Philosophyand ModelTheory
TimButtonandSeanWalsh withahistoricalappendixbyWilfridHodges
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© Tim Button and Sean Walsh 2018 © Historical Appendix D Wilfrid Hodges
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Preface
Philosophyandmodeltheoryfrequentlymeetoneanother.Thisbookaimstounderstandtheirinteractions.
Modeltheoryisusedinevery‘theoretical’branchofanalyticphilosophy:inphilosophyofmathematics;inphilosophyofscience;inphilosophyoflanguage;in philosophicallogic;andinmetaphysics.Butthesewide-rangingappealstomodel theoryhavecreatedahighlyfragmentedliterature.Ontheonehand,manyphilosophicallysignificantresultsarefoundonlyinmathematicstextbooks:theseare aimedsquarelyatmathematicians;theytypicallypresupposethatthereaderhasa seriousbackgroundinmathematics;andlittleclueisgivenastotheirphilosophical significance.Ontheotherhand,thephilosophicalapplicationsoftheseresultsare scatteredacrossdisconnectedpocketsofpapers.
Thefirstaimofourbook,then,istoconsiderthe philosophicalusesofmodeltheory.Westateandprovethebestversionsofresultsforphilosophicalpurposes.We thenprobetheirphilosophicalsignificance.Andweshowhowsimilardialectical situationsariserepeatedlyacrossfragmenteddebatesindifferentareas.
Thesecondaimofourbook,though,istoconsiderthephilosophyofmodeltheory. Modeltheoryitselfisrarelytakenasthesubjectmatterofphilosophising(contrast thiswiththephilosophyofbiology,orthephilosophyofsettheory).Butmodel theoryisabeautifulpartofpuremathematics,andworthyofphilosophicalstudy initsownright.
Bothaimsgiverisetochallenges.Ontheonehand:thephilosophicalusesof modeltheoryarescatteredacrossadisunifiedliterature.Andontheotherhand: thereisscarcelyanyliteratureonthephilosophyofmodeltheory.
Allofwhichistosay: philosophyandmodeltheoryisn’treally‘athing’yet.This bookaimstostartcarvingoutsuchathing.Wewanttocharttherock-faceand traceitsdialecticalcontours.Butwepresentthisbook,notasafinalwordonwhat philosophicallyinclinedmodeltheoristsandmodel-theoreticallyinclinedphilosophersshoulddo,butasaninvitationtojoinin.
So.Thisisnotabookinwhichasingleaxeisground,pagebypage,intoan increasinglysharpblade.Nofundamentallineofargument—archingfromChapter 1throughtoChapter17—servesasthespineofthebook.Whatknitsthechapters togetherintoasinglebookisnotasinglethesis,butasequenceofoverlapping, criss-crossingthemes.
Topicselection
Preciselybecausephilosophyandmodeltheoryisn’tyet‘athing’,wehavehadto makesomedifficultdecisionsaboutwhattopicstodiscuss.
Ontheonehand,weaimedtopicktopicswhichshouldbeoffairlymainstream philosophicalconcern.So,whenitcomestothephilosophical uses ofmodeltheory,wehavelargelyconsideredtopicsconcerning reference, realism,and doxology (atermweintroduceinChapter6).But,evenwhenwehaveconsideredquestions whichfallsquarelywithininthephilosophy of modeltheory,thequestionsthat wehavefocussedonareclearlyinstancesof‘bigquestions’.Welookatquestions of sameness oftheories/structure(Chapter5);oftaking diverseperspectives onthe sameconcept(inChapter14);ofhowtodraw boundaries oflogic(inChapter16); andof classification ofmathematicalobjects(Chapter17).
Ontheotherhand,wealsowantedtogiveyouadecentbangforyourbuck.We figuredthatifyouweregoingtohavetowrestlewithsomenewphilosophicalidea ormodel-theoreticresult,thenyoushouldgettoseeitputtodecentuse.(Thisexplainswhy,forexample,thePush-ThroughConstruction,thejust-moretheorymanoeuvre,supervaluationalsemantics,andtheideasofmoderationandmodelism, occursoofteninthisbook.)Conversely,wehavehadtosetasidedebates—no matterhowinteresting—whichwouldhavetakentoolongtosetup.
Allofwhichistosay:thisbookisnotcomprehensive.Notevenclose. AlthoughweconsidermodelsofsettheoryinChapters8and11,wescarcely scratchthesurface.AlthoughwediscussinfinitarylogicsinChapters15–16,weonly usetheminfairlylimitedways.1 WhilstwementionTennenbaum’sTheoremin Chapter7,thatisascloseaswegettocomputablemodeltheory.Wedevoteonly onebriefsectiontoo-minimality,namely§4.10.AndalthoughweconsiderquantifiersinChapter16andfrequentlytouchonissuesconcerninglogicalconsequence, weneveraddressthelattertopicheadon.2
Thegraveenormity,though,isthatwehavebarelyscratchedthesurfaceofmodel theoryitself.Asthetableofcontentsreveals,thevastmajorityofthebookconsiders modeltheoryasitexistedbeforeMorley’sCategoricityTheorem.
