Singular Propositions and Modal Logic - Philebus
University of Arkansas Press Singular Propositions and Modal Logic Author(s): Christopher Menzel Source: Philosophical Topics, Vol. 21, No. 2, Philosophy of Logic (FALL 1993), pp. 113-148 Published by: University of Arkansas Press Stable URL: http://www.jstor.org/stable/43154156 Accessed: 10-05-2015 16:00 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. University of Arkansas Press is collaborating with JSTOR to digitize, preserve and extend access to Philosophical Topics. http://www.jstor.org This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions TOPICS PHILOSOPHICAL 21 NO.2,SPRING VOL. 1993 Singular and Modal Propositions Logic ChristopherMenzel TexasA&M University THE RUSSELLIAN PUZZLE Accordingto theprevailingview in thephilosophyof languagenothing mediatessemantically betweena propername(in a speaker'smouth)and :; thereis ofa nameis determined thereference itsreference. Rather, directly A that referred to. thesis name than the ofa no moretothesemantics object of a sentence view is that the shared oftenaccompanies thiswidely meaning theproposition [9] itexpresses.If9 containsa name, cpis an abstract entity, theproposition expressedis said to be singular.Thisthesisin turnis often thesisthatpropositions, withthemetaphysical singular proposupplemented arestructurally sitionsin particular, ; thatis, roughly, (i) thatthey complex to rather thatcorresponds structure have,in somesense,an internal directly the that and the sentences that structure of thesyntactic expressthem, (ii) arethesemantic orconstituents , ofthatstructure metaphysical components, ofthose values themeanings ofthecorresponding components syntactic orill, for better related theses refer to these three Let us sentences.1 jointly, as Russelliansemantics. Russellian semanticshas an importantconsequence.Considera a name- thesentence'Quine is a philosopher',say. sentencecontaining it followsthatthesingularproposition If Russelliansemanticsis correct, - contains thissentence [Quineis a philosopher] expresses theproposition 113 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions corstructure inthatpartofitsmetaphysical as a constituent Quinehimself wholes in the realm the name to 'Quine'.Now,although physical responding theirconstituent seemtopresuppose do notgenerally parts(it'sstillthesame car despitethenew set of wipers),thesame does notseemto be trueof ofa Russellianproposition constituents theindividual Rather, propositions. butessentialtoit. tothenatureoftheproposition, seemnotat all incidental Itis hard,forexample,to see howtheproposition [Quineis a philosopher], ifRussellian,couldpossiblyhaveexisted,couldpossiblyhavebeenwhatit it fromthe is, sans Quine.(What,forinstance,wouldhave distinguished had neither is a existed?)Thus,Russellian [Russell philosopher] proposition are ontologically semanticsappearsto entailthatpropositions dependent that i.e.,moreexactly, upontheirconstituents; ofa proposition OD IfXis a constituent y y,then, necessarily, existsonlyifx does. ButOD raisesa puzzle.Intuitively, nothaveexisted; (1) Quinemight thatis,theproposition [Quinedoesnotexist]couldhavebeentrue.However, couldnothavebeentrue.For,as Adamsnotes,2 givenOD, thisproposition mustbe in orderto be true."But by OD therewouldhave "a proposition as [Quinedoes notexist]ifQuinehadn'texisted, beenno suchproposition therewouldhavebeen henceitwouldn'thavebeentheretobe true;rather, Sinceofcourse[Quinedoesnotexist] noinformation aboutQuinewhatever. also failsto be truewhenQuine does exist,it followsthat,necessarily, thathe doesn'tis nottrue.Thus, whether he existsor nottheproposition to ouriniis notreallypossible,andhence,contrary Quine'snonexistence tialintuition, (1) isn'ttrueafterall. Call thistheRussellianpuzzle.My goal inthispaperis todevelopsevviewof boththestructured tothispuzzlethatpreserve erallogicalsolutions - includingitsapparentconsequenceOD - and theintuition propositions inthewritings ofArthur oftheseis suggested that(1) is true.Thefirst Prior, of A simplemodification is theleastattractive. and,duetoitsawkwardness, thesemantical basisofPrior'ssolutionsuggested byRobertAdamsguides of the second,muchmoredesirable,solution.Finally,a generalization mostattractive leads to whatI findto be thesimplest, Adams'suggestion solutiontotheRussellianpuzzle. FRAMEWORK FOR A SOLUTION theRussellianpuzzleis to proThe first orderof businessin approaching one capable of expressingthe vide a properrepresentational framework, 114 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions and makingthedistinctions thatappearin thepuzzle. The propositions of is modal bothcounterfactual con, essentially puzzle, course, involving ditionalsandthenotionsofpossibility andnecessity. Assuming, reasonably, thatthecounterfactuals in thepuzzlearenecessarytruths iftrueat all,the canbe castintoanequivalent, somewhat morestilted form argument though thatreplacesthecounterfactual conditionals withstrict entailment. Wethereforeneed on thesegroundsadoptno morethana standardpropositional modalbase. Althoughquantifiers arenotexplicitly involvedin thepuzzle, does involvea distinguished existencepredicate'E!', and theargument to characterize itslogiccorrectly, we willwantto be able to clarify surely, itsconnection withtheexistential Hence,an accuraterepresenquantifier. tationof thepuzzle requiresthefullexpressivecapacitiesof a first-order modallanguage. Of coursewe cannotstopthere.For theheartof thepuzzleconcerns Russellianpropositions. More specifically, thepuzzlerequiresthatvfebe able to talkaboutsuchpropositions, and in particular, at least,thatwe be ablesensiblytoascribebothtruth andexistence tothem.Furthermore, since thepuzzlehas to do withtheontologicaldependency of singularpropositionson theirconstituents, we shallhave to have someway of indicating thata givenentity is a constituent of a givenproposition. The