterizesthepurportofthepassageasskepticalinacontemporaneouslettertoNeurath:IgavealectureinParisabouttheconceptoflogicalconsequence;thereIcontested(amongotherthings)theabsolutecharacterofthedivisionofconceptsintologicalanddescriptiveaswellasofsentencesintoanalyticandsynthetic,andIendeavoredtoshow,thatthedevisionofconceptsisquitearbitrary,andthatthedivisionofsentenceshasCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01toberelatedtothedivisionofconcepts[Tarski,1992,28].Herethedistinctionbetweenanalyticandsyntheticsentencesandthedelimitationofthelogicalconstantsare“quitearbitrary”andTarski“contests”theabsolutecharacteroftheputativedistinctions;matterscouldhardlybeclearer.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE217—#37\n218AlfredTarski:PhilosophyofLanguageandLogicIndenyinganyprincipleddistinctionbetweenlogicandmathematics,orbetweenthemandscience,TarskiwassimplyfollowingtheLvov–Warsawschool.Murawskiwritesthat“Thesemanticalpointofviewimpliedtherejectionoftheanalyticalconceptoflogic,i.e.,therejectionofthethesisthatlogicisacollectionoftautologiesthatarecontentu-allyempty(thisisathesisonlogicandmathematicsversusreality).Le´sniewski(andKotarbi´nski)claimedthatlogicdescribesthemostgen-eralfeaturesofbeing,logicplaysaroleofageneraltheoryoftherealworld”[Murawski,2004,330–1],butunfortunatelyprovidesnocita-tions.Lookingtothetexts,Kotarbi´nski,thoughhespillssomeinkoverthedoctrinethatlogicconcernsthe“form”ofreasoningishappytocallLe´sniewski’s“Ontology”a“generaltheoryofobjects”[Kotarbi´nski,1966,211].25LookingoverthemaintextsofKotarbi´nski,Le´sniewski,ŁukasiewiczandAjdukiewicz,onedoesnoticeaconspicuousabsenceofworryaboutthestatusofmathematics—theworrythatotherwisedominatedanalyticalphilosophyfromFregeonwards.Grantingthat,theLvov–WarsawschoolwasindirectconflictwiththeViennaCircleonthispoint;seestatementssuchasthisonefromHansHahn:Iflogicweretobeconceived—asitactuallyhasbeenconceived—asatheoryofthemostgeneralpropertiesofobjects,asatheoryofobjectsassuch,thenempiricismwouldinfactbeconfrontedwithaninsuperabledifficulty.Butinrealitylogicdoesnotsayanythingatallaboutobjects;logicisnotsomethingtobefoundintheworld;ratherlogicfirstcomesintobeingwhen—usingasymbolism—peopletalkabouttheworld,andinparticular,whentheyuseasymbolismwhosesignsdonot(asmightatfirstbesupposed)standinanisomorphicone–onerelationtowhatissignified[Hahn,1980,40].ThoughthereisasmallindustrydevotedtosupportingTarski’ssup-posedcommitmenttoanaccountofthedistinctionbetweenlogicalandnon-logicalexpressions,matterscouldhardlybeclearer:Tarskineverbelievedinanysuchdistinction.TheevidencefromtheletterstoNeurath[Tarski,1992]iscompellinghere,andtheconcludingremarksto[Tarski,1986j]showthatheneverchangedhismindonthetopic.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01WhatisworthyofnoteisthatTarskibringsthisskepticismtobearintheformulationofaconceptionofconsequencemuchmoreathomeintheViennaCircleconceptionoflogicthanthatoftheLvov–Warsawschool.Carnap’sconceptionofL-consequencesitsresolutelywithintheframeworkoftheTractarianconceptionoflogic.Tarskitakesitbackoutofthatconceptionwhensubstitutionisreplacedwithsemanticsand10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE218—#38\nLogicalConsequence219provabilitywithtruth,butinthiscontexttheviewthatthereisnoradicaldiscontinuitybetweenlogicandsciencetakestheformofskepticismaboutthepossibilityofaconceptionoftheconstantsthatwouldsupporttheViennaCircleconceptionoflogic.InthisregarditisnotsurprisingthatwhenTarski30yearslaterdoesformulateacriterionoflogicalityinaconstant,itisan“ontological”one,accompaniedagainbyemphasisonthecontinuitiesrunningfromlogicthroughmathematicstoscience;seehereespeciallythediscussionof“∈”intheconclusionof[Tarski,1986j].Thelaterarticlemakesgoodonthesuggestionoftheearlieronethatnonon-representationalunderstand-ingoflogicalitywillbefoundbyproposingacharacterizationoflogicthatisunabashedlyrepresentational:logicaltermsaresimplythosethatdenotegeneralstructuralfeaturesoftheuniverseofdiscourse.Wecannotealso,inthecontextofTarski’sresponsetotheTractariandoctrinestowhichCarnapwasattracted,thatthe1966lectureupholdsexactlytheconceptionoflogicalityasgeneralitythatWittgensteincriticizedinRussell(e.g.TractatusLogico-Philosophicus6.1231,6.1232).7.6EvaluatingTarski’saccount7.6.1TheanalyticproblemWithalloftheforegoinginhand,wecanaskourselvesabasicquestionaboutTarski’sconceptionofwhathewasdoing.SinceEtchemendyisquiterightthatlogicalconsequenceasdefinedbyTarskilacksmanyfea-turesoftheordinarynotionof“validity”or“followingfrom”,weneedtoaskourselveswhatwastherelationTarskiconceivedtoholdbetweenhisanalysisandtheordinarynotion.Now,Tarskibeginsthearticlebyinsistingthattheconnectionisimportant:Theconceptoflogicalconsequenceisoneofthosewhoseintroductionintothefieldofstrictformalinvestigtionwasnotamatterofarbi-trarydecisiononthepartofthisorthatinvestigator;indefiningtheconcept,effortsweremadetoadheretothecommonusageofthelanguageofeverydaylife[Tarski,1983h,409].Andjustafterofferinghisdefinition,hewrites:Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Itseemstomethateveryonewhounderstandsthecontentoftheabovedefinitionmustadmitthatitagreesquitewellwithcommonusage[Tarski,1983h,417].ThephrasingherecorrespondsquitewelltotheremarksinotherpassageswhereTarskiintroducesadefinitionaltask—forinstancetheremarkson10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE219—#39\n220AlfredTarski:PhilosophyofLanguageandLogic“definablesetsofrealnumbers”in[Tarski,1983f]andofcourseontruthin[Tarski,1983a].InviewoftheworkwehavealreadydoneonintuitivemeaninganddeductivetheoriesinTarskithestrongdefaultassumptionisthatTarskiwasdoingin[Tarski,2002]exactlywhathewasdoingintheseotherworks,showinghowtoexpressan“intuitive”,“everyday”conceptwithinadeductivetheory.NeverthelessthegrossdivergencebetweenwhatTarskioffersandthefeaturesthatEtchemendyrightlystresseshastemptedsomecommenta-torstoproposethatTarskisimplywasn’ttryingtocapturethe“commonusageofthelanguageofeverydaylife”.Jané,forinstance,writes:Tarski’scommonconceptisnotsomegeneral,all-purposenotionofconsequence,butaratherpreciseone,namelytheconceptofconsequenceatplayinaxiomatics[Jané,2006,2–3]where“axiomatics”wasthestudyofaxiomaticsystemsfollowingHilbertandespeciallythePeanists,Pieriinparticular.Theattractionhereisobvious:sinceTarskiclearlydoesn’tcaptureessentialfeaturesoftheordi-narynotion,oneproposesthathewasn’ttryingtocapturetheordinarynotion.26However,Tarskiistryingtocapturetheordinarynotion—therearejusttoomanypassagesinthearticlethatsaythis,andcapturingordi-narynotionswithexpressionsconstrainedbythestructureofdeductivetheoriesiswhathehadbeentryingtodoallalong.AndwhatshouldourowntakeonTarski’sconceptionofconse-quencebe?Themodalandextensionalfailingsoftheanalysisaremanifest.Noaccountofthelogicalconstantswillcorrectthis,thoughacceptanceofsomethinglikelogicismwillatleastblunttheforceofovergeneration.WhydoesTarskiacceptCarnap’sconception,despiteitsobviousflaws,anddespiteitspoorfitwiththeLvov–Warsawschool’sgeneralitycon-ceptionoflogic?First,theextensionalismoftheLvov–Warsawschoolwouldpreventtheinclusionofanygenuinelymodalnotioninthedef-initionofconsequenceanyway.InthisrespectTarskisimplycouldn’thavebeenbotheredbywhattoEtchemendy,andtomostofus,aretheclearfailuresofaconceptionofconsequencethatis,atbottom,mate-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01rial.Butwhy,wecanstillask,wasTarskiattractedtoCarnap’sconceptionofconsequence?Herethehistorymatters:Carnap’sconceptionsimplywasanimprovementoverthegoingderivationalconceptionsofconse-quencethatTarskihadaccepteduntilsometimeacoupleofyearsafterGödel’sresultsandwhichwecanfindinKotarbi´nski,Le´sniewskiandothersamonghisteachers.Aswehaveseen,CarnapreactedtoGödel’s10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE220—#40\nLogicalConsequence221resultsnotbyholdingthattheyshowedthatthereweretruthsofthelanguageofatheorywhichweren’tconsequencesofit,butthattherewereconsequencesofmanytheoriesthataren’tderivable(cf.[Gómez-Torrente,1996,146]).Thoughnotthecontemporaryglossontheresults,giventhatGödelsentencesaretrueifthetheoriesforwhichtheyareGödelsentencesare,andgivenabasic“everyday”glossonconsequencethatTarski’ssemanticshadputhiminapositiontoaccept,namelythataconsequenceofasetofsentencesisasentencethatistrueiftheyare,thiswasareactiontoGödelthatwouldhaveseemednaturaltoTarski,andthatwouldhaveled,preciselybyTarski’stakinghimselftohavetamedsemanticnotionsforserioustheory,tohisconceptionofconsequence.7.6.2EliminatingtransformationrulesCarnap’sapproachtologicalvocabulary,asisoftennoted,comesintwoforms.InthetreatmentsofLanguagesIandIIthelogicalexpressionsaresimplylisted.In“general”syntax,Carnap’scriterionoflogicalityisasabove:logicalexpressionsarethosesuchthateverysentenceinvolvingonlythemiseitherprovableorrefutableonnoassumptions,giventhetransformationrulesofthelanguage.Nowthethemostimportantaspectofthisisthatitretainsanessentialreferencetothetransformationrules.Generalsyntaxisgeneralinthatitmakesclaimsabouttherelationshipsamongtransformationrulesandotheraspectsofalanguage,butallofitsnotionsareneverthelesstherebyrelativetoachoiceoftransformationrules,andabouttheseCarnaphasnothingtotellus;indeedthewholepointofthePrincipleofToleranceisthatnothingofacriticalnaturecanbesaidaboutthechoiceoftransformationrules.Itisthisfocusonstated(if,inGeneralSyntax,generalizedover)trans-formationrulessyntacticallyconceivedthatweakensCarnap’sresponsetoGödelintworespects,bothofwhicharecorrectedbyTarski’sseman-tictreatment.First,sinceCarnapdemandsthateverythingbedoneintermsofsyntax,hecannotstateacommonintuitiveupshotofthefirstincompletenesstheorem,namely,thattherearesentencesthataretrueinthelanguageofanyextensionofaweakarithmeticthatdon’tfollowCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01fromtheaxioms.Theonlywayhehastosaythatanythingthatcomesclosetotheforbiddentalkoftruthtothisistosaythatsomethingisa“consequence”oftheaxioms.SothetransformationruleshavetobestrongenoughtostateGödel’sresultastheclaimthatthereare“conse-quences”oftheaxiomsthatneverthelessaren’tderivable,andtheω-ruleistherebyrequired.