Partiallycorrectingforthis,WilfridHodges’wonderfulhistoricalessayappears asPartDofthisbook.Wilfrid’sessaytreatsMorley’sTheoremasapivot-pointfor thesubjectofmodeltheory.Helooksbackcriticallytothehistory,touncoverthe notionsatworkinMorley’sTheoremanditsproof,andhelooksforwardtothe richesthathavefollowedfromit.WehavebothlearnedsomuchfromWilfrid’s ModelTheory, 3 andwearedelightedtoincludehis‘shorthistory’here.
1 Wedonot,forexample,considertheconnectionbetweeninfinitarylogicsandsupervenience,as Glanzberg(2001)andBader(2011)do.
2 Forthat,wewouldpointthereadertoBlanchette(2001)andShapiro(2005a).
3 Hodges(1993).
Still,concerningallthosetopicswhichwehaveomitted:weintendnoslight againstthem.Butthereisonlysomuchonebookcando,andthisbookisalready much(much)longerthanweoriginallyplanned.Wearesincereinourearlierclaim, thatthisbookisnotofferedasafinalword,butasaninvitationtotakepart.
Structuringthebook
Havingselectedourtopics,weneededtoarrangethemintoabook.Atthispoint, werealisedthatthesetopicshavenonaturallinearordering.
Assuch,wehavetriedtostrikeabalancebetweenthreeaimsthatdidnotalways pointinthesamedirection:toorderby philosophicaltheme,toorderbyincreasing philosophicalsophistication,andtoorderbyincreasing mathematicalsophistication. Thebook’sfinalstructureofrepresentsourbestcompromisebetweenthesethree aims.Itisdividedintothreemainparts: Referenceandrealism, Categoricity,and Indiscernibilityandclassification
Eachparthasanintroduction,andthosewhowanttodipinandoutofparticular topics,ratherthanreadingcover-to-cover,shouldreadthethreepart-introductions aftertheyhavefinishedreadingthispreface.Thepart-introductionsprovidethematicoverviewsofeachchapter,andtheyalsocontaindiagramswhichdepictthe dependenciesbetweeneachsectionofthebook.Incombinationwiththetableof contents,thesediagramswillallowreaderstotakeshortcutstotheirfavouritedestinations,withouthavingtostoptosmelleveryrosealongtheway.
Presuppositionsandproofs
Sofaraspossible,thebookassumesonlythatyouhavecompleteda101-levellogic course,andsohavesomefamiliaritywithfirst-orderlogic.
Inevitably,therearesomeexceptionstothis:wewereforcedtoassumesome familiaritywithanalysiswhendiscussinginfinitesimalsinChapter4,andequally somefamiliaritywithtopologywhendiscussingStonespacesinChapter14.Wedo notproveGödel’sincompletenessresults,althoughwedostateversionsofthemin §5.a.Abookcanonlybesoself-contained.
Byandlarge,though,thisbook is self-contained.Whenweinvokeamodeltheoreticnotion,wealmostalwaysdefinethenotionformallyinthetext.When itcomestoproofs,wefollowtheserulesofthumb.
Themaintext includesbothbriefproofs,andalsothoseproofswhichwewanted todiscussdirectly.
The appendices includeproofswhichwewantedtoincludeinthebook,but whichweretoolongtofeatureinthemaintext.Theseinclude:proofsconcerningelementarytopicswhichourreadersshouldcometounderstand(atleastone
day);proofswhicharedifficulttoaccessintheexistingliterature;proofsofcertain folk-loreresults;andproofsofnewresults.
Butthe bookomits allproofswhicharebothreadilyaccessedandtoolongtobe self-contained.Insuchcases,wesimplyprovidereaderswithcitations.
Thequickmoralforreaderstoextractisthis.Ifyouencounteraproofinthemain textofachapter,youshouldfollowitthrough.Butwewouldaddanoteforreaderswhoseprimarybackgroundisinphilosophy.Ifyoureallywanttounderstanda mathematicalconcept,youneedtoseeitinaction.Readtheappendices!
Acknowledgements
Thebookarosefromaseminarseriesonphilosophyandmodeltheorythatweran inBirkbeckinAutumn2011.Weturnedtheseminarintoapaper,butitwasvastly toolong.Ananonymousrefereefor PhilosophiaMathematica suggestedthepaper mightformthebasisforabook.Soitdid.