mostnatural of this last is to show as in way meeting requirement simply constituency theinformal of the the formation of presentation puzzlesimplybyallowing s out of the formulas of the forproposition denoting expressions language, mulascontaining namesin particular. One mightthenexpressontological dependency bymeansoftheaxiomschema[II(E!e =) E!t), wheret is a term thatoccurs(free)in s. in theinformal Propositiondenotingexpressionswere constructed above and sentences, something exposition bysimplybracketing alongthose lineswilldo here.However,bracketing ofsentences is notsufficiently genandrelationsas eral,foranalogousRussellianpuzzlesariseforproperties well.Justas therearesingular therearealso singular propositions, propertiesandrelations thatinvolveindividual forexample,thepropconstituents; fromQuine].Thus,itappearsthatOD generalizes to erty[beingdistinct OD* IfXis a constituent ofa property, orproposition relation, y, then, yexistsonlyifx does. necessarily, Butthenwe can arguemuchas above.Forinstance, intuitively, existed I wouldnonetheless havebeendis(1*) IfQuinehadn't tinct from him; thatis, I would nonethelesshave had theproperty [beingdistinctfrom As with a must in order tobe exemplified. be Quine]. propositions,property ButbyOD*, therewouldhavebeenno suchproperty as [beingdistinct from 115 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions Quine]ifQuinehadn'texisted,andhenceitwouldn'thavebeentheretobe Hence,ifQuinehadn'texisted,I wouldn'thavehadthepropexemplified. of fromhim,andhenceitappearsthat(1*) isn'ttrueafter erty beingdistinct a conclusion no less thantheone in ouroriginalpuzzle. all, discomfiting As withpropositions in theoriginalpuzzle,in thisversionwe ascribe existence toproperties. And,analogoustoourtalkintheoriginalofa proposition'sbeingtrue,herewe talkabouta property's Thus, beingexemplified. a completesolutionto theentireclass ofRussellianpuzzleswillrequireof us themeansfortalkingdirectly aboutproperties, and proposirelations, tions(PRPs) generally. Forthispurposewe willformalize ourinformal use ofthebracket notation abovetoconstruct a classofPRP-denoting or terms, wherecpis anyfirst-order intensional abstracts.Specifically, formulaand Vj,. . . , vnanyvariables,theexpression[Xvj... vn9] is a termthat,intudenotesthe«-placerelation itively, expressedbycp.Thus,whereq is Quine, the 1 ^ is fromQuine]. [Xjcèc q] (i.e., -placerelation)[beingdistinct property Whenn = 0, intensional abstracts denotepropositions. Thus,whereE! is ~E the is the existence, [X 'q] (or,suppressing X,[~E!#]), [Quine proposition doesn'texist].Its ontologicaldependenceon Quinecan thenbe expressed as D(E! [X~E!g] z> E !#),andthedependence of[beingdistinct fromQuine] on Quineby D(E![Xjcjc ï q] 3 E!#).Usingthisapparatus,then,we can as DE![Xvcp]z>E!t, wheret is any expressOD* forPRPs [Xv<p] generally in [Xvcp].3 constant orfreevariableoccurring tobeingabletoasserttheirexistence, As noted,inaddition we also need to be able to expressthatcertainpropositions aretrue,and,in thegeneral PRPsareexemplified. As withexistence, we couldjustadd case,thatcertain truth andexemplification to our But since we areonly predicates language. interested in exploringthelogic of theRussellianpuzzle,it wouldbe far ifwe couldavoidembroiling morepreferable ourselvesin theformidable of these logicalcomplexities predicates.4 Towardthisend,notethatPRPsplaytwometaphysical roles:an objectualroleanda predicative a role in of role, theyplay beingtrulypredicated otherobjects.In theirobjectualrole,PRPs canbe referred toandquantified overno less thananyotherkindofobject.Thisroleis reflected in ouruse of that-clauses, and other infinitives, gerunds, PRP-denoting expressions, andin thelegitimacy oversuchexpresof,e.g.,existentially generalizing sions,as in theinference from,say,'JohnbelievesthatBach nevercomfor to 'John believes . In theirpredicative rolethey posed guitar' something' arepredicated ofotherobjects(or,in thecase ofpropositions, theyare,so to say,predicated oftheworld),a rolereflected in ordinary Englishin our use ofverbphrasesanddeclarative sentences.In ourlogicallanguage,the former roleis ofcoursereflected in ourcomplexterms, andthelatterin the usualwayin thepredicates ofthelanguage.Now,withpossibleexceptions to lightby Russell'sparadox(whichwe shallstudiously brought ignore), 116 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions areexactlythoseofthecorresponding thetruth conditions fora predication if V is a predicateexpressing someproperty statement exemplification term 'A' a a p, ^[Xjccpf complex denoting /?, predi2-placeexemplification '"tit'and cate,andaTtheobjectdenotedbya termt, thenboththepredication are truejust in thecorresponding statement ^A(['jccp],T)^ exemplification in since the involvesonly case a7is theextension ofp. However, predication of the theproperty orrelation and the beingpredicated object(s) predication without themediationof theexemplification we can avoida sperelation, inexpressing thegeneralRussellianpuzzleif cial exemplification predicate ofcomplexPRPs in their we have a robust only sufficiently representation role. predicative , our However,as thingsstand,becausethereare onlycomplexterms in no direct of the has logicallanguage general way expressing predication of a givencomplexPRP of a givenobject;in particular, one cannotin any naturalwaypredicate theproperty distinct from Quine]ofme.What [being we need,ofcourse,is a representation oftheunnominalized verbphrase'is to the term x distinct fromQuine'corresponding complex ['x * q] thatrepWe couldofcourse resentsthenominalization distinct from Quine'. 