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE221—#41\n222AlfredTarski:PhilosophyofLanguageandLogicSecond,andmoregenerally,iftheaimoftheenterpriseistocapturetheordinarynotionofonething“followingfrom”anotherthenTarski’streatment,whateverotherweaknessesitshareswithCarnap’s,isonerespectamajorimprovementpreciselybecauseitisnolongerrelativizedtoasetoftransformationrules,arelativizationsimplynottobefoundinordinarythoughtaboutconsequence.UltimatelyCarnapisstillstuckinthepredicamentofbeingunabletodoanythingmoreinthetreatmentoflogicalconsequencethantoassumealistofformalrules.Withthemovetotolerancetheserulesarenolongersupposedtobeintuitivelyvalid,butCarnapfailstoreplacethatappealtointuitivevaliditywithanyotheranalyticconnectiontoordinarythoughtaboutconsequence,beyondvariousofficiallyforbiddenappealstoordinarythoughtabouttruthandconsequence.SinceLanguagesIandIIdohappentohavetrans-formationrulesthatare,ifcomplicatedinplaces,apparentlyvalidoncestated,Carnap’svariousdefinitionsof“analyticity”andhispumpingoftheintuitionthatthecriterionof“L-consequence”matchesintuitionsaboutwhatistrue“inapurelylogicalway”at[Carnap,2002,180–1]givethereaderamisleadingsenseofsecuritythat“analytic”sentencesarereallyinsomesenseanalytic,and“L-consequences”arereallyinsomesenselogicalconsequences.ImaginingCarnap’sdefinitionsappliedtoalanguagewithconsistentbutintuitivelybizarretransformationrulesremindsusthat“analyticity”and“L-consequence”onlycoincidewithanalyticityandlogicalconsequenceinfortuitouscases.InthecontextofadebatewithintuitionistsandfinitiststhePrincipleofToleranceseemsacceptablesinceallofthecompetingsystemshaveatleastsomeintuitivebacking.Weshouldnot,however,mistaketolerancebetweenchoicesallofwhichhavesomeintuitivebackingforaproofthatintuitivebackingisn’trequired.WecanseesomeoftheeffectsofTarski’seliminationofthetrans-formationrulesinhisremarkthatlogicalconsequencedegeneratestomaterialconsequencewhenallsentencesoftheexpressionsofthelan-guageareincludedinF.Carnap’sconceptionofL-consequencedoesnotdegeneratetomaterialconsequenceifallexpressionsofthelanguagearelogical.Toseethis,consideralanguagewithasingletransformationruleaccordingtowhicheverythingfollowsfromeverything.AllsentencesofCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01thislanguageare“analytic”andallitsexpressionsare“logical”,butitsurelyisn’talanguageinwhichconsequencesoftruesentencesaretrue.Aswehaveseen,Carnapfirstintroducesthenotionofconsequenceingeneralintermsoftransformationrules;hethendistinguishesL-consequenceasconsequencethatispreservedunderuniformsubsti-tutionsofdescriptivevocabulary[Carnap,2002,180–1].Thedifference10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE222—#42\nLogicalConsequence223fromTarskiisthata“transformationrule”accordingtowhichonesen-tencefollowsfromothersjustincaseeitheritistrueoroneofthemisfalsecouldnothavebeencontemplatedbyCarnap,sinceitgoesbeyondtheboundsof“syntax”inbeingformulatedintermsoftruthratherthanthestructuralfeaturesofsentences.(Here“analytic”can’tbesubstitutedfor“true”,since“analytic”isitselfdefinedbywhatfollowsaccordingtothetransformationrules.)Soinmentioningthedegenerationoflogicalintomaterialconsequencewhenallexpressionsare“logical”,TarskiisnotcommentingonCarnap’sview,butpointingoutaconsequenceofhismodificationofit.ThedependenceontransformationrulesinCarnap’sviewhelpsartificiallytosupporttheViennaCircle’sTractariandistinc-tionbetweensentencesthat“saynothingaboutreality”andsentencesthatdo;inthematerialconsequencepassageTarskishowsthathiscon-ceptionofconsequenceunderminesthedistinctionandreinstatesthecontinuityoflogic,mathematicsandsciencethatwasacceptedintheLvov–Warsawschool.Tarskielsewherecharacterizessemanticsintermsof“thetotalityofconsiderationsconcerningthoseconceptswhich,roughlyspeaking,expresscertainconnexionsbetweentheexpressionsofalanguageandtheobjectsandstatesofaffairsreferredtobytheseexpressions”[Tarski,1983b,401].ThearticleisanadvertisementforTarski’smethods.Noticethat,inthetermsofESS,onewayofresistingtheimportanceof“sci-entificsemantics”wouldbetorestonaTractariandistinctionbetweensentencesthatdoandthosethatdonotbear“connexions”toobjectsandstatesofaffairs.Readasapair,ESSandCLCmakeanextendedargu-mentinfavorofTarskiansemantics:thefirstarticlesketchesthebasicsofhowitworks,whilethesecondundoestheprimarysourceofresistanceheexpectedfromhisaudience,namely,TractariandoctrinesaboutlogicastheywerefoundintheViennaCircle.Inparticular,Tarski’saccountofconsequencein[Tarski,2002]makeslogicalconsequenceasemanticconceptinthesenseof[Tarski,1983b,401],sinceconsequenceisdefinedintermsofmodels,whicharedefinedintermsoftheexplicitlysemanticconceptofsatisfaction.7.6.3EpistemicandgeneralityconceptionsoflogicCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01IntuitionisticFormalismcentrallyinvolvedanepistemicconceptionoflogicalconsequence.RecallthepassagefromLe´sniewskithatTarskicitesat[Tarski,1983c,61]:Havingendeavoredtoexpresssomeofmythoughtsonvariouspartic-ulartopicsbyrepresentingthemasaseriesofpropositionsmeaningful10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE223—#43\n224AlfredTarski:PhilosophyofLanguageandLogic(sinnvollerSätze)invariousdeductivetheories,andtoderiveonepropositionfromothersinawaythatwouldharmonizewiththewayIfinallyconsideredintuitivelybinding(welcheIch“intuitiv”alsfürmichbindendbetrachte),Iknownomethodmoreeffectiveforacquaintingthereaderwithmylogicalintuitions(“logischenIntuitio-nen”)thanthemethodofformalizinganydeductivetheorytobesetforth[Le´sniewski,1992e,487][Le´sniewski,1929,78].InthisrespectIntuitionisticFormalisminvolvedanaccountofconse-quencethatAristotlehimself,inhisaccountoftheperfectsyllogismat24b18ff,couldhaverecognized:Asyllogismisdiscourseinwhich,certainthingsbeingstated,some-thingotherthanwhatisstatedfollowsofnecessityfromtheirbeingso…Icallthataperfectsyllogismwhichneedsnothingotherthanwhathasbeenstatedtomakeplainwhatnecessarilyfollows.AsEtchemendynotes,Tarski’saccountsimplylackstherelevantfeatures:Surroundingtheintuitiveconceptsoflogicalconsequenceandlogicaltruthareahostofvagueandphilosophicallydifficultnotions—notionslikenecessity,certitude,aprioricity,andsoforth.Amongthecharacteristicsclaimedforlogicallyvalidargumentsarethefollow-ing:Ifanargumentislogicallyvalid,thenthetruthofitsconclusionfollowsnecessarilyfromthetruthofthepremises.Fromourknowl-edgeofthepremiseswecanestablish,withoutfurtherinvestigation,thattheconclusionistrueaswell.Theinformationexpressedbythepremisesjustifiestheclaimmadebytheconclusion.Andsoforth.Thesemaybevagueandill-understoodfeaturesofvalidinference,buttheyarethecharacteristicsthatgivelogicitsraisond’être.Theyarewhylogicianshavestudiedtheconsequencerelationforovertwothousandyears…Thepropertyofbeinglogicallyvalidcannotsimplyconsistinmem-bershipinaclassoftruthpreservingarguments,howeverthatclassmaybespecified.Forifmembershipinsuchaclasswereallthereweretologicalconsequence,validargumentswouldhavenoneofthechar-acteristicsdescribedabove.Theywould,forexample,beepistemicallyCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01impotentwhenitcomestojustifyingaconclusion.Anyuncertaintyabouttheconclusionofanargumentwhosepremisesweknowtobetruewouldtranslatedirectlyintouncertaintyaboutwhethertheargumentisvalid.Allwecouldeverconcludeuponencounteringanargumentwithtruepremiseswouldbethateithertheconclusionistrueortheargumentisinvalid.Forifitsconclusionturnedouttobe10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE224—#44\nLogicalConsequence225false,theassociatedclasswouldhaveanon-truth-preservinginstance,andsotheargumentwouldnotbelogicallyvalid.Logicalvaliditycan-notguaranteethetruthofaconclusionifvalidityitselfdependsonthatself-sametruth[Etchemendy,2008,265–6].NowCarnapfeignedtorelinquishanepistemicconceptionofconse-quencewiththeprincipleofTolerance,buthiscontrivedselectionofexamplesoflanguageswithtransformationrulesthatareinfactatleastprimafacieintuitivelyvalidmaskstherealconsequencesofthis.TotheextentthatCarnap’spositionisplausible,itisbecauseoftheairofintu-itivevaliditythatcontinuestopervadethetransformationslicensedinLanguagesIandII.Tarski,ontheotherhand,givesusatruegeneralityconceptionofconsequence.WhatarewetomakeofTarski’sprogressfromthederivationalcon-ceptionofconsequencestillpresentinCTFLtothemodel-theoreticconception?Neitherconceptionissatisfying:thederivationalconcep-tionisn’taconceptionatall,butsimplyofferslistsofrulesthatdisplaythefeatureweseektounderstand,whilethemodel-theoreticconcep-tionisplaguedbyjusttheproblemsEtchemendystresses:theonlycaseinwhichitofferssecurityistheoneinwhich,asstressedbyKreisel[Kreisel,1967,152ff]Gödel’scompletenesstheoremassuresusthatthemodeltheoreticconceptiondoesn’tdeclarevalidanyargumentsthatthederivationalconceptiondoesn’t.Inothercasesthereisnoreasontothinkthatthemodel-theoreticconceptionevengetstheextensionoftheconsequencerelationright,whilethederivationalconceptionman-agesthisinhand-craftedcasesbuttellsusnothingaboutwhatitisthatmakescertainrulesgoodones,nothingabouthowitisthatweknowthatcertainformsofinferencepreservetruthevenwhenweareignorantofthetruth-valuesmostofthesubstitutionvariantsofthecomponentsentences.Thisfundamentalquestioninthephilosophyoflogicremainsopen.WhatwecansayinthishistoricalstudyisthatonceTarskidiscoveredhiswayofreinstatingtheconnectionbetweenconsequenceandtruthwhileretainingthebasicsofCarnap’sdefinitionofL-consequence,heacceptedthegeneralityconceptionoflogicthatfollowed.AstheidentificationofCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01logicandmathematicsthisinvolvedforSTTwassomethinghealreadyacceptedanyway,andsincehisextensionalismlefthimunperturbedbyanyapparentfailureoftheconceptionheofferedtocapturetheapparentmodalityoftheconsequencerelation,Tarskiwasperfectlyhappywiththeresult.AndwiththismovetherewasnothingleftofIntuitionisticFormalism,hisadherencetowhichhadalreadybeenweakenedbyhis10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE225—#45\n226AlfredTarski:PhilosophyofLanguageandLogicwillingnesstogiveupthetheoryofsemanticalcategories.AswewillseeintheConclusion,bythetimeof[Tarski,1986e],allappealtoIntuition-isticFormalistnotionsofintuitivemeaning,intuitivelyvalidinference,linguisticconventionsfortheexpressionofthought,andsoon,hadbeenexpungedfromhisthinkinginfavorofsemantics,whichnowstoodonitsown.