Wehavepresentedtopicsfromthisbookseveraltimes.Itwouldnotbethe bookitis,withoutthefeedback,questionsandcommentswehavereceived.So weowethanksto:ananonymousrefereefor PhilosophiaMathematica,andJames StuddforOUP;andtoSarahActon,GeorgeAnegg,AndrewArana,BahramAssadian,JohnBaldwin,KyleBanick,NeilBarton,TimothyBays,AnnaBellomo,Liam Bright,ChloédeCanson,AdamCaulton,CatrinCampbell-Moore,JohnCorcoran,RadinDardashti,WalterDean,NataljaDeng,WilliamDemopoulos,Michael Detlefsen,FionaDoherty,CianDorr,StephenDuxbury,SeanEbels-Duggan,Sam Eklund,HartryField,BrandenFitelson,VeraFlocke,SalvatoreFlorio,PeterFritz, MichaelGabbay,HaimGaifman,J.EthanGalebach,MarcusGiaquinto,PeterGibson,TamaravonGlehn,OwenGriffiths,EmmylouHaffner,BobHale,Jeremy Heis,WillHendy,SimonHewitt,KateHodesdon,WilfridHodges,LucaIncurvati,DouglasJesseph,NicholasJones,PeterKoellner,BrianKing,EleanorKnox,JohannesKorbmacher,ArnoldKoslow,Hans-ChristophKotzsch,GregLauro,Sarah Lawsky,ØysteinLinnebo,YangLiu,PenMaddy,KateManion,TonyMartin,GuillaumeMassas,VannMcGee,TobyMeadows,RichardMendelsohn,Christopher Mitsch,StellaMoon,AdrianMoore,J.BrianPitts,JonathanNassim,FredrikNyseth,SaraParhizgari,CharlesParsons,JonathanPayne,GrahamPriest,Michael Potter,HilaryPutnam,PaulaQuinon,DavidRabouin,ErichReck,SamRoberts, MarcusRossberg,J.Schatz,GilSagi,BernhardSalow,ChrisScambler,Thomas Schindler,DanaScott,StewartShapiro,GilaSher,LukasSkiba,JönneSpeck,SebastianSpeitel,WillStafford,TrevorTeitel,RobertTrueman,JoukoVäänänen,Kai Wehmeier,J.RobertG.Williams,JohnWigglesworth,HughWoodin,JackWoods, CrispinWright,WesleyWrigley,andKinoZhao.
WeowesomespecialdebtstopeopleinvolvedintheoriginalBirkbeckseminar.
First,theseminarwasheldundertheauspicesoftheDepartmentofPhilosophyat BirkbeckandØysteinLinnebo’sEuropeanResearchCouncil-fundedproject‘Plurals,Predicates,andParadox’,andweareverygratefultoallthepeoplefromthe projectandthedepartmentforparticipatingandhelpingtomaketheseminarpossible.Second,wewereluckytohaveseveralgreatexternalspeakersvisittheseminar,whomwewouldespeciallyliketothank.Thespeakerswere:TimothyBays, WalterDean,VolkerHalbach,LeonHorsten,RichardKaye,JeffKetland,Angus Macintyre,PaulaQuinon,PeterSmith,andJ.RobertG.Williams.Third,manyof theexternaltalkswerehostedbytheInstituteofPhilosophy,andwewishtothank BarryC.SmithandShahrarAliforalltheirsupportandhelpinthisconnection.
AmoredistantyetimportantdebtisowedtoDenisBonnay,BriceHalimi,and Jean-MichelSalanskis,whoorganisedalovelyeventinParisinJune2010called‘PhilosophyandModelTheory.’Thateventgotsomeofusfirstthinkingabout‘PhilosophyandModelTheory’asaunifiedtopic.
Wearealsogratefultovariouseditorsandpublishersforallowingustoreusepreviouslypublishedmaterial.Chapter5drawsheavilyonWalsh2014,andthecopyrightisheldbytheAssociationforSymbolicLogicandisbeingusedwiththeir permission.Chapters7–11drawheavilyupononButtonandWalsh2016,published by PhilosophiaMathematica.Finally,§13.7drawsfromButton2016b,publishedby Analysis,and§15.1drawsfromButton2017,publishedbythe NotreDameJournalof FormalLogic.
Finally,though,awordfromus,asindividuals.
FromTim. IwanttooffermydeepthankstotheLeverhulmeTrust:theirfunding,intheformofaPhilipLeverhulmePrize(plp–2014–140),enabledmetotake theresearchleavenecessaryforthisbook.ButImostlywanttothanktwoveryspecialpeople.WithoutSean,thisbookcouldnotbe.AndwithoutmyBen,Icould notbe.
FromSean. IwanttothanktheKurtGödelSociety,whosefunding,intheform ofaKurtGödelResearchPrizeFellowship,helpedusputontheoriginalBirkbeck seminar.IalsowanttothankTimforbeingamodelco-authorandamodelfriend. Finally,IwanttothankKariforhercompleteloveandsupport.