'being tomeetthisneed,butthatwould introduce a classofnewcomplexpredicates be unnecessary. Forifwe acceptthatitis thesamePRP indicated bya verb indicated in different then there and its albeit roles, nominalization,5 phrase serve as both is no reasonnottoletourcomplexPRP-denoting expressions nomandthereby the roles of both termsandpredicates, play grammatical Call thisdualinalizedverbphrasesandtheirunnominalized counterparts. role syntax.Extendingthedutiesof complextermsin thisfashion,we it distinctness fromQuinebysimplypredicating myexemplifying represent as indicated, anexplicitexemofmethus:[kxx & q]c; andinthisfashion, difficulties areavoided.6 plification predicateanditsattendant Truth of as a speWhataboutthetruth is naturally thought predicate? An « + 1-placeexemplification cial case ofexemplification. predicate ' +l is trueofitsarguments t, tv . . . , Tnjustin case t denotesan «-placerelationR thatis trueoftheobjectsav . . . ,andenotedbytv . . . , Tn.In thecase wheren = 0, the 1-placeexemplification predicateAj takesonlya single termt as itsargument, andis trueoft justin case t denotesa 0-placerela- thatis truesimpliciter. tionP- i.e., a proposition By allowing«-place we avoidedtheneedforspecial terms to serve also as predicates, complex thesecondformof theRussellian exemplification predicatesto represent - suffices: The alone indicated puzzleabove;predication byconcatenation iff . . . . . t is true ... . , anexemplify n-placepredication av ^[Xvļ vn(p]jļ J . . . vncp]lTruthis just a special (standin) therelationR denotedby case. In thelimiting case wheren = 0, a 0-placetermstandingalonewill P itdenotes suffice: The0-placepredication ^[Xcp]lis trueifftheproposition thata certainproposition P is true,as is true.Thus,assertions to theeffect 117 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions in theoriginalformof thepuzzle,in ourlogicallanguagesimplytakethe form^[Xcp]!, where^[Xcp]! expressesP. We can thushaveall theexpressive we need for power examiningthelogic of theRussellianpuzzle without ourselves intheformidable difficulties oftruth andexemplificaembroiling tion.Forreadability, we introduce an eliminable however, "pseudo"truth T' in predicate suchthat7ļcp]=df[9], so thatthepuzzle thegarbof dualrolesyntaxcorresponds morecloselyto thelogicalformof thepuzzle as expressedin ordinary language. THELOGICAL STRUCTURE OFPRPS As notedabove,a sufficiently robustnotionof a singularproperty or that PRPs exhibit some sort of structural corproposition requires complexity moreor less to thegrammatical structure of thesentencesthat responding expressthem.We shall cash thisnotionhereby adoptingtheview that ofas being"builtup"logicallyfromlogically complexPRPsarebestthought less complexconstituents These bymeansof a groupoflogicalfunctions.7 can be into classconsistsoflogical operators grouped threeclasses.Thefirst to theclass of familiarsyntactic functions corresponding operatorsfrom modalpredicate the modal and boolean andthequantilogic;viz., operators theproperty fiers.So, forinstance, nonexistence is the [Xjc~Elx] negationof theproperty ofexistence;i.e., [Xjc~E!jc]= neg([Xjc we E!jc])(note areusing rather thanmentioning ourabstracts the here); property [Xjc0~E!jc]possible nonexistence is thepossibilization of [Xjc~E!jc]; therelation beingobjects Xandy suchthatxis richandy is unhappyis theconjunction ofthepropertiesbeingrichandbeingunhappy;i.e., [Xjcy Rx A~Hy] = conj([XjcRx], [Xy~//y])= conj([XjcRx],neg([Xy/fy]));andtheproperty beingmarried of(thesecondargument (to someone)is theexistential quantification place = exist2([Xjcy relation; i.e.,[XjcByMxy] of)themarriage Mxy]). in thesecondclass reflect certainpossiblerelations Logicalfunctions betweentheX-bound variablesin an abstract andoccurrences [Kvļ. . . vn<p] lack of in in a complete those variables (or thereof) cp.Thoughimportant these functions are irrelevant for our account, presentpurposes,and so we won'texaminethemany further. Not so our finalclass, thepredication functions. These are thefunctions thatgive risemostdirectly to singular These fall into two propositions. operators naturally camps: "simple" functions thatarereflected in abstracts like'[X[Xjc0~E!jc]#]'and predication a freeoccurrence ofany(toplevel) '[XjcBx[KSa]]' inwhichnotermcontains and inabstracts X-bound functions reflected variable; "complex"predication like'[XjczBx[' &]]' and '[Xjc[kyCyx'b' inwhichthereis sucha term. As it both and functions are instances of a single,general happens, simple complex of the type.For purposeshere,however,we can avoid thecomplications and think in their terms of brethren. only complexoperators simpler 118 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions In thesimplestcase, a predication function pred.takesa singleobject a and"plugs"itintotheIthargument orrelation, placeofa property yielding a PRP R withone fewerargument place,and witha as a newconstituent. inthecase ofa property theresultis a singular Thus,inparticular, proposition about a. So, for example,pred1 takes the propertyof possible - [Xjc0~E!;t]- and plugsQuine,say,intoitsfirst(and only) nonexistence thatQuine is possibly argument place, therebyyieldingtheproposition = [X[Xjt more 0~E !*]#].In nonexistent; i.e., formally, predj([X*O-Eîjt],#) a can involve an general,however, simplepredication «-placerelationand forexample,suchmulti-place anyof itsargument places; consider, predicationsas theproperty [XjcSaxb] of beingan x suchthata saysx to b. To this,letor= (iv . . . , im)be anyfinite, capture increasing sequenceofnumbers than 0. The is for such idea,then, that, any o' we definea predicagreater tionfunction suchthatforanyn ifR is an «-placerelation, and predCT . . . m then . . . is the of , , ) predication av amany objects, predCT(Ä, av am R ofav . . . , amrelativeto the/^, . . . , iJ* argument of R , respecplaces tively;thatis, to expressthisusingourlanguage,pred^/?,av . . . , am) = M.x. +. ,l . . . x. MX. ['x .i ... xn-m L ~J. Thus,inpartiJRx.l . . . x.iļ-ll/ļ lm~ m cular,pred^ 3)(5,a , b) = [kxSaxb]. Henceforth we willassumea variety ofaxiomsthatguarantee thefineof PRPs. It is rather tediousto lay thegroundwork forstating grainedness theseaxiomsprecisely. clarbe