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-0110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP07”—2011/11/29—14:26—PAGE226—#46\n8Conclusion8.1Paris1935andthereceptionofsemanticsWhenSeptember1935rolledaround,TarskiwentoutofhiswaytopleasehisaudienceattheUnityofScienceCongressinParis.Aswehavenoticed,hepitchedthediscussionofconsequencelargelyasamodifica-tionofCarnap’snotionofL-consequence,andherelatedmisgivingshegenuinelyhadaboutaxiomaticsemanticstoaconcernwithphysicalismthatheprobablydidn’t.LikePopper,whowasmade“joyful”[Popper,1974,398]byTarski’sviews,Carnaphadalreadybeenconvinced[Car-nap,1963,60],andKokoszy´nskawasalsopresentinsupportofTarski’sviews.However,noteveryonewasenamoredwith“scientificsemantics”.HereisCarnap’srecollection:WhenImetTarskiagaininViennainthespringof1935,IurgedhimtodeliverapaperonsemanticsandonhisdefinitionoftruthattheInternationalCongressforScientificPhilosophytobeheldinParisinSeptember.Itoldhimthatallthoseinterestedinscientificphilosophyandtheanalysisoflanguagewouldwelcomethisnewinstrumentwithenthusiasm,andwouldbeeagertoapplyitintheirownphilosophicalwork.ButTarskiwasveryskeptical.Hethoughtthatmostphilosophers,eventhoseworkinginmodernlogic,wouldbenotonlyindifferent,buthostiletotheexplicationoftheconceptoftruth.IpromisedtoemphasizetheimportanceofsemanticsinmyCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01paperandinthediscussionattheCongress,andheagreedtopresentthesuggestedpaper.AttheCongressitbecameclearfromthereactionstothepapersdeliveredbyTarskiandmyselfthatTarski’sskepticalpredictionshadbeenright.Tomysurprise,therewasvehementoppositionevenonthesideofourphilosophicalfriends.[Carnap,1963,60].22710.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP08”—2011/11/28—13:58—PAGE227—#1\n228AlfredTarski:PhilosophyofLanguageandLogicAdditionaldiscussionwasarranged,andthereadebatethatwastocon-tinueforseveralyears,inprintandincorrespondence,wascarriedoutamongTarski,CarnapandKokoszy´nskaontheoneside,andNeurath,Naessandcomradesontheother.As[Mancosu,2008]providesanexcellentdiscussionoftheensuingdebate,Iwillonlyindicateithereinoutline.Neurath,whohadalreadylongobjectedtotheideathatsentencesareevercomparedwithreal-ityratherthanothersentences,andwhohadadvocatedreplacementof“true”with“accepted”(inparticular“intheEncyclopeida”ofunifiedsci-ence)[Mancosu,2008,201],wasadamantlyopposed.ConfrontedwithTarski’sproposal,Neurathhadlittletosayonitstechnicalmeritsbutratherseemsoveraperiodofyearsrelentlesslytohavepoundedthetableoveritspotentialtomisleadtheuninitiatedintothinkingtherewassuchathingasacomparisonofastatementwithreality.Kokoszy´nskahadtakenNeurathtotaskovertheobviousincompatibilitybetweentheiden-tificationof“true”with“accepted”andtheT-sentences[Mancosu,2008,212]and,tellingly,Neurathatonepointina1937debatewithCarnapsuggestedthattheconceptionwasn’tthusincompatible,since,taking“theEncyclopedia”tobethesetofacceptedsentences,onecouldadopttheusageof“istrue”onwhich“‘p’istrue”isacceptedintotheEncyclo-pediaifandonlyif“p”is[Mancosu,2008,216].Neurath’sconfusionisthecommonarisingfromthefactthatanyonewhodefendsananalysisoftruthwill,iftheyremainconsistent,acceptsentencesifandonlyifthesesentenceshavethepropertywhichbytheiraccountreducesthepropertyoftruth.Thisdoesn’tshowthatanyanalysisiscomptiblewiththeT-sentences—thatis,withthetruthconditionsthatsentencesactu-allyhave—anymorethanthefactthatthedefenderofanethicaltheorywithrepugnantconsequencescanremainconsistentbyactingontheseconsequencesshowsthatallethicaltheoriesarecompatiblewithwhatisactuallyright.1Sympatheticinterpreterslike[Mormann,1999]tieNeu-rath’sreactionintohisoverallsetofviews,butforourpurposesthisisthefundamentalconfusion.CarnaphimselfgrewfrustratedwithNeu-rathovertheyears,writingtoNeurathin1943thatasearlyas1935“therewerenorationalargumentsleftonyourside”[Mancosu,2008,220].Tarski,ontheotherhand,seemstohavebeenconvincedbytheCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01episodethatthelong-standingpolicyoftheWarsawlogiciansdictat-ingprofessionalsilenceonphilosophicalmatterswastherightone[Murawski,2004,329].Thoughasdetailedinthesetofconferencenotestowardtheendofv.4ofhisCollectedWorksTarskicontinuedsporadicallytoparticipateinsomephilosophicalevents,hispublicationsceasedtohaveeventhemodestphilosophicalcastofhisearlywork;e.g.[Tarski,10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP08”—2011/11/28—13:58—PAGE228—#2\nConclusion2291986d]simplypresentsthestandardlimitativeresultsinarecogniz-ablycontemporaryformwithoutanyhintoftheIntuitionisticFormalistaspectsoftheoriginaldiscussioninCTFL.Thelate“TruthandProof”,publishedinScientificAmerican,confirmsthis;clearlyasetofmothballedremarksfromthe1930s,thearticleshowsthat,totheextentthathehadgiventhetopicsanythoughtatall,hisviewshadnotchanged.Theoneexceptionhereis,ofcourse,Tarski’smostwidely-readpubli-cation,1944’s“TheSemanticConceptionofTruthandtheFoundationsofSemantics”.Tarskiintroducesthetopicasgivinga“satisfactorydef-inition”,thatis,a“materiallyadequateandformallycorrect”one,ofthe“notion”oftruth[Tarski,1986e,665].Interestingly,note4notesthat“notion”and“concept”canmeananythingfromatermitself,to“whatismeant”byit,toitsdenotation(ausagefoundin[Tarski,1986j]),butthenwavesofftheissuewiththecommentthatTarski“share[s]theten-dencytoavoidthesewordsinanyexactdiscussion”[Tarski,1986e,694].Tarski’spreviousinterestsinconceptsandcapturingthemisclearlygone;whatwehaveleftisdirectinterestinsemanticsinitsownright.Tarskithenannouncesthattheaimofthedefinitionisto“catchholdofthemeaningofanoldnotion”,notesthattruthappliestosentencesandistherebyrelativetoalanguage(withnomoreexplicitconcernforhowthisrelativizationiscompatiblewiththerebeingasingleconceptthanheshowedinthe1930s)andrepeats,withsomeelaboration,theremarksonthe“semanticaldefinition”from[Tarski,1983a,155],thistimecall-ingtheconceptionthe“classicalAristotelian”conceptionandnotingitsimprecision.Withnomoreexplanationthanin[Tarski,1983a],capturingthemeaningof“true”accordingtotheclassicalAristotelianconceptionisthenequatedwithimplyingtheT-sentences;theideaisthat“ifwebaseourselvesontheclassicalconception”wewillsaythat“snowiswhite”istrueifsnowiswhite,andfalseifitisn’t.ConventionTisthenintroducedbutleftunnamed.AnothercentralbitofIntuitionisticFormalismisthereforemissingfromSCT:theconceptionoflanguageintermsofconventions.TherequirementthattheT-sentencesbeimpliedbythedefinitionisnowsimplyacceptedinitsownright,ratherthanbeingunderwrittenbyatheoryofthecaptureofconceptsintheusuageofatermasdeterminedbyadefinitionoranyothersortofconvention.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01SemanticsisdiscussedasinESSandtheusualpointsareenteredaboutanexactstudybeingfocusedonformalizedlanguages.AfterthatTarskirehearsesLukasiewicz’spresentationoftheliarparadoxandthenremarks,following[Tarski,1983a,157–8]thatitshowsthat“semanti-callyclosedlanguages”mustbeleftoutofconsideration.Tarskinotesinawayconsonantwith[Tarski,1983a,165]that“everydaylanguage”10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP08”—2011/11/28—13:58—PAGE229—#3\n230AlfredTarski:PhilosophyofLanguageandLogicseemstobethesortoflanguagethatistobedisregardedandfollowsCTFLinattributinginconsistencytosuchlanguage.2Tarskithenexplainsthedistinctionbetweenobjectlanguageandmetalanguageandnotesthatanadequatedefinitioncanonlybeformulatedinan“essentiallyricher”metalanguage.Thisisthenfollowedupbyathumbnailsketchofrecursiononsatisfactionandthedefinitionoftruthassatisfactionbyeverything.Tarskithenassertsthe“material”adequacyofthedefinition,andnotesthat“wecanshowthatoneandonlyoneofanytwocontra-dictorysentencesistrue”.AsinCTFL,thisistreatedasarequirementoftheAristotelianconceptiononceitismentionedeventhoughitwasn’tbuiltintotheearlierdefinitionofmaterialadequacy.Itaketherepetitionofthisinconsistenttreatmentof“material”adequacytobeconfirmationoftheextenttowhichTarskiisrunningonauto-pilotinsummarizingCTFL.InSCTTarskiistryingtogetinthebareminimumofsemanticswithouttheIntuitionisticFormalistbackgroundfromwhichitoriginallysprang,buthedoesn’totherwiseappeartohavebeenthinkingaboutthetopicsintheinterveningyears.Mostimportantly,missingfromSCTisanyconceptionofmeaningotherthanthatprovidedbysemantics.Innote20to[Tarski,1986e,678]Tarskisaysthatallofthesemanticconceptsmentionedinthesection(§13)canbedefinedintermsofsatisfaction,notesthatconse-quencecanbedefinedintermsofitasin[Tarski,2002]andthen,citing[Carnap,1942,55]mentionsthatsynonymycanalsobedefinedintermsofsatisfaction.SincethroughoutCTFL“translates”and“meansthesameas”areusedinterchangeably,thisisasignificantchange.WhileinCTFLtranslationwastiedtotheexpressionofintuitivemeaning,inSCTitisanordinarysemanticconceptlikeanyother;inaccordwithCarnap’sdefinition,twoexpressionsaresynonymousjustincasetheyhavethesameextensionalsemanticvalue.IntuitionisticFormalismhasnowcom-pletelybeenabandonedasaconceptionofmeaninginfavorofsemanticsitself.3HavingseenthatthelastpieceofIntuitionisticFormalismthatheaccepted,theepistemicconceptionofconsequenceembodiedinthedefinitionofconsequenceintermsofstatedrulesofderivation,couldgo,Tarskishiftedtounderstandingmeaningingeneralintermsofthesemanticshehadintroduced.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Tarskithenturnstohis“PolemicalRemarks”inthesecondpartofthearticle.His§14disclaimsanyintentiontoadjudicatebetweentheclassi-calconceptionoftruthandvariouscompetitors,suchas“thepragmaticconception,thecoherencetheory,etc.”[Tarski,1986e,680].Thoughoffi-ciallythisagnosticismandtheconcomitantthesisoftheambiguityoftheeveryday“true”amongthesenotionsfiguredalreadyat[Tarski,1983a,10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP08”—2011/11/28—13:58—PAGE230—#4\nConclusion231153]onesomehowgetsthefeelingthatTarski’sclaimhereisn’tsincere,especiallygivenhisscornfulremarksondenialsoftheT-sentenceslaterinthearticle.Then§§15–16dispensewithsomeconfusionsaboutthesententialconnectivesandthesuggestionthatTarski’stheoryisaformofwhatKotarbi´snkicalled“nihilism”;Tarskimakestheobviouspointsandrefersthereaderto[Kokoszy´nska,1936].Tarskithenturnstothequestionofwhetherthesemanticconceptionreallyoffersa“preciseform”oftheclassicalconception.Asnotedabove(§4.1.4),thisisagoodquestion.IhavemaintainedthatTarskihas,inprinciple,ananswerwhentheclassicalconceptionisstatedintheformofCTFL’s“semanticaldefinition”,while,bycontrast,thereisgoodrea-sontothinkthatthesemanticconceptionisn’tinanysignificantsenseacorrespondencetheory(§4.3.4).Moreinteresting,though,inaddi-tiontotheratherplaininconsistencyofTarski’sinsistenceherethatthesemanticconceptiondoes“conformtoaveryconsiderableextentwithcommon-senseusage”with§14,isTarski’sdiscussionofArneNaess’searlyexercisein“experimentalphilosophy”.Naess’ssurveyshowedthatonly15%ofpeopleagreedthat“true”means“agreeingwithreality”,while90%assentedtoatypicalT-sentence.Tarski“waives”considera-tionofwhetherthetwoconceptionsarethesame,thoughsurelytheresultindicatesjustthedisconnectbetweencorrespondenceandtheT-sentenceswehavenoted.TheconclusionIwoulddrawhereisthatTarskineverquestionedwhetherKotarbi´nskihadthingsright,andalwaysconcernedhimselfratherwithhowtoexpresstheconceptioninadeductivetheory.Thebeginningof§18isinterestinginthatTarskiseemstodisclaimthephilosophicalrelevanceofhisworkthathestressedinCTFL:“ingeneral,Idonotbelievethatthereissuchathingas‘thephilosophicalproblemoftruth’”.Tarskihadhaditwithphilosophicalcontroversyafterthedis-cussionsfollowingtheParisconference,andtheensuingpagesmanifesthisfrustration.Ratherthancontinuingthephilosophicalproject,TarskiinSCTpresentshisworkontruthasofformalinterest.Tarskimakestheperfectlycorrect,thoughoftenmisunderstoodpointthatitispartofthesemanticconceptiontoequatetheassertabilityconditionsof“‘p’istrue”and“p”,butnotesthatthissaysnothingabouttheconditionsCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01underwhicheithercanbeasserted.Hestressesthecompatibilityoftheconceptionwithany“epistemologicalattitude”,thoughhealsoscornsanyconceptionoftruthitselfthatdeniestheT-sentences.Theapparentinconsistencyofthiswith§14isagainleftunexplained.Movingon,§19concernsitselfwithNeurath’schargethattheseman-ticconceptionis“metaphysical”.Tarskinotesthat“metaphysical”inthe10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP08”—2011/11/28—13:58—PAGE231—#5\n232AlfredTarski:PhilosophyofLanguageandLogicobjector’susageseemstobenothingmorethananemptytermofabuse,suggestsseveralreadingsof“metaphysical”,andarguesthatonallofthereadingsonwhichitexpressessomethingbad,thesemanticconceptionisn’t“metaphysical”.Inparticular,Tarskidefendstheincompatibilityofhisdefinitionswiththeverificationistrequirementthatadefinitionofatermgivecriteriaforitsapplicability[Tarski,1986e,687–8].Then§20discussestheroleofsemanticstudyinempiricalscience,§21ascribesalimitedroletosemanticsinthemethodologyofempiricalscience,and§22discussesitsgreaterroleinmetamathematics,andthewayinwhichmetamathematics,viaarithmetization,isreallyabranchofmathemat-ics.Finally§23closesbyrejectingtheideathatthevalueofabodyofresearchistobejudgedbyitsapplicability.Inlightofitshaphazardcompositionandinparticulartheblatantinconsistencyof§14with§18—aninconsistencythatcouldonlybeeliminatedbytakingseriouslyTarski’sfeignedadoptionoftheviewthatanyonewhodisagreedwithhimabouttruthhadchangedthetopic—SCTcanonlybereadasanintendedfinalattempttogetphilosopherstostopbotheringhim.Onecanhardlyblamehim:thoughmorecogentobjectionshaveplayedasignificantroleinthecontemporaryliterature,theobjectionssurveyedinSCTscarcelymeritattention.8.2FinalremarksFromthispointonecoulddiscusstheimportantroleTarski’sworkplayedinthedevelopmentofanalyticalphilosophy,butsincethisstory,unliketheoneIhavetold,isrelativelywellknown,ourstudycomestoanendhere.Tarski’sphilosophicalcareerendedwithsomethingofawhimper:theconceptionthatgothimstarted,IntuitionisticFormalism,containedtheseedsofitsowndestructionwhenTarskiturneditonsemanticconcepts.Hisapproachtotruthwasgreetedinitiallywithincompre-hensionbyitscriticsandtosomeextentitssupporters.Thetechniquesheintroducedwentontoanillustriouslifeinlogicandthephilosophyoflanguage,buthehimselfwentoninlogicandmathematics;foranoverviewofhiscareeronemayconsult[Vaught,1986]and[FefermanandFeferman,2004].Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Thoughtheconceptionofconsequenceisproblematic,truth-theoreticsemanticshasmanyadvantagesovertheIntuitionisticFormalistaccount,especiallywhenitisdetachedfromTarski’sdefinitionalprojectandtreatedasissuinginaxiomatictheoriesthatstatethetruthcon-ditionsofthesentencesofalanguageasrecommendedin[Davidson,1990]and[Etchemendy,1988].Intheplaceofhand-wavingaboutthe10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP08”—2011/11/28—13:58—PAGE232—#6\nConclusion233associationofcomplexthoughtsorthought-constitutentswithcom-plexexpressionsonehasastraightforward,genuineaccountofsemanticcompositionalityinTarski’srecursiononsatisfaction.Intheaccountofmeaningthiscanonlyberegardedasenormousprogress,althoughtheproblemsofmovingbeyondthesimplestockofmodesofcompositionextensionalmethodscanhandlearewellknown.Asmaintainedintheintroduction,neithertherepresentationalnortheexpressiveconceptionsofmeaningstandsaloneintheoverallaccountoflanguageandcommunication.Truth-theoreticsemanticsabstractsmeaningfromtheattitudesofspeakersinahelpfulway,butitremainsthecasethatexpressionshavethetruth-theoreticpropertiesthattheydobecauseoftheattitudestowardthemandusemadeofthembyspeakersintheirenvironments.BuildingonTarski’scontributionswehave,aswedidnotbeforethem,aclearandworkableconceptionofwhatitisthatthesefactorsdeterminewhentheydeterminemeaning.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-0110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“CHAP08”—2011/11/28—13:58—PAGE233—#7\nNotes0Introduction1.Oneimportantconnectiontothismathematicalworkneedstobenoted:Tarski’sworkintheperiodwas,onmanyviews,motivatedinpartbythedesiretogiverigortothenotionofadefinableset.Idiscussthismattertosomeextentin3.3.2below,butseealso[Feferman,2008],aswellas[Vaught,1974]and[Vaught,1986]aswellasworksreferredtotherein.2.Forsurveysofthemathematics,see[Duda,2004],[BlokandPigozzi,1988],[DonerandHodges,1988],[vandenDries,1988],[Levy,1988],[McNulty,1986],[Monk,1986].3.ForcommentsonthemanuscriptIthankGregFrost-Arnold,WilfridHodgesandTerjeLohndal.Ialsothankaudiencesatseveralpresentationsofmaterialfromthebook:atthemeetingoftheSocietyforExactPhilosophyinKansasCity,inparticularStewartShapiro,PhilipKremer,MichaelGlanzbergandPaulPietroski;attheSchoolforAdvancedStudyattheUniversityofLondon,inparticularBarrySmith,IanRumfittandOysteinLinnebo;attheUniversityofPaderborn,inparticularmyhostVolkerPeckhaus;attheUniversityofAlbany,inparticularBradleyArmour-Garb;attheLudwig-MaximillianUni-versityinMunich,inparticularHannesLeitgeb,JeffreyKetland,OlivierRoyandFlorianSteinberger;attheTruthatWorkconferenceinParis,inpartic-ularHenriGalinonandhisfellowhosts,aswellasStephenRead,GilaSher,VolkerHalbachandMichaelGlanzberg;andattheUniversityofSiegen,inparticularmyhostRichardSchantz.ThisbookwaswritteninpartduringthetermofaForschungsstipendiumfürErfahreneWissenschaftlergrantedbytheAlexandervonHumboldtFoundationforstudyattheUniversityofLeipzig.Ithankthefoundationandespeciallymykindhost,ProfessorDr.PirminStekeler-Weithofer.1IntuitionisticFormalism1.OnecanalreadyfindtheviewinarticulatedwellinPasch,e.g.[Pasch,1912,98].Detlefsen[Detlefsen,2008]tracesthefundamentalideasbackmuchfurther.2.OneWarsawfigurewhowasaformalistinHilbert’smoderatherthanCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Le´sniewski’swastheonewhospenttimeinGöttingen:theAjdukiewiczofthehabilitationdissertation,whorelegatestheappealtointuitivemeaningtotheprimitivestateofadeductivescience,whileinamatureaxiomatiza-tionbeingmeaningfulisamatterofasign’sroleinatheory,ratherthansomeexternalrelationofdenotation[Ajdukiewicz,1966,4–6].3.See[Betti,2004,278]foradiscussionoftheissuesinvolvedininterpretingLe´sniewski’slaterworkintermsoftheearlywork.23410.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“NOTES”—2011/11/28—14:07—PAGE234—#1\nNotes2354.Le´sniewski’s“adequately”(adekwatnosci´)hereanticipatesTarski’sownnotionofintuitiveadequacyinatheory.5.OnTwardowski,founderoftheLvov–WarsawSchoolandLe´sniewski’smentor,see[Wole´nski,1989]and[Smith,1994].6.AsWole´nskinotes[Wole´nski,1989,252],aHusserliantheoryofmeaningintentionwasfairlyuniversallyacceptedintheLvov–WarsawSchool.ForonecomparativediscussionconcerningHusserl,MeinongandtheearlyRus-sell,withsomereferencealsototheBritishEmpiricists,seethetranslator’sintroductiontoTwardowski[1977].7.Indeed,aswewillseein§2.2.1,atleastinTarski’shandstheviewisincompatiblewithcompositionality.8.[Betti,2004,269ff]discussesthetraditionallogicalframeworkofLe´sniewski’searlypapers.9.ThequotationisprecededimmediatelybytheratherincoherentclaimthatLe´sniewskidoesn’t“assert”hisaxiomsbutonlythatthetheoremsfollowfromthem;thishiccupofcommitmentto“if-then-ism”abouthisdeductivetheoriesseemsnottohavesurvivedthetransitiontothelaterwork.10.OntherootsofLe´sniewski’sattitudehereinTwardowski’soppositionto“symbolomania”,see[Smith,1994,182–3].11.CarnapnotestheinfluenceofFregeonLe´sniewskiat[Carnap,1963,31].ThecomparisonbetweenFregeandLe´sniewskiwouldrepaymoreattentionthanIhavebeenabletogiveithere.12.See[Smith,1994,184]forasimilardiscussionofLe´sniewskiwithcommentaryontherelationbetweenhisviewsandTwardowski’s.13.HenceLe´sniewski’speculiarsententialconnectives[Le´sniewski,1992f,669–71].Thegraphicdesignmeansthateachconnectiveactuallyisadisplayofitsowntruth-table,andthatsimplemereologicalrelationshipsamongtheconnectivesdisplayrelationsofimplicationandincompatibilityamongsen-tencesformedwiththem.Thislogicallyuselessinnovationembedsremindersofthemeaningsoftheconnectivesandtheinferentialrelationstherebydeterminedwithineveryformula.Thisisotiosewithrespecttothelogicitself,butisquitepertinentifwhatoneisinterestedinisensuringsuccessfulcommunication.14.Le´sniewskinoteshisownaffinityforKotarbi´nskiat[Le´sniewski,1992h,372].15.See[Rojszczak,2002,45–6]onconventionintheBrentanoschool,especiallyinKotarbinskiandMarty.16.Onefindstheviewalsoat[Ajdukiewicz,1978a,44ff].17.Theview,then,isn’ttheviewcriticizedinvanOrmanQuine[1966],sincewhatisheldtobeconventionalisafunctionfromexpressionstothoughtsandthoughtconstituents;themeaningsandlogicalrelationsamongthelat-terareheldbyneitherKotarbi´nskinorthelaterLe´sniewskitobesomehow“assigned”tothem,inturn,byconvention.Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-0118.OnemightwonderherehowthisappealtotypesistobereconciledwithKotarbi´nski’snominalism,butforourpurposesitwillsufficetonotethatKotarbi´nski,forwhateverreason,hasnoparticularqualmsaboutspeakingof“equiform”sentences[Kotarbi´nski,1966,6].19.OntheBrentanianBackground,see[Wole´nski,1989],[Wole´nskiandSimons,1989],and[Smith,1994].10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“NOTES”—2011/11/28—14:07—PAGE235—#2\n236Notes20.Theviewsurvivesintothelaterworkonactionsandproducts;seee.g.[Smith,1994,177–8]whereTwardowski’s1912viewisstillthatthesignificanceofasignliesinitspowertocauseideasinpeople’sminds;asSmiththerewrites,“Communicationandmutualunderstandingarepossibleonthisaccount,notbecauseourwordsorsentencesrelatetoPlatonicmeaning-entitiescapableofbeingentertainedsimultaneouslybydifferentsubjects,butbecauseourwordsareabletoevokeinothersmentalprocesseswhichareinrelevantrespectssimilartothosementalprocesseswhichtheywereusedtoexpress”.21.OnefindsthissortofviewinAjdukiewicz,too:e.g.[Ajdukiewicz,1978b,6]:“SpeakingPolish…consistsnotonlyinproducingspeech-soundswhichbelongtoPolishphonologybutalsoinexperiencing,simultaneouslyandinconnectionwiththosespeech-sounds,mentalactsinamannerassigned,whetheruniquelyorambiguously,bytherulesofPolishtothosespeechsounds”.AjdukiewiczdiffersfromKotarbi´nskiinfinding,viaHusserl[Ajdukiewicz,1978a,37]atighterconnectionbetweenlanguageandthought,aviewthatwasacentralaspectofhis“radicalconventionalism”,thoughlikeKotarbi´nskiherejectssimplisticassociationistaccountsoftherelationsbetweenthoughtandlanguage[Ajdukiewicz,1978b,7–20]22.ThispointofviewwascommoninthePeanoschool;seee.g.