3.6TheNewman-conservation-objection................60
3.7Observationvocabularyversusobservableobjects.........63
3.8TheNewman-cardinality-objection.................64
3.9Mixed-predicatesagain:thecaseofcausation............66
3.10Naturalpropertiesandjustmoretheory...............67
3.aNewmanandelementaryextensions.................69
3.bConservationinfirst-ordertheories.................72
4Compactness,infinitesimals,andthereals 75
4.1TheCompactnessTheorem.....................75
4.2Infinitesimals..............................77
4.3Notationalconventions........................79
4.4Differentials,derivatives,andtheuseofinfinitesimals.......79
4.5Theordersofinfinitesmallness....................81
4.6Non-standardanalysiswithavaluation...............84
4.7Instrumentalismandconservation..................88
4.8Historicalfidelity............................91
4.9Axiomatisingnon-standardanalysis.................93
4.10Axiomatisingthereals.........................97
4.aGödel’sCompletenessTheorem...................99
4.bAmodel-theoreticproofofCompactness..............103
4.cThevaluationfunctionof§4.6....................104
5Samenessofstructureandtheory 107
5.1Definitionalequivalence........................107
5.2Samenessofstructureandanteremstructuralism.........108
5.3Interpretability.............................110
5.4Biinterpretability............................113
5.5Fromstructurestotheories......................114
5.6Interpretabilityandthetransferoftruth...............119
5.7Interpretabilityandarithmeticalequivalence............123
5.8Interpretabilityandtransferofproof.................126
5.9Conclusion...............................129
5.aArithmetisationofsyntaxandincompleteness...........130
5.bDefinitionalequivalenceinsecond-orderlogic...........132
6Modelismandmathematicaldoxology 143
6.1Towardsmodelism...........................143
6.2Objects-modelism...........................144
6.3Doxology,objectualversion......................145
6.4Concepts-modelism..........................146
6.5Doxology,conceptualversion.....................148
7Categoricityandthenaturalnumbers 151
7.1Moderatemodelism..........................151
7.2AspirationstoCategoricity......................153
7.3Categoricitywithinfirst-ordermodeltheory............153
7.4Dedekind’sCategoricityTheorem..................154
7.5Metatheoryoffullsecond-orderlogic................155
7.6Attitudestowardsfullsecond-orderlogic..............156
7.7Moderatemodelismandfullsecond-orderlogic..........158
7.8Clarifications..............................160
7.9Moderationandcompactness.....................161
7.10Weakerlogicswhichdelivercategoricity...............162
7.11Applicationtospecifickindsofmoderatemodelism........164
7.12Twosimpleproblemsformodelists.................167
7.aProofoftheLöwenheim–SkolemTheorem.............167
8Categoricityandthesets 171
8.1Transitivemodelsandinaccessibles.................171
8.2Modelsoffirst-ordersettheory....................173
8.3Zermelo’sQuasi-CategoricityTheorem...............178
8.4Attitudestowardsfullsecond-orderlogic:redux..........179
8.5Axiomatisingtheiterativeprocess..................182
8.6Isaacsonandincompletestructure..................184
8.aZermeloQuasi-Categoricity.....................186
8.bElementaryScott–Potterfoundations................192
8.cScott–PotterQuasi-Categoricity...................197
9Transcendentalargumentsagainstmodel-theoreticalscepticism 203
9.1Model-theoreticalscepticism.....................203
9.2Mooreanversustranscendentalarguments.............206
9.3TheMetaresourcesTranscendentalArgument...........206
9.4TheDisquotationalTranscendentalArgument...........210
9.5Ineffablescepticalconcerns......................214
9.aApplication:the(non-)absolutenessoftruth............217 10Internalcategoricityandthenaturalnumbers 223
10.1Metamathematicswithoutsemantics................224
10.2Theinternalcategoricityofarithmetic................227
10.3Limitsonwhatinternalcategoricitycouldshow..........229
10.4Theintoleranceofarithmetic.....................232
10.5Acanonicaltheory...........................232
10.6Thealgebraic/univocaldistinction.................233
10.7Situatinginternalisminthelandscape................236
10.8Moderateinternalists.........................237
10.aConnectiontoParsons........................239
10.bProofsofinternalcategoricityandintolerance...........242
10.cPredicativeComprehension......................246
11Internalcategoricityandthesets 251
11.1InternalisingScott–Pottersettheory.................251
11.2Quasi-intoleranceforpuresettheory................253
11.3Thestatusofthecontinuumhypothesis...............255
11.4Totalinternalcategoricityforpuresettheory............256
11.5Totalintoleranceforpuresettheory.................257
11.6Internalismandindefiniteextensibility...............258
11.aConnectiontoMcGee.........................260