expressed withsufficient Theycan,however, forpurposeshererather ityandcompleteness easily.First,itis assumedthat P ^ Q, whereP is an «-placerelation, and Q is an m-placerelation, andn ^ m.Next,itis assumedthattheclassesofmodal,boolean,quantified, and are all PRPs where a modal PRP is one that is predicative pairwisedisjoint, in therangeofthepossibilization ornecessitation a boolean PRP function, is in therangeofone ofthebooleanfunctions, andso on.Thisaxiomguarthattheproposition antees,forinstance, [X[kx 0~E!jc]#]Quine is possibly fromtheproposinonexistent, thoughlogicallyequivalentto it,is distinct tion[X 0~E!g] Possibly,Quine does notexist.The latter is a modalpropoto sition,derivedby applyingthepossibilization operator theproposition thatQuinedoes notexist,whiletheformer is predicative, derivedas noted above.8The finalgroupof axiomscapturesthemostintuitive elementof that PRPs built from different constituents must viz., fine-grainedness; up themselves differ. Thus,fromtheseaxiomsitfollows,forexample,thatthe resultofpredicating theproperty beinga philosopherofQuine,[X Pq], is a different thantheone, [X Pg], thatresultsfrompredicating proposition thatproperty ofPeterGeach,say,orofpredicating theproperty existenceof [X Quine, E!#]. 119 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions THE PUZZLE REVISITED Now thatwe havean appropriate medium,letus use it to representational the the We will do so out ordinary languagerenbylaying express puzzle. the in its form of followingeach explicitly, dering puzzle propositional formal with its formalized Our languagehas no councounterpart. premise the subjunctive terfactual conditional=>, and hencewe can't represent all the are elementoftheargument since However, subjunctives precisely. z> to and since =» is D(9 i|>),9 i|/) logicallyequivalent '3(<p necessary, thelogicof entailment insteadwithout we can substitute affecting ordinary we let r[cp]=df[9]. Call the theargument. Also, as noted,forreadability, RP. following argument nothaveexisted. (1) Quinemight (Assumption.) (10 0~E'q. (2) theproposition [Quinedoesnotexist]is true. Possibly, (From(1).) (20 0r[~E!tf]. (3) ifQuinehadn't existed, [Quinedoesnot Necessarily, haveexisted. exist]wouldn't (ByOD.) (30 CK~mqz>~m[~B'qì). if[Quinedoesnotexist]hadn't it existed, Necessarily, wouldn't havebeentrue.(Assumption.) (4) (40 D(~mi~mq]z>~T[~E'q'). ifQuinehadn't existed, (5) Hence,necessarily, [Quine havebeentrue.(From(3) and doesnotexist]wouldn't (4).) (50 □(~E!^rz>~r[~E!#]). ifQuinedoesexist,[Quinedoesnotexist] (6) Necessarily, is nottrue. (Bylogic.) (60 n(E'qz>~T[~E'q]). ornot, whether (7) Hence,necessarily, Quinehadexisted [Quinedoesnotexist]wouldnothavebeentrue.(From (5a) and(6a).) (70 HQElqv ~E 'q 3 ~r[~E!tf]). either (8) Necessarily, Quineexistsorhedoesn't. (By logic.) (80 D(ß'q v ~E!tf). (9) that[Quinedoesnotexist]be true, Itis notpossible contradiction. (2).) (From(7) and(8),contradicting 120 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions (90 ~07[~E!tf]. itis falsethatQuinemight nothaveexisted. (10) Therefore, (1).) (From(2) and(9),discharge (100 THEUNDERLYING LOGIC Beforeexamining solutionsto thepuzzle,itwillbe useful prospective tomakeexplicitthecentral theargument. First, logicalprinciples underlying themovefrom(1) to(2), though is nottrivial. Itis based,ofcourse, intuitive, directionof the intuitiveequivalence0~E!<? = upon the left-to-right . Thisis 071-E!#],whichis an instanceofthegeneralschemaO9 = ÖTfXcp] a modalizedversionof theprinciple9 = T['q>] that,roughlya statement holdsif and onlyiftheproposition it expressesis true.This principlein - bearingin mindourstipulation turn that7[cp]is nothing otherthan[cp] : is justa specialcase ofk-conversion X-con: cpi= [kx <p]T, wherex is an «-tupleofpairwisedistinct variablesandt an«-tupleofterms. direction ofX-conappearsto be one ofthegenSo at leasttheleft-to-right erallogicalprinciples theargument. How thendo we getfrom the underlying ~E 3 to its modalized instance 'q T[~E'q] 0~E!#3 left-to-right counterpart ~E 3 The easiest route would seem to be this. From Ö7T-E!#]? 'q T[~E'q], assumingtheruleofNecessitation □I: h (p=> h Dep, we have D(~E!^ 3 r[~E!#]). Fromthebasic modalpropositional schema K: D(cp3 iļj)3 (Dtp3 Q|j), some propositional logic,and theusual principlethat□ and 0 are interdefinable, i.e., □/0: dep= ~0~9, we can derivetheschemaD(9 3 iļ/)3 (O9 3 Oiļi).Thus,pluggingin ~E 'q and 7ļ~E!#] for9 and iļ/,respectively, we have Q~E!#3 Ô71-E!#],as desired. Now alreadyat thisearlystageofthegamedifficulties beginto loom. It is, presumably, a logicaltruth thatEbe = 3;y(y= x). Hence,because Vjc3y(y = x) is a theorem of classicalquantification theory(CQT) with DI andD/0 identity, E!#is as well.So inthecontextofCQT withidentity, alone are enoughto yield(100- Exploringthisproblemnow will getus ahead of ourselves.So forthemomentwe will simplyassumesomereasonablefix,e.g.,switching fromCQT to a freequantification (i.e.,a theory thatallowsnondenoting termsandhencedoesnothave quantification theory E!jcas a theorem). 121 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions ofontological Next,itwouldbe welltomaketheprinciple dependence - explicit;in fact,let us statea slightlymore of which(3) is an instance of a PRP are generalprinciplethatcapturestheidea thattheconstituents forits existenceas well as individually also jointlysufficient necessary. letE!t abbreviate Wheret = Tj . . . Tmis a nonrepeating sequenceofterms, in questionis: E!tj A ... A EÎTm;thentheprinciple T contains all andonlythenoncomEx: E!