[Pieri,1901].23.Thepassageisfrom1937,butgivenitsclearconnectiontoPadoa,Le´sniewskiandKotarbi´nski,wecanfeelsafequotingithereasrepresentativeofTarski’sviewssomeyearsearlier.24.[Mancosuetal.,2009,323]agreethatthenotionofinterpretationinPadoaisn’tthecurrentsemanticoneandgiveareadingintermsofsentenceswithmeaningthatislargelyconsonantwiththeonehere.25.Le´sniewskialsomakesreferencetothe“well-knownmethodof‘interpreta-tion’”at[Le´sniewski,1992h,261].26.As[Murawski,2004,330]notes,followingTwardowskimembersoftheLvov–Warsawschoolwereopposedto“psychologism”butwhattheywereopposedtowaspsychologismaboutlogic,theviewfamouslycriticizedbyFrege,onwhichlogicallawsareempiricalgeneralizationsabouthowpeopleactuallythink.Thepsychologismatissuehereispsychologismaboutsemantics—thefarmoreplausibledoctrinethatmeaningisamatteroftheattitudesofspeakerstowardsymbols.Seealso[Wole´nski,1989,262–3].27.See[Ferreirós,1999,348ff]and[Reck,2004,164ff]fordiscussionofthesimpletheoryoftypesanddevelopmentsintypetheoryinthe1920s.28.ThedistancetravelledfromTwardowski,whoisquitehappytocomparecontentsof“presentations”(Vorstellungen)topaintingsoflandscapes[Twar-dowski,1977,11–17],isperhapssignificant,then,thoughitgoesbysubtraction.29.Similarremarkspersistintothetextbook,e.g.[Tarski,1995,118].30.OnereadilyassociatestheviewrejectedherewithHilbert,butTarskiprobablyCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01alsohasAjdukiewiczinmind,e.g.[Ajdukiewicz,1966,14]:“Sincenosuchele-mentwithintuitivemeaningisacomponentofformalizedaxioms,noneofthemmayberegardedasasentenceintheintuitivesense…Sinceaxiomsarenotsentencesintheordinarysense,andintheirordinarymeanings[the]words‘true’and‘false’refertosentencesexclusively…axiomscannotbejudgedfromthispointofview”.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“NOTES”—2011/11/28—14:07—PAGE236—#3\nNotes23731.Interpreterswhoagreethatthenotionofinterpretationatissueinthispas-sageisroughlytheoneIhavesuggestedinclude[Raatikainen,2003,41ff],[Fernández-Moreno,1997]and[Milne,1997].[Hodges,1986,147ff]offersthemorestandardinterpretationoftheseremarksbecauseHodgestakestheonlypossiblenotionofinterpretationtobethemodel-theoreticone.32.Onefindsasimilarviewofconnotation(alsowithreferencetoMill)anditscaptureintheinferentialstructureofadeductivetheoryat[Ajdukiewicz,1978b,21]:“AtermNconnotesinEnglishpropertiesC1,…,CnifandonlyifeveryEnglish-speakingpersonisprepared,onthebasisofhisbeliefthatanobjectAhaspropertiesC1,…,CntoacceptasanimmediateconclusionansingularsentenceofEnglishwhosesubjectisanameofA,whosepredicateisNandwhosejoiningcopulaistheword‘is’;andviceversa,onthebasisofhisacceptanceofsuchasingularsentenceheispreparedtoacceptthatAhasthepropertiesC1,…,Cn”.Ajdukiewiczgoesontorejectthisview(duetoperceivedproblemswiththeextensionalismaboutpropertieshetakesittoinvolve)andtoreplaceitwithanentirelyintra-systemicconceptionofmeaningofthesortthatwehaveattributedabovetoHilbert—compare,forinstance,hisdefinitionofsynonymyat[Ajdukiewicz,1978b,31-2]withCar-nap’sat[Carnap,2002,42].Thus,notsurprisingly,giventhathewaswithHilbertinGöttingen[Ajdukiewicz,1978c,xiii]whileLe´sniewskiwaswork-ingwiththeviewexpressedat[Le´sniewski,1992b,16-7],Ajdukiewiczisin1931astandardratherthananIntuitionisticFormalist;indeed,[Ajdukiewicz,1978b,33-4]explicitlynotestheincompatibilityofhisconceptionofmean-ingwithanynon-trivialformofthethesisthattheuseofasentenceexpressesathoughtandsuggestsreducingourconceptionofahavingathoughttoadis-positiontoassenttoasentence.Aswewillnotebelow§6.2therearethereforecorrespondingquestionsaboutwhathisconceptionhastodowithintuitivethought.ThatAjdukiewiczrejectstheviewisfurtherevidencethatitwasthereforTarskitoaccept,thoughwemightspeculatethatTarski’sownfailuretoexplicatethenotionofthecontentofaconceptintermsofconnotationconceivedinatraditionalwayboresomesignificantrelationtoAjdukiewicz’scriticism.2TarskiasIntuitionisticFormalist1.[Suppes,1988,81]attributesthefirstaxiomatizationofthenotionofconse-quenceto[Tarski,1983g].See[Wybraniec-Skardowska,2004]foradiscussionofTarski’sapproachhereandsomelaterextensionsofit.2.See[Arpaia,2006]forahistoricaldiscussionofTarski’sapproachanditsconnectionswithalgebraiclogic.3.Seealso[Tarski,1983d,39,55]forsimplerdefinitionsthatarestillhardlyCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01“intuitive”.4.IamthusinclinedtodisagreewithDavid’s[2008]attributionofanimplicittreatmentof“true”asindexicalinTarski’smetatheories.5.SmithconsidersproblemsraisedbyindexicalityforTwardowskianviewsaboutmeaningandcommunicationat[Smith,1994,180–1].6.Theideathatrelativizationalsomakesfor“ambiguity”isfoundat[Tarski,1983f,110].However,theODSpassagecanbetakensothatametalinguistic10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“NOTES”—2011/11/28—14:07—PAGE237—#4\n238Notesexpressionis“ambiguous”ifitsextensionvariesbythelanguagesitisappliedto,whichistheviewofferedhere.7.Inafulleraccountwecouldtracethehistoryofthetwoconceptionsofdefi-nitionfurtherbackandoutsideofPoland;butfortwoobviousstatementsoftheabbreviativeconception,see[Russell,1903,429]and[vanOrmanQuine,1966,71].8.[Ajdukiewicz,1978a,54]sideswiththetheory-relativeconception,thoughtheearlier1921dissertationseemstoendorsetheabbreviativeconception[Ajdukiewicz,1966,16n.2].9.AtonepointKotarbi´nskicomparesaxiomatic“pseudo-definitions”tolearn-inglanguagebythe“Berlitzmethod”.Thecomparisonfails,sincewhatmightbedifficultaboutlearningthemeaningofaprimitiveexpressionwithoutmet-alinguisticcommentarydoesnotcarryovertolearningadefinedone.Whenitcomestoprimitives,evenLe´sniewskiagreesthattheyneedmetalinguisticcommentary.10.Tarski’saccountappearstohaveenjoyedsomesuccessbackinWarsaw;notesonLe´sniewski’slecturesfromthe1933–34termincludeaversionofTarski’saccount[Le´sniewski,1988,168].11.HereIdisagreewithotherreadingsofPadoa,e.g.[Suppes,1988,84]and[Hodges,2008,105ff].12.NoteherethatKotarbi´nski’sownexplanationofPadoa’smethodistrans-parentlypsychologistic,inthatgivinganinterpretationis“attachingthemeaning”ofaspecifiedtermtoasymbol,andforKotarbi´nskithemean-ingofatermisthecontentofaconcept[Kotarbi´nski,1966,255].Likewise,Ajdukiewicz’s1921discussionofproofsofconsistencyofaxiomsystemsbythegivingofexamples,withreferencetoPieri,takesan“example”tobeasetofinterpretedsentencessothat,asAjdukiewiczconstruesit,“givinganexample”foranaxiomsystemisn’tsufficientforprovingthatitiscon-sistentwithoutaguaranteethattheexampleitselfisn’t“self-contradictory”[Ajdukiewicz,1966,22].Nobodythinkinginmodel-theoretictermswouldsaythis.Ajdukiewiczascribes“intuitivemeaning”[Ajdukiewicz,1966,6]tosentences,butdoesn’tlinkthistoanythingmodel-theoretic,either.(WatersaremuddiedherebytheHusserelianandMeinongian§3ofAjdukiewicz[1966]withitsinconsistentobjects,etc.Thiswouldbearattention,butnothere.)13.HodgesreadsPadoaasofferinganimplementationofhis“method”intermsofanotionofamodelwhichIsimplycannotfindin[Padoa,1967],offer-ingthejustificationthatthisis“themoststraightforwardreadingofPadoa’sclaim”[Hodges,2008,105].HethennotesthatTarski’sprocedureisessen-tiallymorecomplicatedthanPadoa’ssounderstoodandisrestrictedbyfeaturesofsimpletypetheoryonwhichTarskirelies.ThisisquiterightaboutPadoaasHodgestakeshim,butdependsonignoringPadoa’sessentialagreementwithCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Tarskithatan“interpretation”isanideamentallyassociatedwithasymbol.14.Afairamountofink,agooddealofitcritical(e.g.[DevidiandSolomon,1999]),hasbeenspilledoverwhatTarskimeansbyan“essentiallyricher”language,butwearetoldexactlywhatitishere.See[Rogers,1963,28]foranearlydiscussionofthenotion,and[Ray,2005]inresponsetoDevidiandSolomon.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“NOTES”—2011/11/28—14:07—PAGE238—#5\nNotes23915.AcomparisontoCarnap’sworkon“GeneralAxiomatics”inthelate1920smightbefruitfulhere,sinceCarnapwasinterestedin“monomorphism”,thatis,categoricity(indeed,Tarskiusesbothterms)butfailedtodistinguishsyntaxfromsemanticsorobject-frommetalanguage[Reck,2004,169];sinceheheldinthesameperiodthatBedeutungisaScheinrelation,hisconceptionmaywellhavebeenclosetotheoneTarskioffers.16.Tarskicanstillaccountfortheapparentdifferencebetween“human”and“featherlessbiped”inatheorythatimpliestheirextensionalequivalence:thelatter,butnottheformer,haspartsthatbeartheintuitivemeaningsoffeatherlessnessandbipedality.Inthetermsof[DouglasPatterson,2005,326–52]thetheoryisneithercompositionalnorreversecompositional.17.AdifferentissueiswhatTarskiwouldorshouldsayaboutextensionsofatheorythatinvolvenotjustnewexpressions,butnewcontexts—e.g.modalorattitudecontexts.InPatterson[2008b]Iassumethathewouldholdthetheviewthattwotermsequivalentinanextensionaltheorymightnever-thelesshavemeaningsthatwoulddistinguishtheminanintensionaltheory.Thisis,roughly,theanalogueoftheviewthattheoriesadmittingofthispos-sibilityaren’tmonotransformable:thereareextensionsofthetheoriesthatdistinguishexpressionsthataredefinableintermsofoneanotherinthethe-oriesthemselves.SinceTarskicertainlyholdsthatgoodtheoriescanfailtobemonotransformable,hemightaccepttheanalogousclaimabouthisownmetatheoriesonceintensionalcontextswereinview,butnowthatIhaveabettergrasponwhatisgoingoninTarski[1983k]Iamabitlessconfidentofthis.18.Addingaxiomstothiseffectis,historically,acquiredfromLangford,whoalsoincludessuchaxiomsinthetheoriesheinvestigates(cf.[Scanlan,2003,318],[Mancosu,2006,221]).19.Cf.[Mancosu,2006,222–3].Thedistinctionbetweenintrinsicandabso-lutecategoricity,asMancosunotes,isimportanttoassessingtheclaimsofsomeinterpreters.e.g.Gómez-Torrente[1996],thatTarski’suseofdomain-specifyingpredicatesshowsthathehadavariabledomainconceptionofamodel.See§7.3.1below.20.ThereisaworthwhilecomparisonheretoAjudkiewicz’s1934remarkson“open”and“closed”languages.ForAjdukiewiczaclosedlanguageisonesuchthatanewexpressionaddedtoitwithameaningnotsharedwithanyoftheoldexpressionsproducesalanguagethatis“disconnected”inthesensethatthemeaningofthenewexpressionbearsnoconstitutiverela-tionshipswiththemeaningsoftheoldones;bycontrast,anopenlanguagecanhaveanexpressionaddedtoitwhosemeaningisinpartdependentonthemeaningsofoldexpressions[Ajdukiewicz,1978a,50–52].Theoriescom-pletewithrespecttotheirspecifictermscorrespondtoAjdukiewicz’sclosedlanguages,andthepointthatalanguageiscompletewithrespecttoitsspe-Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01cifictermsjustincaseithasnocategoricalessentiallyricherextensionsthencorrespondstoAjdukiewicz’sthesisthatatermwithanewmeaningaddedtoaclosedlanguageproducesalanguagethatisdisconnected[Ajdukiewicz,1978a,51].TheimportanceofcompletenesssoconceivedthencorrespondstoAjdukiewicz’sthesisthatclosedlanguagescompletelydeterminemeaninginthesensethatanytwoclosedlanguagesthatshareanopenpartperfectlytranslateoneanother.