11.bConnectiontoMartin.........................262
11.cInternalquasi-categoricityforSP...................263
11.dTotalinternalcategoricityforCSP..................266
11.eInternalquasi-categoricityofordinals................268
12Internalcategoricityandtruth 271
12.1Thepromiseoftruth-internalism...................271
12.2Truthoperators.............................273
12.3Internalismaboutmodeltheoryandinternalrealism........276
12.4Truthinhigher-orderlogic......................282
12.5Twogeneralissuesfortruth-internalism...............284
12.aSatisfactioninhigher-orderlogic...................285
13Boolean-valuedstructures 295
13.1Semantic-underdeterminationviaPush-Through..........295
13.2ThetheoryofBooleanalgebras....................296
13.3Boolean-valuedmodels........................298
13.4Semantic-underdeterminationviafilters...............301
13.5Semanticism..............................304
13.6Bilateralism...............................307
13.7Open-ended-inferentialism......................311
13.8Internal-inferentialism.........................314
13.9Suszko’sThesis.............................316
13.aBoolean-valuedstructureswithfilters................321
13.bFullsecond-orderBoolean-valuedstructures............323
13.cUltrafilters,ultraproducts,Łoś,andcompactness..........326
13.dTheBoolean-non-categoricityofCBA................328
13.eProofsconcerningbilateralism....................330
14TypesandStonespaces
14.7Propositionsandpossibleworlds...................350
14.aTopologicalbackground........................354
14.bBivalent-calculiandbivalent-universes................356
15Indiscernibility
15.1Notionsofindiscernibility......................359 15.2Singlingoutindiscernibles......................366 15.3Theidentityofindiscernibles.....................370 15.4Two-indiscerniblesininfinitarylogics................376
15.5 n-indiscernibles,order,andstability.................380 15.aChartingthegradesofdiscernibility.................384
16Quantifiers
16.1Generalisedquantifiers........................387
16.2Clarifyingthequestionoflogicality.................389
16.3TarskiandSher.............................389
16.4TarskiandKlein’sErlangenProgramme...............390 16.5ThePrincipleofNon-Discrimination................392 16.6ThePrincipleofClosure.......................399
16.7McGee’ssqueezingargument.....................407
16.8Mathematicalcontent.........................408
17.1Thenatureofclassification......................413
IntroductiontoPartA
ThetwocentralthemesofPartAare reference and realism
Hereisanoldphilosophicalchestnut: Howdowe(evenmanageto)representthe world? Ourmostsophisticatedrepresentationsoftheworldareperhapslinguistic. Soaspecialised—butstillenormouslybroad—versionofthisquestionis: Howdo words(evenmanageto)representthings?
Entermodeltheory.Oneofthemostbasicideasinmodeltheoryisthatastructureassignsinterpretationstobitsofvocabulary,andinsuchawaythatwecanmake excellentsenseoftheideathatthestructuremakeseachsentence(inthatvocabulary)eithertrueorfalse.Squintslightly,andmodeltheoryseemstobeprovidingus withaperfectlyprecise,formalwaytounderstandcertainaspectsoflinguisticrepresentation.Itisnosurpriseatall,then,thatalmostanyphilosophicaldiscussion oflinguisticrepresentation,orreference,ortruth,endsupinvokingnotionswhich arerecognisablymodel-theoretic.
InChapter1,weintroducethebuildingblocksofmodeltheory:thenotionsof signature,structure,andsatisfaction.Whilstthebaretechnicalbonesshouldbe familiartoanyonewhohascovereda101-levelcourseinmathematicallogic,we alsodiscussthephilosophicalquestion: Howshouldwebestunderstandquantifiers andvariables? Hereweseethatphilosophicalissuesariseattheveryoutsetofour model-theoreticinvestigations.Wealsointroducesecond-orderlogicanditsvarioussemantics.Whilesecond-orderlogicislesscommonlyemployedincontemporarymodeltheory,itisemployedfrequentlyinphilosophyofmodeltheory,and understandingthedifferencesbetweenitsvarioussemanticswillbeimportantin manysubsequentchapters.
InChapter2,weexaminevariousconcernsaboutthedeterminacyofreference andso,perhaps,thedeterminacyofourrepresentations.HereweencounterfamousargumentsfromBenacerrafandPutnam,whichweexplicateusingtheformalPush-ThroughConstruction.Sinceisomorphicstructuresareelementarily equivalent—thatis,theymakeexactlythesamesentencestrueandfalse—this threatenstheconclusionthatitisradicallyindeterminate,whichofmanyisomorphicstructuresaccuratelycaptureshowlanguagerepresentstheworld.
Now,onemightthinkthatthereferenceofourword‘cat’isconstrainedbythe causallinksbetweencatsandourusesofthatword.Fairenough.Butthereareno causallinksbetweenmathematicalobjectsandmathematicalwords.So,oncertain conceptionsofwhathumansarelike,wewillbeunabletoanswerthequestion: Howdowe(evenmanageto)refertoanyparticularmathematicalentity? Thatis,we willhavetoacceptthatwe donot refertoparticularmathematicalentities.