['x (p]= E!t, where andprimitive variables, (i.e.,constants, prediplexterms cates)thatoccurfreein['x cp]. is clearly direction theright-to-left bytheproof, Thoughnotstrictly required andhencewillhavetoholdin a coma partoftheunderlying metaphysics, we neednotexplicitly pletelogicforit.Notealso that,givenNecessitation, is calledEx becauseitis a reasonably Theprinciple modalizetheprinciple. ofourlanguage)ofthe boundaries (withintheexpressive precisestatement .10 orworse)as existentialism doctrine thathascometobe known(forbetter as theview thattheonly Existentialism is sometimescharacterized - i.e., arethosethatareeither PRPs thatexistnecessarily purelyqualitative individuals thosethatcountnoconcrete amongtheirconstituents roughly, all existnecessarily. themselves constituents orwhoseconcrete Thoughnot - i.e., in theformexpressedby entailedby existentialism strictly proper andexistencearegenerSinceidentity Ex- theviewis a naturalcorrelate. tocapturethisviewinourlogicwe must allytakentobe purelyqualitative, matterswe will To simplify expresstheirnecessaryexistenceexplicitly. E !, withtheproperty existence, [Xjc3y(x = j)] ofbeingidentical identify withsomething. Thenwe needonlyadd theaxiom E!=: E!=. relationnowfollowsfromE!= and The necessary existenceoftheidentity K.11 from and and that of existence Ex, DI, □I, let us the (4) a littlemoreexplicassumption Finally, unpack important cannotbe truewiththat a of the is an instance proposition itly.(4) principle outexisting: (11) 7Tcp]=)E![cp], or more generallystill,the principleused in the generalformof the without Russellianpuzzlethata PRP cannotbe exemplified existing(where is justforittobe true): tobe exemplified fora proposition (12) tttidE!TT, whereit is any«-placepredicateandT a sequenceofn terms.Intuitively, hascalledseriousactuofwhatPlantinga (12) expressesa classofinstances or standin a relation alism, theviewthatan objectcannothavea property without existing: 122 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions SA: 7tt=>E!T. fromPtx.. . tnitfollowsthatP standsintheexemplification For,intuitively, relationA withtv . . . , tn, APtv . . . ,řn, (orwheren = 0, thattthasthepropand hence that of and E!P, bySA. However,becausetruth erty beingtrue), in our PRPs are not equivalanguage,nothing expressible exemplification fromSA. GivenEx, howlentto either(11) or (12) followsin thismanner followfromthegeneralizedseriousactualism ever,all threestatements principle GSA: 9 3 E![9],where 9 is atomic;12 cannotholdunlessitexpressesan existing an assertion i.e.,roughly, propooftheexpressedproposisition,andhencebyEx unlesseveryconstituent of(12) we haveas a theorem a succinctexpression tionexists.In particular, andSA: (13) ittz) (E!ttAE!T). Notealso thatbyGSA, Ex, andX-conwe havein general (14) 9 d E![<p]foranycp. ForbyX-conwe have9 z> [cp],andso since[9] is a 0-aryatomicformula, of Ex in bothdirecwe have [9] z> E![[9]] by GSA. Severalapplications tionsandsomepropositional logicyieldthedesiredresult. I willtakeGSA tobe thegeneralmetaphysical (4). underlying principle AN ACTUALIST ACCOUNT OF POSSIBLE WORLDS In oursearchfora solutionto theRussellianPuzzle,itwillbe illuminating notionofa possibleworld.Possibleworldsare to makeuse ofthefamiliar forourdisanditwillbe important sortofentity, ofcoursea controversial cussion that whateveraccountof worlds we adopt be actualistically FamiliaraccountsfromthelikesofAdams,Plantinga, Pollock, respectable. 1find orstatesofaffairs.13 andotherstakeworldstobe (setsof)propositions andso willuse insteada moremodel-theoretic theseaccountsproblematic, constructions accountofpossibleworldsthattakesthemtobe set-theoretic have been. as the structure of things theymight exhibiting As a firstcutat thisaccount,considera setD = u{D p D0, Dp . . .}, whereD j is a setofindividuals (i.e.,non-PRPs)and,forn > 0, Dnis a setof existence E! in Dr Let extbe an extension relations that includes w-place of themembers ofD0 intotruth andfalsity, function thatmapsthemembers > memfor n into of the members of 0 into subsets of and D, ^-tuples Dw Dj inparticular thatext(E') = D. Thensaythatthepair bersofD. We stipulate 123 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions S = (D, ext)is a stateofaffairs ofthePRPs justincase theactualextensions inD, restricted toD, areexactlyas extdepictsthem,andsaythatS is a possiblestateofaffairs i.e., justincase itis possiblethatS be a stateofaffairs; in case it is ofD exist(together) andthat just possiblethatall themembers extcorrectly of thePRPs in D. Notethatsinceit is depictstheextensions constructed outofactuallyexisting S objects,everypossiblestateofaffairs even if it not even if extenis ext does not the actual exists, actual;i.e., depict sionsofthePRPs in S. We assumein additionthatpossiblestatesofaffairs exhibit ourunderlying Russellianmetaphysics withrespecttothePRPsthey containas embodiedin theprinciple so that an n-placePRP P e Dniff Ex, all itsconstituents arein D. Giventhis*notealso thatsinceextmapsPRPs intoD, eachpossiblestateofaffairs is onein whichthegeneralized serious actualismprincipleGSA holds.Finally,say thatS is a possibleworld, or worldstructure, itis possible justincase,amongthepossiblestatesofaffairs, thatS be thelargest . Thatis, letA be theclass ofpossiblestatesof affairs; thenS is a possibleworldjustincase,possibly, andfor S is a stateofaffairs = e S' if is a state of then D/ for all z'.14 S' ç D¿ affairs, (D', exť) A, any A limitation of thisconstruction arisesfromthefactthat,intuitively, therecouldhavebeenthingsotherthanthethingsthatactuallyexist;John Paul II, forexample,underquitedifferent circumstances, mighthavehada What state of affairs this grandson. possible Though represents possibility? we willnotexplorethisissueindetailinthispaper,thecrucialidea is toset aside someclass of actualistically acceptableobjects- puresets,say- to serveas surrogates forsuch"possibilia"as JohnPaul IPs grandson. These, in turn,can serveas constituents of "surrogate PRPs"; thatis, PRPs conthattherefore serveas surrogates taining surrogates amongtheirconstituents forPRPs thatwouldexistiftheindividuals bythosesurrogates represented existed.Thus,iftheemptyset0 wereto be a surrogate fora grandsonof thentheproposition JohnPaul II in