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“NOTES”—2011/11/28—14:07—PAGE239—#6\n240Notes21.MorecouldbedonethanhasbeendoneheretounearththesourcesofTarski’sremarksoncategoricity,especiallygiventhediscussionofgeome-tryandmechanicsthatclosesthearticle.TarskicitesFraenkel[?,347–54]oncategoricity[Tarski,1983i,390],buttherelevantpassagesinFraenkelservemerelytodisambiguatecategoricityfromothermeaningsof“vollstandigkeit”currentatthetime,thoughFraenkel’sreferencestoMachontheunityofscienceindicatethatsomethingmorephilosophicalisintheairaswell.OneavenueofexplorationwouldbetheconnectionbetweenthissectionofTarski[1983k]andCarnap’sworkinthe1920sonaxiomatizationsforspace-time,sinceCarnapfromthedissertationof1921throughthedecadesinglesout“Eindeutigkeit”asanimportantfeatureofanaxiomsystem,amarkofitshav-ingcapturedreality,andthetermanditscognatesappearinTarski’sGerman.Forastartonthistopic,seeHoward[1996]and[AwodeyandReck,2002,25–7].TheconnectiontoHilbert’svarious“axiomsofcompleteness”wouldalsomeritfurtherstudy,thoughasnotedat[AwodeyandReck,2002,12–14]thoughtheaxiomsimplycategoricitytheythemselvesarenotmetalinguistic(apointalsomadebyFraenkel[Fraenkel,1928,347]).22.Itakemyself,then,toprovidesomeexplanationforafactthatHodgesnotes[Hodges,2008,113],thatTarskineverclaimsthatthetermsinfavorofwhichadefinedtermiseliminatedaresomehowclearerormorebasicthanthedefinedtermitself.23.OnesmallbitofevidencethatideaslikethiswereintheairinWarsaw:acritic,discussingŁukasiewicz’smaniaforshorteninghisaxioms,writesthat“Usuallytheshortestones…havenoapplicationinpractice…Łukasiewiczwasevenatheoristofthatsport;heclaimedthattheintuitivecontentofanaxiomsystemisnotimportant,sinceitisonlythesetofthosetheoremswhichareitsconsequencesthatcount”[Wole´nski,1989,91].3Semantics1.Foradiscussionofthemainlineofdevelopmentofthoughtaboutquantifi-cationalongsemanticlines,bycontrast,see[Goldfarb,1979].2.CompareherethediscussionofthecorrespondencebetweenGödelandCar-napin1932whereGödelpointsoutamanifestationofthisprobleminaparticularcaseat[Coffa,1987,552–3].(Foraproposedwork-aroundalongCarnap’spre-1934lines,see[deRouilhan,2009,139–40].)3.SettingasideLe´sniewski’sputativealternativediscussedbelow.4.See[Simons,1993,215ff]foronediscussion.5.ByZermelo,asnotedbyFraenkelat[Fraenkel,1928,210]6.ThedistinctionisoftenerroneouslycreditedtoRamsey.Seee.g.[Ferreirós,1999,351].Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-017.Thepeculiarsuggestionthattheparadoxeshavesomethingtodowithinten-sionalityisboundupwithCarnap’sunderstandingofthelatter.TheAbrissrefersthereaderto§§43-45oftheAufbauonthispoint.8.Foramorethoroughhistory,see[Mancosuetal.,2009,419ff].9.See[Mancosuetal.,2009,326–8].10.[Fraenkel,1928],whichTarskicites,isalsocompletelysemantic;see,e.g.,thediscussionofcategoricityat349ff.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“NOTES”—2011/11/28—14:07—PAGE240—#7\nNotes24111.SometimestheinspirationforTarski’sdevelopmentofsemantics,inparticulartheuseofthenotionofsatisfaction,isattributedtoAjdukiewicz’s1921habil-itationdissertation(e.g.[Betti,2008,61ff],[Mancosuetal.,2009,435]).Theattributionisproblematic,however;considerthetreatmentofan“example”intendedtoshowasetofaxiomstobeconsistentasbeingasetofsentencesthatitselfmightbeinconsistent[Ajdukiewicz,1966,22,32].Nomodelinthesemanticsensecanbe“inconsistent”.Thingsarecomplicatedhere,how-ever,bytheHusserlianandMeinongiansection3ofthework,inwhichAjdukiewiczmaintainsthat“inconsistentobjects”existandare“createdbyourminds”.Clearer,perhaps,isAjdukiewicz’sdiscussionoftruthandsatis-factioninthatsection,butinthiswegetsomethingthatis,thoughclear,certainlynotsemantic,namely“thedefinitions:‘ObjectPsatisfiesthepropo-sitionalfunctionf(x)means—the[constructing]definitionofPentailsf(P)’and‘thesentencef(P)istruemeans—thereexistsanobjectPsatisfyingf(x)’.”Satisfactionhere,suchasitis,isanentirelyintra-linguisticmatterofacertainprivilegeddefinitionofatermimplyingtheclosureofanopensentencebythatterm([Betti,2008,63]agrees).Bythisdefinition,theencouragingdic-tum“thedomainofatheoryisthesetofallobjectssatisfyingtheaxioms”[Ajdukiewicz,1966,41]can’tbereadsemantically,sincetheeffectofthedefi-nitionofsatisfactionistoequatetheapparentlysemantic“satisfies”withtheexistenceofcertainimplicationsamongsentencesofatheory.PerhapsthesepassagesgotTarskithinking,buttheyaresoconfused—notetherampantuse–mentionsloppinessandtheinsouciantphenomenologicaldoctrinesthatleadtotheviewthatthereare“objects”“inthedomain”ofatheorythatnei-therexistnorhavedefinitionsconsistentwithitsaxioms[Ajdukiewicz,1966,43]—andaresofarfromclearlysemanticthatlittleanticipationofTarski’ssemanticscanbeattributedtothem.12.[Heck,1997,545–6]understatestheimportanceofexplicitdefinitionforguaranteeingconsistency,takingthedistinctionbetweenobjectandmetalan-guagetobesufficient.Buttheproblemwiththisisthatitisn’talwaysobviouswhenwehaveadistinctionbetweenobjectandmetalanguageintherelevantsense.Ascendingtoametalanguageblocksoneknownsufficientconditionforinconsistency,butaslongasthetheoryremainsaxiomatictheworryaboutinconsistencyremains.Onlyexplicitdefinitionguarantees(relative)consistency,anditwasforthisthatTarskiwaslooking.13.See[Addison,2004]formorehistoricalbackgroundandasurveyoflaterdevelopmentsofthemathematicalaspectsofthearticle.14.See[Hodges,2008]foragooddiscussionofthemathematicaldevelopmentshereandtheirrelationtoTarski’sworkonquantifierelimination.15.OnarelatednoteFeferman,takinghimselftodisagreewiththeclaimof[Hodges,1986]claimsthat“thenotionoftruthinastructureispresentimplic-itly”thearticle[Feferman,2008,80–2],but,liketheclaimof[Hodges,2008]Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01Feferman’sreadingsuffersfromfollowingTarskiintakingsemanticdefinitionofthesetofthingsdefinableinLforaformaldefinitionof“definesinL”.16.Cf.[Hodges,2008,111–12],whonotescorrectlythatthepassageisoddinthatwhatistobeshownisn’tevenexpressedinthetheory.17.“Theempiricalmethod”isaphraseTarskiusesforomittingaproofbyinductionandsimplyconsideringafewcases[Hodges,2008,98].10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“NOTES”—2011/11/28—14:07—PAGE241—#8\n242Notes18.[Sinaceur,2001,54]notesTarski’sclaimsaboutwhathehasaccomplishedbutdoesnotcommentonthedistancebetweentheseclaimsandwhatTarskiactuallydoes.See[Vaught,1986,871]forapositivetakeonthisasmathematics.4Truth1.[Ajdukiewicz,1966,21]callsatheory“adequate”(“adekwatny”)ifitsaxioms,whengivenanintuitiveinterpretation,sufficetoproveallintuitivetruthsexpressibleinthetherebyinterpretedlanguage.Tarski’susagebearssomeresemblancetothis,aswellastoLe´sniewski’suseofthesamePolishcognateforanotionclosertoTarski’sat[Le´sniewski,1992b,16–17].2.Icouldn’tfindadefinitivesourceonthis,butsinceTarskiotherwisewrotehisownGerman(ashissonisreportedassayingintheintroductionto[Tarski,1992])itseemsmorelikelythathewrotethePostsciptinGermanhimselfthanthathewroteitinPolishandhadBlausteintranslateit.3.TheclassiconthetraditionleadinguptoTarskiis[Wole´nskiandSimons,1989].SomeusefulsourcesonthegeneralsituationbeforeTarskiinclude[Niiniluoto,1999b]onthesituationintheGerman-speakingworldinthe1910sand1920s,and[Sluga,1999]onFrege,Russell,WittgensteinandtheirinfluenceonCarnapinhisreceptionofTarski(althoughSluga’sreadingsuf-ferssomewhatfromtheconfusionofTarskiwithDavidson,andofTarski’s“theory”withaTarskiandefinition(§4.3.2)).4.Ileaveforlaterthequestionastowhetheritreallyisacorrespondencetheory.5.Onwhomsee[Wole´nski,1989].6.AdvocatingredundancywouldseemtohavebeenapostaticalintheLvov–Warsawschool,sinceTwardowskihimselfendorsedvariouscorrespondenceformulationsand,at[Twardowski,1977,96]hasadiscussionof“affirma-tionsofaffirmations”that,likeKotarbi´nski’sdiscussionof“verbal”and“real”interpretationsof“true”,clearlydistinguishesonesenseinwhichtheyareredundantfromanotherinwhichtheyarenot.7.Especiallyafterthepublicationof[Tarski,1986e]asignificantdebatedevel-opedastowhetherthe“classicalconception”TarskitakesoverfromKotarbi´nskiasthe“semanticconception”isreallyincompatiblewiththe“utilitarian”conception.Atleastinthemid-1940stherewasasignificantmovementintheliteraturetodenythis—e.g.[Black,1948,61],[Perkins,1952,581ff].Thisissubjecttonow-familiarcounterexample,asTarskihim-selfexplainsat[Tarski,1986e,686]:if∀x,xistrueiffxcohereswithasetofbeliefsorfacilitatessuccessfulactionwhenbelieved,wecan’talsoholdthat∀x,xistrueiffx=
pandp,sincethisgivesustheobviouslyfalseclaimthat∀x,acceptingxfacilitatessuccessfulaction(etc.)iffx=
pandp,Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01sincethepsychologicalandotherempiricalfactsmayconspiretomakefalsebeliefsfacilitatesuccessfulaction(etc.)ortrueonesfailtodoso.See[Haack,1976b]foramorerecentconsiderationoftheissues,especially[Haack,1976b,240]wherethesupposedcompatibilityof“bizarre”theoriesoftruthwiththeT-sentencesiswronglyarguedforbypointingoutthatanyonewhothinksthatallandonlytruthsareassertedintheBiblewillacceptthat“p”isassertedintheBibleiffheacceptsthatp,since,“ifheiswise”doingthefirstwillleadto10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“NOTES”—2011/11/28—14:07—PAGE242—#9\nNotes243hisdoingthesecond;tosaythisissimplytopointoutthatthereisavalidargumentfromtheclaimthatallandonlytruthsareassertedintheBibletotheclaimthatsomeparticularthingassertedintheBibleistrue.[Keuth,1978,320]alsoadvocatesHaack’sclaim,andsaysthatTarskidid,too.Butthisistoconfusetheremarksin[Tarski,1986e]on“epistemologicalattitudes”withthetheoryoftruth.8.IamawarethatTarskisaysthat“InthisconstructionIshallnotmakeuseofanysemanticalconceptifIamnotablepreviouslytoreduceittootherconcepts”[Tarski,1983a,153].Buthere,asinotherpassages,Tarskiisbeingsoloosewith“concept”thathetreatsitassynonymouswith“term”:seethesentencesthatprecedetheoneIhavequoted.Ifaparticulardefinitionreallyreducesasemanticalconcepttootherconcepts,thentheviewthateachdef-initionintroducesadifferentconceptiscorrect.ThiswouldbeincompatiblewithTarski’sconstantlywritingasthoughthereisoneandonlyoneconceptoftruth.Giventhetension,Idiagnoseverbalsloppinessratherthanblatantinconsistency.9.