Whilstdiscussingtheseissues,weintroducePutnam’sfamous just-more-theory manoeuvre.Itisimportanttodothisbothclearlyandearly,sincemanyversionsof thisdialecticalmoveoccurinthephilosophicalliteratureonmodeltheory.Indeed, theyoccurespeciallyfrequentlyinPartBofthisbook.
Now,philosophershaveoftenlinkedthetopicofreferencetothetopicofrealism.Onewaytodrawtheconnectionisasfollows:Ifreferenceisradicallyindeterminate,thenmyword‘cabbage’andmyword‘cat’failtopickoutanythingdeterminately.SowhenIsaysomethinglike‘thereisacabbageandthereisacat’,I have atbest managedtosaythatthereareatleasttwodistinctobjects.Thatseems tofallfarshortofexpressinganyrealcommitmenttocatsandcabbagesthemselves.1 Inshort,radicalreferentialindeterminacythreatenstoundercutcertainkindsof realismaltogether.Butonlycertainkinds:wecloseChapter2bysuggestingthat someversionsofmathematicalplatonismcanlivewiththefactthatmathematical languageisradicallyreferentiallyindeterminatebyembracingasupervaluational semantics.
Concernsaboutreferentialindeterminacyalsofeatureindiscussionsaboutrealismwithinthephilosophyofscience.InChapter3,weexamineaparticularversion ofscientificrealismthatarisesbyconsideringRamseysentences.Roughly,these aresentenceswhereallthe‘theoreticalvocabulary’hasbeen‘existentiallyquantifiedaway’.Ramseysentencesseempromising,sincetheyseemtoincurakindof existentialcommitmenttotheoreticalentities,whichischaracteristicofrealism, whilstmakingroomforacertainlevelreferentialindeterminacy.WelookattherelationbetweenNewman’sobjectionandthePush-ThroughConstructionofChapter2,andbetweenRamseysentencesandvariousmodel-theoreticnotionsofconservation.Ultimately,bycombiningthePush-ThroughConstructionwiththese notionsofconservation,wearguethatthedialecticsurroundingNewman’sobjectionshouldtrackthedialecticofChapter2,surroundingPutnam’spermutationargumentinthephilosophyofmathematics.
ThenotionsofconservationweintroduceinChapter3arecrucialtoAbraham Robinson’sattempttousemodeltheorytosalvageLeibniz’snotionofan‘infinitesimal’.Infinitesimalsarequantitieswhoseabsolutevalueissmallerthanthatofany givenpositiverealnumber.Theywereanimportantpartofthehistoricalcalculus; theyfellfromgracewiththeriseof ε–δ notation;buttheyweregivenanewlease oflifewithinmodeltheoryviaRobinson’snon-standardanalysis.Thisisthetopic ofChapter4.Hereweintroducetheideaof compactness toprovethattheuseof infinitesimalsisconsistent.
RobinsonbelievedthatthisvindicatedtheviabilityoftheLeibnizianapproach tothecalculus.Againstthis,BoshasquestionedwhetherRobinson’snon-standard analysisisgenuinelyfaithfultoLeibniz’smathematicalpractice.InChapter4,we
1 Cf.Putnam(1977:491)andButton(2013:59–60).
offeranoveldefenceofRobinson.BybuildingvaluationsintoRobinson’smodel theory,weprovenewresultswhichallowustoapproximatemorecloselywhatwe knowabouttheLeibnizianconceptionofthestructureoftheinfinitesimals.Indeed,weshowthatRobinson’snon-standardanalysiscanrehabilitatevarioushistoricalmethodsforreasoningwithandaboutinfinitesimalsthathavefallenfarfrom fashion.
Thequestionremains,ofwhetherweshould believe ininfinitesimals.Leibnizhimselfwastemptedtotreathisinfinitesimalsas‘convenientfictions’;Robinsonexplicitlyregardedhisinfinitesimalsinthesameway;andtheirmethodofintroductioninmodeltheoryallowsforperhapsthecleanestpossibleversionofa fictionalist-cum-instrumentalistattitudetowards‘troublesome’entities.Indeed, wecanprovethatreasoning asif thereareinfinitesimalswillonlygenerateresults thatonecouldhaveobtained without thatassumption.Onecanhaveanti-realism, then,withaclearconscience.
InChapter5,wetakeastepbackfromthesespecificapplicationsofmodeltheory,todiscussamoremethodologicalquestionaboutthephilosophicalapplication ofmodeltheory: underwhatcircumstancesshouldwecalltwostructures‘thesame’? Thisquestioncanbeposedwithinmathematics,whereitsanswerwilldependupon thesimilaritiesanddifferencesthatmatterforthemathematicalpurposesathand. Butthequestioncanalsobegivenametaphysicalgloss.Inparticular,considera philosopherwhothinks(forexample)that:(a)thereisa single,abstract,entity whichis‘thenaturalnumberstructure’,andthat(b)thereisa single,abstractentitywhichis‘thestructureoftheintegers’;butthat(c)thesetwoentitiesaredistinct.Thenthisphilosophermustprovideanaccountofidentityanddistinctness between‘structures’,soconstrued;andweshowjusthowhardthisis.