somepossiblestateofaffairs, [0 is sitwill the that would exist ifthat ting] represent corresponding proposition were to A exist.15 state of is a S affairs, then, pair = (D, grandson possible such that it for there is to be some ext) possible mappingmfromthesurroin D into the set of such ext objects16 that,underthatmapping, gates existing D the extensions in of the PRPs note thereof; (orsurrogates correctly depicts thatonourtype-free thePRPsforma subsetofD). Thefollowing picture figureillustrates theidea in a simplefashion.WhereS = (D, ext),thepicture indicatesa setDj ofproperties anda setD j consisting of fiveindividuals, threeofwhicharesurrogates, A of 0. including portion extis indicatedby thelinesfromthreeproperties in thesetofPRPs to subsetsofD r M indicatesthesetofindividuals thatwouldhaveexistedwereJohnPaulII tohave hada grandchild, andmis themappingtherewouldhavebeenfromD { into mmapsJohnPaul II tohimselfand0 to hisgrandchild. M; in particular, 124 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions n- ( ' P ' Q ' R ^ J cit(Py' / ' V 'ral(Q) n- f( * ■ )} 1 vvWfeTy »<mJ V m(0)1 ,„(■> A possibleworld,then,is a possiblestateofaffairs thatis possiblythelargest understateofaffairs inthesensedefined above(exceptwith'stateofaffairs' ofcourse).The important stoodas in thecurrent pointtonoteis paragraph, to playthe thatby adoptingan actuallyexistingmodel-theoretic structure roletypically accordedto possibleobjectsandpossibleworlds,we can use ourselvesto an theoftenhelpfullanguageofpossibiliawithout committing ungainly possibilistmetaphysics. SOLUTION 1: PRIORE AN INTERN ALISM The first solutionto thepuzzleI wantto consideris suggested bythework While ofArthur Prior.17 Prior'ssolutionis, at firstsightanyway,extreme: ifwe take thelogicunderlying theRussellianpuzzleis flawed,nonetheless, itsconclutheimplications ofthenatureofsingular seriously, propositions sionis stillunavoidable. As Priornotes: 125 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions 4Itcouldbethatcp'thatitistrueifandonlyif Ifwesointerpret could then is something that [9] couldbetrue, mynon-existence that couldbetrue.18 notbe,since[I donotexist]... isnota thing withtheunderlying therearetworelatedproblems So, forPrior, logicofthe Russellianpuzzle.The firstis thatfromthetrueconclusionthatQuine's nonexistence is impossible,it followsby D/0 thathis existenceis necesofcourse,appliestoevery sary, thatheis boundtoexist.A similarargument, andthat,Priornotes,"makesgodsof existing thing, ostensibly contingently ina modallogic an inappropriate us all."19D/0 is therefore logicalprinciple forcontingent beings. The problemsdon'tstopthere,however.Considernextthefollowing Fromthepropositional tautology argument. (15) E!#v ~E'q itfollowsfrom'-con thattheproposition expressedis true, (16) T'E'q v ~E 'q]è As an instanceof(11), we have (17) 71E!?v~E!tf]z>E![E!tf v~E!tf], byEx we have (18) E![E!#v ~E!g]z>E!g, andhencebypropositional logic (19) T'E'q v ~E!#]z>E!#, andK we have andso byNecessitation (20) 'I'T[E'q v ~E 'q] z>'jE'q. onceagain,from(16), we also have By Necessitation (21) an^lq v ~E 'ql fallsoutonceagain: thenecessity ofQuine'sexistence andsobyModusPonens, (22) DE 'q. is here.Since(15) is a tautology, whattheculprit Itis nomystery (16) should in anylogic withtermsdenotingthepropositions be a theorem expressed by its sentencesand some meansof expressingtruth;forif a sentence of logicthatthe how couldit notalso be a truth expressesa logicaltruth, so expressedis true?20 That,in the0-placecase, is just what proposition itis a truth oflogic,Priorargues,(15) '-con guarantees. However,although thatwouldhavebeentrueno matis nota necessary truth ; i.e.,a proposition it appearsthatthatcouldbe terwhat.Forby ourunderlying metaphysics, so onlyifitwerea necessary being,andhenceonlyifall ofitsconstituents, inregardto a As Priorhimself remarks werenecessary. Quineinparticular, 126 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions "ifit is necessarythatifI am a logicianthenI am a similarlogicaltruth, in theguiseof DI, is logician,it is necessarythatI am."21Necessitation, therefore no moreappropriate thanD/0. : In termsoftheapparaPrioris thuswhatwe mightterman internalist of a proposition tusabove,theonlybasis forevaluatingthetruth Q with = intheworld;i.e., respecttoa possibleworldw (Dw,extw)is itstruth-value thevalueofextw on Q. Becauseextw is defined onlyonthePRPs inw,itfolcan be evaluatedas true(orfalse)onlywithrespect lowsthata proposition to thoseworldsin whichitexists.Let us makethismoreexplicitbydrawbetweena proposition's ingouttheconnections logicalformanditsextension.For Prior,in accordancewithGSA, thefollowing principlegoverns theatomiccase: = T, PI extw([Paļ . . . tfj)= T justincase(i) n = 0 andextw(P) or(ii) n > 0 and(av...,an) e extw(P). Thatis,foran atomicproposition [Pax. . . an' (= [P] whenn = 0) tobe consideredtruein w,av . . . , anmuststandin therelationP (or,in the0-place of extw , theargucase, P mustbe true)in w, and hence,by thedefinition mentsto therelationPav . . . , an (hence,byEx, P itself)mustall existin workas expected. w. The cases ofconjunctive andquantified propositions The negatedandmodalcases areworthconsidering overtly: = T justincaseextw([ P2 ejciw([~<p]) 9]) = F. Giventhedefinition ofextw , onceagainitfollowsthata negatedproposition mustbothexistin a givenworldandbe suchthatitsunnegated counterpart is falsetherein orderforit to be truein theworld;thisis crucialto the Prioreanpicture. we needclausesforeachofthem: Since□ and0 arenotinterdefinable = T justincaseextw,([<p]) = T forsomepossible P3 £xiw([0<p]) worldw'' = T justincase¿*/^([9])= T forallpossible P4 ex^flOcp]) worlds w'. Giventhedefinition ofextwitis clearthat,inaccordancewithPI, thepropositionthatQuinedoes notexist,[~E!#],is nottruein anyworld,andhence that[~Q~E!g] is true.But,as expected,[DE!#]is nottrue,since[~E!