[FineandMcCarthy,1984]discussestheextenttowhichtherecursiononsatisfactionisnecessary,respondingtoquestionsraisedin[Wallace,1970],[Tharp,1971]and[Kripke,1976].10.Onthisphrase,see§3.3.2.11.See[Wole´nski,1989,147–9]and[Ferreirós,1999,355–6].12.Wewillseethisissueagainin§7.4.1.13.Hereisafarfromcompletelist:[Davidson,1990,295],[Etchemendy,1988,54],[FefermanandFeferman,2004,10],[Gómez-Torrente,2004,28],[Heck,1997,541–2],[Hodges,2008,115],[Ray,1996,625].Therearesomeexcep-tionsintheliterature,butnotmany:[Rogers,1963,26],[García-Carpintero,1996,114–5],[Mou,2001,103–4].14.SomethingclosetothispracticecanbefoundinLe´sniewski,thoughinthefirstpersonsingular,e.g.[Le´sniewski,1992e,471].15.Interestinglytheoppositeviewisalsoattested:[Levison,1965,386]holdsthatConventionTstatesanecessarybutnotasufficientcondition.16.AsomewhatrelatederrorhereisfoundinspeculationsometimesheardastowhetherornotthenotionofconsequenceatissueinConventionTisstrongerthanmaterialconsequence.Thequestionhasafalsepresupposition,sincetheonlynotionofconsequenceTarskihasinCTFLisderivationaccordingtostatedrules.17.ForbrevityIlistsomeofthebest-knownreferencesonly.18.Donot,ofcourse,bedistractedbythefactthatatheorystatedinthatcoun-terfactualsituationwouldhaveneededtohave“‘snowiswhite’istrueiffsnowiswhite”asatheorem.19.[Milne,1999,150]and[Ray,1996,662]beatmetothebasicpoint,somethingIregrethavingoverlookedthere:adefinitionaddedtoalogicaltheoryisn’tCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01itselfalogicaltruth.20.“genuinely”:non-disjunctive,statedinthevocabularyofsuch-and-suchaphysicaltheory,supportingcounterfactuals,etc.,asthereaderpleases.21.Incaseitneedsmentioning,Fieldlaterabandonedtheseviewsfordefla-tionism,inawaythatamountstoacceptingthecriticismat[Leeds,1978,120–3].10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“NOTES”—2011/11/28—14:07—PAGE243—#10\n244Notes22.[Cummins,1975]remainsaninterestingearlyresponsetoField’sconceptionofaphysicalisticallyacceptabletheoryofreferencethatparallelsdevelop-mentsinthephilosophyofmindandphilosophyofscienceconcerningeliminationandreductioningeneral.Soames[Soames,1984,419ff]andoth-erspointoutthatbyField’sstandardsTarski’srecursionclausesarejustasobjectionable,sincetheytoofailtosupportcounterfactualsandsoforth.[McDowell,1978]acceptsthatTarski’stheoryasheunderstooditisn’tphys-icalist,asheunderstoodthat,butthatitcanbeunderstoodalongbroadlyDavidsonian(e.g.[Davidson,2002a],[Davidson,1990,299–300])linesasinstrumentalistaboutsub-sententialsemanticsandphysicalistattheleveloftheconfirmationofT-sentences.Thereis,inturn,asubstantialliteratureonthissortofDavidsonianview;forarecentdiscussioninthecontextofinterpretingTarski,see[Hoffmann,2007,161–70].[Kirkham,1993,289–302]arguesthat“physicalism”forTarski,followingNeurathandCarnap,requiredreductiontoobservables,notphysicsproper,andthatTarskisucceedsintheprojectsoconceived,thoughitisabitpuzzlinghowKirkhamtakesitthataTarskiandefinitionreducessemanticnotionsto“observable”entities;theonlysecurecontentioninthearticleisthatCarnapandNeurath,andthere-foreTarski,requiredonlythatthe“reduction”ofsemanticsbeextensionallyadequate.23.[Rojszczak,2002,32]suggeststhatViennaCirclephysicalismdidnotmotivateTarski’s“choiceofsentencesfortruth-bearers”,butthisquestionisnarrowerthanthegeneralquestionofwhetherphysicalismmotivateshisattempttoeliminatesemantictermsfromhistheories.24.SCTin1944saysthatthedictum“Thetruthofasentenceconsistsinitsagreementwith(orcorrespondenceto)reality”[Tarski,1986e,666–7]isaversionoftheclassicalAristoteliantheorywhichisalsoexpressedbyCTFL’s“semanticaldefinition”.25.See,e.g.,[HawthorneandOppy,1997]foracatalogueofpossibilities.26.Thethesisthat“‘p’istrue”isequivalentto“p”—somethingliketheRedun-dancytheoryforsentences,perhapsmostfamiliartodayfromprosententialtheories—wasknowninPolandatthetimeas“nihilism”abouttruth.Wolen-skiandMurawskiattributeittoZawirskiandwehavefounditin[Cze˙zowski,1918].Kotarbi´nskidiscussesandrejectsitin[Kotarbi´nski,1966].ItwasrecognizedatthetimethatTarski’sviewwasnotaformofnihilism;see[Kokoszy´nska,1936,162–5]whostressestheineliminabilityoftheTarskian“true”invariouscontexts.5IndefinabilityandInconsistency1.[HintikkaandSandu,1999,218–221]emphasizesthisaspectofTarski’sviewsCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01onindefinability.2.Comparehere[Tarski,1986d],inwhichalloftheIntuitionisticFormalistaspectsofthediscussioninCTFLaregoneandwehaveintheirplaceaclear,concisetreatmentoftheresultsastheyarenowunderstood.3.[Gómez-Torrente,2004,34]notesthepointaboutthediagonallemmaandtriestowriteitoff,butthereasongivenforattributingthediago-nallemmatoTarski’spresentation—that(2)“appliestoallnegationsof10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“NOTES”—2011/11/28—14:07—PAGE244—#11\nNotes245predicates”—doesn’ttakesufficientaccountofthepointthatTarskiassumesthataclassratherthanapredicateisdefinedatthetopof250.4.Moreover,nothinglikethecurrentversionoftheresult,whereTarski’sthe-oremabouttruthisdistinguishedfromthemoregeneralresultthatnopredicateofthelanguageofatheorythatextendsQcandefinethesetofsentencesthatitsemanticallyimpliescanbeatissueinthepassage,sincein1933Tarskihasnonotionofconsequenceotherthanprovability.5.IfIamnotmistakenthisistheonlyone-sentenceparagraphinthework.6.Unlikeinthediscussionat[Tarski,1983a,197](§4.2.1)Tarskidoesn’therespecificallysaythatprovingsuchtheoremsisrequiredforintuitiveadequacy.7.Certainlyavailableatthetime,since[Ajdukiewicz,1978a,38]utilizesit.8.Myownapproachtotheparadoxisalongtheselines.See[Patterson,2008c],[Patterson,2009].9.Sometimescommentators[Levison,1965],[Ray,2003]trytododgethestraightforwardreadingofTarski’sclaimsbyarguingthatheholdsthat“colloquial”languageistooill-formedreallytobeinconsistent(orconsis-tent).InadditionfortherebeingnoneedtoabsolveTarskihere,thisstrategydoesn’tsquarewithTarski’swillingnesstocallsuchlanguage“inconsistent”[Sinisi,1967].10.TheviewIofferedtherewasroughlythatfoundat[Herzberger,1970,164],anditwasmistakenasareadingofTarski’snotionofa“universal”lan-guageforthesamereason,inthatbothreadingstakeuniversalityintermsofsetsdefinedratherthanconceptsexpressed.Martin’sresponsetoHerzberger[Martin,1976]alsoretainsthestandardbutmisguidedconceptionofwhatTarskitakes“universality”tobe.11.Aninterestingcomparison:[Ajdukiewicz,1978e,81]deniesthepossibilityoflanguagesthatareuniversalinTarski’ssensepreciselybecausehelackstheIntuitionisticFormalistdistinctionbetweenlanguageandthethoughtsexpressedinit;hispositioncorrespondstotheoneTarskiholdsforformalizedlanguagesand,tosomeextent,tothepositionIattributedtoTarskiin2006.6Transitions:1933–19351.ThesamepointthereforegoesforTarski’sremaksonaxiomatictruththeoryanditssupplementationbytheω-rule.2.ThusthewholepointofthePostcriptisthatthedifferencebetweenthetheoryofsemanticalcategoriesandCarnap’stheoryofLevelsmatters,contrarytotheconflationofthetwoat[Coffa,1987,560].Coffabasicallyfallsintotheerrorofthinkingthattheonlythinglackingin§5relativetothePostscriptisthenotionofanexpressionofinfiniteorder[Coffa,1987,563];notethatCoffa’sownexplanationofCarnap’stheoryoflevelsat551–2doesn’tmentiontheCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01pointthatwasimportantforTarski,theprovisionthatafunctorcouldtakeargumentsofvariablelevel.3.Again,thesepointsaren’tdirectlyrelevanttothebodyofthework,sincetheyconcernCarnap’stheoryoflevelsratherthanSTT.4.Tarskiconcludesthepostscriptwithanelegant(thoughnon-constructive)pre-sentationofGödel’sfirstincompletenesstheorem;sincetheapplicationisstraightforwarditneednotconcernushere.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“NOTES”—2011/11/28—14:07—PAGE245—#12\n246Notes5.ThisaspectoftherelationshipbetweenTarskiandCarnapismuchstudied;see,e.g.[Wole´nski,1999],[Coffa,1987],[deRouilhan,2009].6.AsimilarimpetustoTarski’sdevelopmentatthetimemayhavecomefromaninterchangewithAjdukiewicz,whose1934publication[Ajdukiewicz,1978a]attempted,inalesssophisticatedbutstrikinglysimilarway,todefinemeaninginpurelysyntacticterms[Ajdukiewicz,1978c,315].Ajdukiewicz’sviewwassubjecttoasimplecounterexamplebyTarski,whopointedoutthatitimplied,foratheoryconsistingonlyofa=bandb=a,thataandbweresynonymous,despitetheirhaving,bythetheoryitself,differentdenotations.ThoughtheworkwasnotassophisticatedasLogicalSyntaxthissurelyprovidedastrikingdemonstrationthatsyntaxalonemightnotbetrustedtodeliveragoodaccountofsynonymyandothermeaning-theoreticnotions.7LogicalConsequence1.AnoteaddedtotheEnglishtranslation[Tarski,1983g,30],distinguishesthearticleasconcernedwithwhatis“derivable”andnotesthatwhatwegetin[Tarski,2002]is“semanticconsequence”.2.StroinskáandHitchcockretain“materialadequacy”intheirannotations,inaccordwiththeirstatedaimofadheringtothestandardEnglishterminology[Tarski,2002,173]3.AsTarskihimselfmentionsinafootnoteaddedlater,theaccountissimi-lartotheoneofferedbyBolzanointheWissenschaftslehre.SinceBolzanohadasignificantinfluenceontheLvov–WarsawschoolthroughTwar-dowski’searlytraining[Smith,1994,157]andalsothoughotherroutes(e.g.Husserl),onecan’truleoutsomeinfluenceonTarskihere.OnBolzano,see[Lapointe,2011].4.[Kneale,1961,94–7]pointedouttheovergenerationproblemwithrespecttobothBolzanoandTarskilongbeforeEtchemendy.Itwouldn’tsurprisemeifthepointhadbeenmadebysomeoneaboutBolzanoevenearlier.5.[Mancosu,2006,215]suggestsdevotingapapertotherelationshipbetweenCarnapandTarskionmodelsandconsequence.[Gómez-Torrente,1996,146]alsonotestheconnectiontoCarnap,butitplaysonlyaminorroleinthepaper’sinterpretation.Bycontrast,forexample,[Sagüillo,1997,235–7]men-tions“logicians”whoseconceptionTarskiwastryingtocaptureanddiscussesRussell,butneglectsthearticle’sovertdiscussionofCarnap.6.Kotarbi´nskilikewiseidentifieslogicasthestudyofsentencesinvirtueoftheirlogicalform,wherethisisunderstoodasbeingbuiltoutofonlyvariablesandlogicalconstants,andtheninturnsimplyliststhelogicalconstants[Kotarbi´nski,1966,131],inhiscasethesententialconnectivesandthecopula.TarskistillfindsanenumerativedefinitionofthelogicalconstantsacceptableCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01in[WhiteandTarski,1987].7.Carnaphimselfisunabletoavoidcharacterizingconsequenceintheusualwayat[Carnap,2002,216].8.ForasystematiccomparisonoftheLvov–WarsawschoolwiththeViennaCircle,see[Wole´nski,1989,295ff].9.[Hodges,1986,138]isoftencitedonthispoint,butnotethatHodgesdoesn’texplicitlyproposetheunderstandingofthearticleasaresponseto10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“NOTES”—2011/11/28—14:07—PAGE246—#13\nNotes247theputativeproblemthatlogicshouldbefreeofcommitmentsastohowmanyobjectsexist.10.[Bays,2001,1710]arguesthatTarskimusthaveheldavariabledomaincon-ceptionsinceheletsVbedifferentindifferentapplicationsofSTT;Mancosu’sresponsetothisiscorrect:Tarski’sconceptionisa“weak”fixeddomainconception.11.