Notionsofsamenessofstructurealsoinducenotionsofsamenessoftheory.Aftersurveyingawidevarietyofformalnotionsofsamenessofstructureandtheory, wediscussthreeambitiousclaimsconcerningwhatsamenessoftheorypreserves, namely:truth;arithmeticalprovability;andproof.Weconcludethatmorephilosophicallyambitiousversionsofthesepreservation-thesesgenerallyfail.
Thismeta-issueofsamenessofstructureandtheoryisagoodplacetoendPart A,though,bothbecause(a)thediscussionisenhancedbythespecificexamples ofstructuresandtheoriesdiscussedearlierinthetext,andbecause(b)questions aboutsamenessofstructureandtheoryinformanumberofthediscussionsand debateswhichwetreatinlaterPartsofthebook.
ReaderswhoonlywanttodipintoparticulartopicsofPartAcanconsultthe followingHassediagramofdependenciesbetweenthesectionsofPartA,whilst referringtothetableofcontents.Asection y dependsuponasection x iffthereis apathleadingdownwardsfrom x to y.So,areaderwhowantstogetstraighttothe discussionoffictionalismaboutinfinitesimalswillwanttoleapstraightto§4.7;but
theyshouldknowthatthissectionassumesapriorunderstandingof§§2.1,4.1,4.2, andmuch(butnotall)ofChapter1.(Weomitpurelytechnicalappendicesfrom thisdiagram.)
5.3 5.4 5.5 5.6 5.7 5.8 5.2
Logicsandlanguages
Modeltheorybeginsbyconsideringtherelationshipbetweenlanguagesandstructures.Thischapteroutlinesthemostbasicaspectsofthatrelationship.
Onepurposeofthechapterwillthereforebeimmediatelyclear:wewanttolay downsomefairlydry,technicalpreliminaries.Readerswithsomefamiliaritywith mathematicallogicshouldfeelfreetoskimthroughthesetechnicalities,asthereare nogreatsurprisesinstore.
Beforetheskimmingcommences,though,weshouldflagasecondpurposeof thischapter.Thereareatleastthreeratherdifferentapproachestothesemantics forformallanguages.Inastraightforwardsense,theseapproachesaretechnically equivalent.Mostbookssimplychooseoneofthemwithoutcomment.We,however,laydownallthreeapproachesanddiscusstheircomparativestrengthsand weaknesses.Doingthishighlightsthattherearephilosophicaldiscussionstobehad fromtheget-go.Moreover,byconsideringwhatisinvariantbetweenthedifferent approaches,wecanbetterdistinguishbetweenthemerelyidiosyncraticfeaturesof aparticularapproach,andthethingswhichreallymatter.
Onelastpoint,beforewegetgoing:traditiondemandsthatweissueacaveat. SinceTarskiandQuine,philosophershavebeencarefultoemphasisetheimportantdistinctionbetween using and mentioning words.Inphilosophicaltexts,that distinctionistypicallyflaggedwithvariouskindsofquotationmarks.Butwithin modeltheory,contextalmostalwaysdisambiguatesbetweenuseandmention. Moreover,includingtoomuchpunctuationmakesforuglytext.Withthisinmind, wefollowmodel-theoreticpracticeandavoidusingquotationmarksexceptwhen theywillbeespeciallyhelpful.
1.1Signaturesandstructures
Westartwiththeideathatformallanguagescanhaveprimitivevocabularies:
Definition1.1: A signature, L ,isasetofsymbols,ofthreebasickinds:constantsymbols,relationsymbols,andfunctionsymbols.Eachrelationsymbolandfunctionsymbol hasanassociatednumberof places (anaturalnumber),sothatonemayspeakofan n-placerelationorfunctionsymbol.
Throughoutthisbook,weusescriptfontsforsignatures.Constantsymbolsshould bethoughtofas names forentities,andwetendtouse c1, c2,etc.Relationsymbols, whicharealsoknownaspredicates,shouldbethoughtofaspickingout properties or relations.Atwo-placerelation,suchas xissmallerthany,mustbeassociated withatwo-placerelationsymbol.Wetendtouse R1,R2,etc.forrelationsymbols. Functionsymbolsshouldbethoughtofaspickingoutfunctionsand,again,they needanassociatednumberofplaces:thefunctionof multiplicationonthenatural numbers takestwonaturalnumbersasinputsandoutputsasinglenaturalnumber, sowemustassociatethatfunctionwithatwo-placefunctionsymbol.Wetendto use f1, f2,etc.forfunctionsymbols.