#]does notexistin anyworldlackingQuine,andthusD/0 fails.Andsince[E'q v ~E!g] does notexistinanysuchworldas well,DI failsas well.P1-P4 thus forpropositions in Prior'sview. conditions appearto capturethetruth If,as just noted,theconclusionof RP is soundforPrior,whatis an abouthowthe forit?Thebestwaytoansweris tothink acceptableargument logicaboveneedstobe revisedin lightofthefailureofD/0 andDI. Letus notefirst thatitis onlyone direction ofED/0thatbreaksdown,theimplicationfrom~0~9 toCkp;as we'vejustseen,fromthefactthattheproposition 127 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions [9] couldnotpossiblyhavebeenfalseitdoesn'tfollowthatitwouldthereforehavebeennecessary; thatexist i.e.,countedamongthetruepropositions ineveryworld.However,equallysurely, if[9] is necessary, ifitwouldhave beenamongthetruepropositions no matter what,thenitcouldnotpossibly havebeenfalse.~0~cpthusexpressesa sortofweaknecessity, impliedby, butnotimplying, itsfull-blooded To simplify notation, then,let counterpart. us introduce a weaknecessity operator explicitly: DefH: Hep =df~0~cp, andnaildownitsrelationtoitsstronger counterpart: Qp =>ftp. Whatabouttheotherdirection? As justnoted,whatprevents a weaklynecv like from essaryproposition [E 'q ~E!g] beingstrongly necessaryis the factthatitis nota necessary It is it being. (looselyspeaking)truewhenever - that'sjustwhatitis to be weaklynecessary - butitis notthecase exists thatit wouldhavebeentrueno matter what.Thus,fora proposition to be it true whenever must be both it and furthermore exists, strongly necessary suchthatitneverfailsto exist;i.e., a littlemoreprecisely, itmustbe both and existent: weaklynecessary necessarily U/m2: BcpADE![(p]3 D(p. Bothconditions can be combinedintothesingleequivalence □/■: Qp = HepA DE![cp]. The reasoningbehind□/■ also determinesthe fate of Necessitation: Because of contingently we are warranted existinglogicaltruths, onlyin that the weak necessitation of an truth is a inferring arbitrary logical logical truth. Hence,in ourPrioreansystem,DI is simplyreplacedby its weak counterpart: ■I: l-<p=>l-H<p. To infermore,we have to knowmoreabouttheontologicalstatusof the [<p],as capturedpreciselyin thederivedrule(from□/■ and proposition ■d DRD: h(p=» hDE![cp]z) dep; ofa necessarily is itselfa logical i.e.,thenecessitation existing logicaltruth truth. Thisraisesthequestionofhowone provesthata givenproposition P existsnecessarily. this is or Generally, accomplished byshowing assuming thatall ofitsconstituents existnecessarily. Thus,one appealsto (therightto-left direction of) a modalizedversionofEx: □Ex: DE![Xx <p]s DE!t, 128 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions whereT is as in Ex. QEx is clearlytruein ourunderlying metaphysics. itis no However,withstrongnecessitation replacedbyweaknecessitation, longerpossibletoproveitfromEx. Hence,IHExtooneedstobe addedas a in itsownright. separatelogicalprinciple Finally,thoughK remainstrueinthisPriorean logic,itis oflimiteduse due toitsapplicability Hence,we onlyto necessarily existing propositions. needa corresponding to governweaknecessity. We cannot,howprinciple "If ever,simplyreplace■ with□. Consider,forexample,theconditional is human,"Hq 3 3xHx.Thisproposition Quineis human,thensomething is weaklynecessaryforPrior:In termsoftheusualjargon,in everyworld in whichit exists,theproposition thatif Quine is human,something is humanis true.Butnowsupposewe add H(cp 3 iļ/)3 (Bcp3 Bv)/)as an axiom.Thenit followsthatMHq 3 ■ 3xHx.It is indeedplausiblethat thatQuineis essentially human.However, itis falsethat MHq; i.e.,ineffect, MBxHx;itsurelycouldhavebeenthattherewereno humansat all. Wherethingsgo wrongis that,unlikethecase ofK, [i(/]canexistwithout [cp],and in suchcircumstances itstruthis notguaranteed. So whatis neededis a simplequalification thatrulessuchcircumstances out: 3 (Hep3 ■(E!t 3 iļ/)), where t contains allthe ■K: H((p3 iļ/) terms thatoccurfreein<pbutnotiļ/. noncomplex ofMK tobe ofthesameformas K, only (WhereT is null,we takeinstances with■ replacing□.) Returning to ourexample,all thatfollowsnowfrom M{Hq 3 3xHx)andMHq is M(E 'q 3 3xHx),andthatis unproblematic. Forinanysituation inwhichQuineexistshe is human(byassumption), and so in anysituation in whichhe existssomething is human,as required.It shouldbe notedthattheoriginalprinciple K nowfollowsfromMK andthe otherprinciples above. theRussellianpuzzle,ifwe simplychangeevery□ to Now,regarding ■, RP is in factvalidin ourPrioreanlogic.However,we neednotgo to suchlengths. Foran advantageofthelogic,withitsweakerbrandofnecesis thatwe arefreetoreturn toCQT. Thus,we caninfer sitation, Quine'sexisandhencebyMl theimpossibility ofhisnonexistence, tence,E!<?,directly, his necessaryexistence,DE!#, in accordancewith ■E!#,butnotthereby Prior'sinitialconception. ifnotparticular Now,thisis perhapsa serviceable, comely,quantified modallogic,anditcertainly behindRP appearsto capturethemetaphysics withgreatprecision.However,something is stillamiss.Forsurelythereis a senseinwhich(1) is true;surelyitis possibleinsomesensethatQuinefail to exist.AndindeedPrioragrees:22 . . . there is a senseofThis might nothaveexisted' inwhich whatitsayscouldbethecase(andgenerally is),i.e.,thesense: 'Itis notthecasethat(itis necessary that(xexists))'~DE!x. 129 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions To accountfortheintuitive truth of (1), then,corresponding to theweak in Prior effect introduces a weak ■, necessity operator possibility operator ♦: Def#: ♦(p=df whilenotstrongly therefore, Quine'snonexistence, possible,is nevertheless that(1) is true.Priorthus weaklypossible,andthataccountsforourintuition ofsingular appearstobe abletohaveitbothways:The metaphysics propositionsdictatesthat(1) mustbe false;semantical intuitions tellus itis true. The twosensesofpossibility appearto letus hangon to boththedemands ofmetaphysics andtheappealsofintuition. But notso. Considera logicallyfalsesingularproposition; that,say, Quinebothis