[Simons,2009,4]attributesTarski’sfocusonabsolute,non-model-relativetruthtotheinfluenceofTwardowski’sattackonrelativism.12.[Kreisel,1967,139]makesadifferentbutsomewhatrelatedpointabout[Gödel,1967a],focusingonGödel’savoidanceevenoftheideaofthegeneralnotionoflogicaltruthastruthinallset-theoreticstructures.13.Or,alternatively,HilbertandAckermannwouldhaveaskedalessgeneralquestionandGödelwouldhaveprovedamorerestrictedresult,commonlyreferredtoas“weak”completeness[Etchemendy,1988,76].14.Again,comparehere[Łukasiewicz,1970a,17–18],whichgivesadefinitionmuchlikeTarski’scondition(F)andthenhappilyacceptstheresultthatvariousintuitivelymathematicalconsequencesare“logical”.15.Ihaveneglectedthecorrespondingproblemof“undergeneration”inthissection.Etchemendyisright,however,thatitisalsoathreat:anyrestrictionofFintendedtoavoidovergenerationthreatenstodeclarecertainintuitivelyvalidinferencesinvalid—e.g.taking“=”outofFwouldridtheaccountoftroublesomemathematicalconsequencesforfirst-orderlogic,butwouldn’tcorrespondtotheconceptionoflogicprevalenttoday.Anotherwayofstress-ingundergenerationisfoundat[Hanson,1999,610].Sincetheaccountrequiresmodelstobesets,consequencesoftheorieswithintendedinterpreta-tionsthataren’tsets(e.g.settheoryitself)won’tbeaccountedconsequencesbytheaccount.Seealso[McGee,1992,279].16.[Jacquette,1994]suggestsnotthatTarski’saccountquantifiesoverpossi-bilia,butthattheobviousfalsehoodoftheReductionPrincipleforordinaryextensionalsemanticsmotivatesa“Meinongian”semanticsthatdoesso.17.AmoreseriouserrorcomesinRay’sresponsetoEtchemendy’sclaimthatTarski’snotiondoesn’tcapturetheintuitiveepistemicfeaturesofdeductiveconsequence,thepointbeingthat,ifallthereistologicalconsequenceiswhatiscapturedbyTarski’sdefinition,then“ingeneral,itwillbeimpossibletoknowboththatanargumentis‘valid’andthatitspremisesaretrue,withoutantecedentlyknowingthatitsconclusionistrue’[Etchemendy,1990,93].RaysuggestsinresponsethatifϕfollowsfrombyTarski’saccount,andifallmembersofaretrue,thenitfollowsbyTarski’sdefinitionthatϕistrue.AsEtchemendyandothershavepointedout,thismakesthemistakeofusingadefinitionasapremiseinanargumentforitsownadequacy.18.ThattheconsequencesoftruesentencesbeingtrueisanimportantfactaboutconsequenceissomethingTarski’scommitmenttowhichantedates[Tarski,Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-012002].Shortlyafterdefiningtruthin[Tarski,1983a],TarskiprovesTheo-rem3ofthearticle,onwhichtheconsequencesoftruesentencesaretrue,andthediscussionoftheω-rulein§5emphasizestheconnectionbetweenconsequenceandtruthjustas[Tarski,2002]does.19.IquotetheWoodgerversionsinceasfarasIcantelltherelevantsentencein[Tarski,2002]isungrammatical.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“NOTES”—2011/11/28—14:07—PAGE247—#14\n248Notes20.JusthowmanydependsondetailsabouttheversionofSTTassumed,especiallywhetheritincludestheaxiomofinfinity.21.[Sher,2008]andherrelatedwritingsarethebestdevelopedpresentationofsuchanapproach.22.Thetreatmentofnon-logicalexpressionsas“empirical”predatesCarnapinmaterialtowhichTarskipaidattention,e.g.[Padoa,1967,151].23.OnemightofcoursechargeCarnap’streatmentofLanguagesIandIIwithbeingadhocitself;see§7.6.2below.24.Comparehere[Wole´nski,1989,269]on“thePolishtheoryofanalyticsentences”.25.Truetoidiosyncraticform,Le´sniewskidisagreedandmaintainedthatOntol-ogyhadnoontologicalcommitments.As[Simons,1993,215]notes,theissueturnsonthedisputedstatusofthequantifiersinLe´sniewski’ssystemthatwenotedat§3.1.1.26.[Sagüillo,2009,25]alsoemphasizestheconnectionbetweenTarski’sdefini-tionandspecificallymathematicalpractice.8Conclusion1.Asnotedabove,thisconfusionisthesourceofbothoftheclaim(e.g.[Haack,1976a,323–4])thattheT-sentencesarecompatiblewithalltheoriesoftruthandtheclaimthattheT-sentencesaretrivialtruthsthat“givenoinformation”(as,e.g.,inthecriticismofdeflationismin[Vision,2004]).2.[Ray,2003]makessomethingsignificantoutofthedifferencebetweenthispassageand§1of[Tarski,1983a],butinlightof[Tarski,1983a,267]Idon’tthinkthedifferencecomestomuch.3.SomepauseheremaybegivenbyCarnap’sunexplaineduseof“translates”uptoandincludingthedefinitionof“designation”at[Carnap,1942,53–4],butthattranslationisamatterofsharingdesignataismadeclearat[Carnap,1942,53]whereCarnapwrites“LetussupposeforthemomentthatweunderstandagivenobjectlanguageS,sayGermanorS3(§8),insuchawaythatweareabletotranslateitsexpressionsandsentencesintothemetalanguageMused,sayEnglish(includingsomevariablesandsymbols).Itdoesnotmatterwhetherthisunderstandingisbasedontheknowledgeofsemanticalrulesorisintuitive;itismerelysupposedthat,ifanexpressionisgiven(saye.g.‘Pferd’,‘drei’inGerman,‘P(a)’inS3),forallpracticalpurposesweknowanEnglishexpressioncorrespondingtoitasits‘literaltranslation’”.Heretherearetwowaystoknowwhattranslateswhat—byknowingexplicitlystated“semanticalrules”orintuitively,butsincewhatisknowninthefirstcaseisasemanticmatterofdesignation,whatisknowninthesecondcasemustbedesignationaswell.Carnap’sappealtotranslationin[Carnap,1942]isn’tanappealtoaCopyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01differentconceptionofmeaning,butisratheranimplicitlymetametalinguisticappealtorelationsofsamenessofextensionalsemanticvaluethatisusedinjudgingsemanticalrulesincaseswheretheyaren’tpurelystipulative,asinhisexamplesofsemanticsforGermanandFrench.10.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“NOTES”—2011/11/28—14:07—PAGE248—#15\nBibliographyAddison,J.W.(2004).Tarski’stheoryofdefinability:Commonthemesindescrip-tivesettheory,recursivefunctiontheory,classicalpurelogicandfiniteuniverselogic.AnnalsofPureandAppliedLogic,126:77–92.Ajdukiewicz,K.(1935).Diesyntaktischekonnexität.StudiaPhilosophica,1(1–27).Ajdukiewicz,K.(1966).Thelogicalconceptofproof:Amethodologicalessay.StudiaLogica,19:12–45.Ajdukiewicz,K.(1978a).Languageandmeaning.InTheScientificWorld-PerspectiveandOtherEssays1931–1963,volume108ofSyntheseLibrary,pages35–66.Reidel.Ajdukiewicz,K.(1978b).Onthemeaningofexpressions.InGiedymin,J.,edi-tor,TheScientificWorld-PerspectiveandOtherEssays1931–1963,volume108ofSyntheseLibrary,pages1–34.Reidel.Ajdukiewicz,K.(1978c).TheScientificWorld-PerspectiveandOtherEssays1931–1963,volume108ofSyntheseLibrary.Reidel.Ajdukiewicz,K.(1978d).Syntacticconnection.InGiedymin,J.,editor,TheSci-entificWorld-PerspectiveandOtherEssays1931–1963,volume108ofSyntheseLibrary,chapter118–139.Reidel.Ajdukiewicz,K.(1978e).Theworld-pictureandtheconceptualapparatus.InGiedymin,J.,editor,TheScientificWorld-PerspectiveandOtherEssays1931–1963,volume108ofSyntheseLibrary,pages67–89.Reidel.Arpaia,S.R.(2006).OnMagari’sconceptofgeneralcalculus:NotesonthehistoryofTarski’smethodologyofdeductivesciences.HistoryandPhilosophyofLogic,27(1):9–41.Awodey,S.andReck,E.H.(2002).CompletenessandcategoricitypartI:Nineteenth-centuryaxiomaticstotwentieth-centurymetalogic.HistoryandPhilosophyofLogic,23:1–30.Bays,T.(2001).OnTarskionmodels.JournalofSymbolicLogic,66(4):1701–1726.Bellotti,L.(2003).Tarskionlogicalnotions.Synthese,135(3):401–413.Belnap,N.(1993).Onrigorousdefinitions.PhilosophicalStudies,72:115–146.Betti,A.(2004).Le´sniewski’searlyliar,Tarskiandnaturallanguage.AnnalsofPureandAppliedLogic,127:267–287.Betti,A.(2008).Polishaxiomaticsanditstruth:OnTarski’sLe´sniewskianback-groundandtheAjdukiewiczconnection.InPatterson,D.,edi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,45,93–94,contentof,35199relativetoalanguage,58–62Hintikka,J.,45connotation,22,23,48Hodges,W.,20,40,44–45,101consequenceHusserl,E.,22derivation,2,52–55epistemic,42incompletenesstheorem,201–202logical,2,7,172ff.indexicality,59–6126010.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“INDEX”—2011/11/29—14:29—PAGE260—#1\nIndex261inference,1,18primitives,theoretical,27epistemic,18Protothetic,16–17intuitionisticformalism,2,16–52,63,quantification,8895,121semanticparadox,89–90intuitiveadequacy,109–111,124,semanticalcategory,124,126129–130semantics,85sententialconnectives,235Jane,I.,220Lewis,D.,31,126justification,1liar,antinomyofKotarbinski,T.,7,31–37,85seesemanticparadoxcommunication,33Lowenheim-SkolemTheorem,128,connotation,34–35198consequence,37Lukasiewicz,J.,115,184–185,240convention,31–33,47Mancosu,P.,196deductivetheory,37materialadequacydefinition,64,65–66seeintuitiveadequacydenotation,35,85McGee,V.,205indexicality,59–60meaning,1,64intention,31intuitive,14linguisticusage,31metalanguage,6,10meaning,33–35metamathematics,57,121Padoa’smethod,39Mill,J.S.,16,35–36statement(directandindirect),model,2,7,40,18431–32inintuitionisticformalism,40truth,113variabledomain,194ff.variables,87–88monotransformability,70–80Kripke,S.,157Neurath,O.,7,228ff.Langford,C.,92languageomegarule,159–160,207,colloquial,15,161214–215everyday,15,161expressingthought,43Padoa,A.,8,38formalized,57independence,39inconsistent,15,160–168methodofeliminatingtheoreticalnatural,7terms,38,67–70semanticallyclosed,168psychologism,38Lesniewski,S.,2,16–30physicalism,138–139axioms,24,27Popper,K.,5communication,27,29Priest,G.,205connotation,23PrincipiaMathematica,26convention,23,24,29psychologism,38Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-01deductivescience,24,28,30Putnam,H.,5,133definition,23,66–67quantification,86–88intuition,16–17,24,26,27quotation,116language,21meaning,23Raatikainen,P.,135non-contradiction,24Ray,G.,166,206Ontology,29representation,110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“INDEX”—2011/11/29—14:29—PAGE261—#2\n262IndexRicketts,T.,42Tractarianism,191–193,215–219Russell,B.,9,26translation,130truthsatisfaction,126–128,140–141,184Aristotelian,112recursionon,2,5,122–123axiomatictheory,123–124,158–160Scanlan,M.,92–93classical,112semanticparadox,6,88,114–115correspondence,140–143semanticalcategory,125–126,145,definitionof,4,122–124170–174indefinable,144ff.semanticsindexical,59–61expressive,1individualdomain,128–129reduction,121Lvov-Warsawschool,111–114relativetoadeductivescience,necessary,135–13955–62relativized,131representational,1T-sentence,117–122truthconditional,1,6Twardowski,K.,22,35–36,111,236seealsomeaningsentence,55–62UnityofScienceCongress,Paris1935,set9abstraction,104validity,1theory,173vanHeijenoort,J.,45Sher,G.,205Veblen,O.,72–73,93Skolem,T.,91verificationism,175–176Soames,S.,5,133ViennaCircle,9structuraldescription,133Sundholm,Göran,45Wittgenstein,L.,9Tarski’sTheorem,147ff.Zawirski,Z.,112theoryofknowledge,12Zermelo,E.,26Copyrightmaterialfromwww.palgraveconnect.com-licensedtoUniversityofSydney-PalgraveConnect-2012-02-0110.1057/9780230367227-AlfredTarski:PhilosophyofLanguageandLogic,DouglasPattersonDOUGLAS:“INDEX”—2011/11/29—14:29—PAGE262—#3