Theexamplesjustgiven—beingsmallerthan,and multiplicationonthenatural numbers—suggestthatwewilluseourformalvocabularytomakedeterminate claimsaboutcertainobjects,suchaspeopleornumbers.Tomakethisprecise, weintroducethenotionofan L -structure;thatis,astructurewhosesignatureis L .An L -structure, M,isanunderlyingdomain, M,togetherwithanassignment of L ’sconstantsymbolstoelementsof M,of L ’srelationsymbolstorelations onM,andof L ’sfunctionsymbolstofunctionsoverM.Wealwaysusecalligraphic fonts M,N,…forstructures,and M,N,…fortheirunderlyingdomains.Where s isany L -symbol,wesaythat sM istheobject,relationorfunction(asappropriate) assignedto s inthestructure M.Thisinformalexplanationofan L -structureis alwaysgivenaset-theoreticimplementation,leadingtothefollowingdefinition:
Definition1.2: An L -structure,M,consistsof:
• anon-emptyset,M,whichistheunderlyingdomainofM,
• anobjectcM ∈ Mforeachconstantsymbolcfrom L ,
• arelationRM ⊆ Mn foreachn-placerelationsymbolRfrom L ,and
• afunctionfM : Mn → Mforeachn-placefunctionsymbolffrom L .
Asisusualinsettheory, Mn isjustthesetof n-tuplesover M,i.e.:1
Mn ={(a1,…,an) : a1 ∈ M and…and an ∈ M}
Likewise,weimplementafunction g : Mn → M intermsofitssettheoreticgraph.Thatis, g willbeasubsetof Mn+1 suchthatif (x1,…,xn,y) and (x1,…,xn,z)areelementsof g theny = zandsuchthatforevery(x1,…,xn)in Mn thereis y in M suchthat (x1,…,xn,y) isin g.Butwecontinuetothinkaboutfunctionsinthenormalway,asmapssending n-tuplesofthedomain, Mn,toelements oftheco-domain, M,sotendtowrite (x1,…,xn,y)∈ g justas g(x1,…,xn)= y.
1 Thefulldefinitionof Xn isbyrecursion: X1 = X and Xn+1 = Xn × X,where A × B = {(a,b) : a ∈ A and b ∈ B}.Likewise,werecursivelydefineordered n-tuplesintermsoforderedpairsbysettinge.g. (a,b,c)=((a,b),c).
Giventheset-theoreticbackground, L -structuresareindividuated extensionally:theyareidenticalifftheyhaveexactlythesameunderlyingdomainandmake exactlythesameassignments.So,where M,N are L -structures, M = N iffboth M = N and sM = sN forall s from L .Toobtaindifferentstructures,then,wecan eitherchangethedomain,changetheinterpretationofsomesymbol(s),orboth. Structuresare,then,individuatedratherfinely,andindeedwewillseeinChapters 2and5thatthisindividuationistoofineformanypurposes.Butfornow,wecan simplyobservethattherearemany, many differentstructures,inthesenseofDefinition1.2.
1.2First-orderlogic:afirstlook
Weknowwhat(L -)structuresare.Tomovetotheideaofamodel,weneedtothink ofastructureasmakingcertainsentencestrueorfalse.Sowemustbuilduptothe notionofasentence.Westartwiththeirsyntax.
Syntaxforfirst-orderlogic
Initially,werestrictourattentionto first-ordersentences.Thesearethesentenceswe obtainbyaddingabasicstarter-packoflogicalsymbolstoasignature(inthesense ofDefinition1.1).Theselogicalsymbolsare:
• variables: u,v,w,x,y,z,withnumericalsubscriptsasnecessary
• theidentitysign: =
• aone-placesententialconnective: ¬
• two-placesententialconnectives: ∧, ∨
• quantifiers: ∃, ∀
• brackets: (, )
Wenowofferarecursivedefinitionofthesyntaxofourlanguage:2
Definition1.3: Thefollowing,andnothingelse,arefirst-order L -terms:
• anyvariable,andanyconstantsymbolcfrom L
2 Apedanticcommentisinorder.Thesymbols‘t1’and‘t2’arenotbeingusedhereasexpressionsin theobjectlanguage(i.e.first-orderlogicwithsignature L ).Rather,theyarebeingusedasexpressionsof themetalanguage,withinwhichwedescribethesyntaxoffirst-order L -termsand L -formulas.Similarly, thesymbol‘x’,asitoccursinthelastclauseofDefinition1.3,isnotbeingusedasanexpressionoftheobject language,butinthemetalanguage.Sothefinalclauseinthisdefinitionshouldbereadassayingsomething likethis. Foranyvariableandanyformula φ whichdoesnotalreadycontainaconcatenationofaquantifier followedbythatvariable,thefollowingconcatenationisaformula:aquantifier,followedbythatvariable, followedby φ.(Thereasonforthisclauseistoguaranteethate.g. ∃v∀vF(v) isnotaformula.)Wecould flagthismoreexplicitly,byusingadifferentfontformetalinguisticvariables(forexample).However,as withflaggingquotation,wethinktheadditionalprecisionisnotworththeugliness.