andis nota logician,[Lq A ~Lq ]. Since [Lq A ~Lq] contains it is notitselfa necessarybeing.Hence, Quine amongits constituents, falsewhenever itexists,theproposition failstobe falseinsituations though in whichitdoes notexist,andhenceitis notnecessarily false.It is thusa theorem ofourPrioreanlogicthat-□ -{Lq A ~Lq)' i.e.,itis a theorem that thecontradiction in questionis weaklypossible,♦(£#A ~ Lq).23Thus,in thislogic,Quine'snonexistence is possiblein precisely thesamesensethat hisbeingbotha logicianand a nonlogician is possible.But surelytheformeris trueinsomesensethatthelatter is not;surely, insomesense,Quine's nonexistence is a waythingscouldhave been and his simultaneous existenceand nonexistence is not.Prior'slogic,however,is unableto distinthat(1) is true.24 Can we guishthem,andhencefailstoexplainourintuition do better andstillremainwithintheboundsofactualism? SOLUTION 2: ADAMS' PERSPECTIVALISM Perhapswe can. In his seminalarticle"ActualismandThisness,"Robert Adamssuggestsan actualistunderstanding ofmodalpropositions thatdoes notgenerally cashtheirtruth conditions intermsofwhatpropositions could or musthavebeentruein Prior'ssense.In particular, on Adams'view,the of his nonexistence is nota matter ofthetruth oftheproposition possibility [Adamsdoes notexist]withinsomepossibleworld.He writes: ... I deny. . . that ^Itis possible that thatthe p^ alwaysimplies couldhavebeentrue. haveoften that-/? proposition Philosophers found itnatural tocharacterize andnecessities in possibilities terms ofwhatpropositions wouldhavebeentrueinsomeorall situations Thisseemsharmless solongas it possible enough isassumed that allpropositions arenecessary Butitismisbeings. if(asI hold)somepropositions existonlycontingently.25 leading 130 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions The properapproachto understanding thetruthof modal propositions, Adamssuggests,involvesa metaphor ofperspective.Interestingly, Prior himselfsuggeststheidea in thefollowing passage: Thereare,then, nopossiblestatesofaffairs inwhichitis the casethat~E!jc, andyetnotallpossible states ofaffairs areones inwhichE'x. Forthere arepossiblestatesofaffairs inwhich there arenofactsaboutx atall;andI don'tmeanonesinwhich itis thecasethatthere arenotfactsaboutx,butonessuchthat itisn'tthecaseinthem thatthere arefactsaboutx. Adamsexpressestheidea thus: A [possible nosingular about world]thatincludes proposition meconstitutes anddescribes a possible inwhich I would world notexist.Itrepresents notbyincludnon-existence, mypossible I donotexistbutsimply that me. ingtheproposition byomitting ThatI wouldnotexistifallthepropositions itincludes, andno other actualpropositions, weretrueis nota factinternal tothe worldthatitdescribes, butanobservation thatwemakefrom ourvantage ofthat pointintheactualworldabouttherelation toanindividual intheactualworld.26 world-story Let w be a worldlackingtheproposition thatQuineexists.The idea,then, is that,eventhoughtheproposition [~E!#]is notpartofw (oranyworldfor thatmatter), fromourperspective intheactualworldwe cansee nonetheless, thatitsayssomething trueaboutw; i.e.,abouthowthingswouldhavebeen hadw beenactual.As Adamsputsit,whileitis nottrueinw, itis nonethelesstrueat w.Itis invirtueofthis- not,perimpossibile ofitspos, invirtue and not simplyin virtueof its siblybeingamongthetruepropositions, - thatwe complement failingtobe necessarily amongthetruepropositions takethemodalproposition can,anddo, intuitively [0~E!#]to be truesimplicités In a nutshell,then,Adams retainsan underlying ontologyof Russellianpropositions RussellianPRPs) alongwithEx, (ormoregenerally, and henceretainsPrior'simplicitconceptionof otherpossiblestatesof affairs.UnlikePrior,however,we determine thetruth-values of modal themat worldsrather thaninworlds. propositions byevaluating To getfullyclearaboutwhatis goingonhere,letus capture therelevant conditionsfortruth-at implicitinAdams'approacha littlemoreformally anda littlemoregenerally. Let w = (Dw, extj be a possibleworld.Adams followsPriorinaccepting echoes SA fortheatomiccase,andso hisprinciple Prior's: = T, Al [Pax. . . an]istrue atwjustincase(i) n = 0 andextw(P) or(ii) n > 0 and(av . . . , an) e extw(P). notethatAl is toholdevenifP is a complexproperty or Now,importantly, relationliketheproperty ofbeinga non-fish: 131 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions that isnota fish; I amsomething means that ... 'I ama non-fish' IfI didnot ofbeinga non-fish. itascribes tometheproperty thatis I be something I havethatproperty? exist, Might might atall,andwouldhaveno nota fish?No,I wouldbe nothing false counted isappropriately Hence'lama non-fish' properties. I donotexist.27 inwhich inworlds at worldsw in whichhe doesn't However,to denyAdams'non-fishiness of ['x -Fx] in w andhenceto he is in the extension that to exist,i.e., deny thattheatomicproposition affirm [['x ~Fx]a] is falseat w,is notto deny thatAdamsis nota fishatthoseworlds;thatis to say,while[[X* ~Fx]a] is falseat w, thenegatedproposition[-Fa] is trueat w, and indeedat any world:Itis truein,henceat,thoseworldsinwhichAdamsexists(assuming andat thoseworldsinwhichhefailstoexistsimheis essentially non-fishy) he is not Either then, amongthefishythings.Moregenerally, way pliciter. is nottrueatw. A2 [~i|i]is trueatwjustincase[iļ/] at a worldgivesus a senseoftruth UnlikePrior,then,theconceptoftruth can turnoutto be truewithrespectto the on whichnegatedpropositions worldsinwhichtheydon'texist.Thisis thesenseinwhichsuchpropositions be countedpossible.Thatis, moregenerally: can nonetheless inparticular) atw(theactualworld, istrue A3 [Ov|i] justincase w' suchthat[iļj]is trueatw' there is a world oftheform itfollowsthata proposition FromA2 andA3 inparticular is trueatw iff[0~i|;]is nottrueatw iffitis notthecase thatthereis a world is trueat w' iffitis notthecase thatthereis a worldw' w' suchthat suchthat[i|;]is nottrueat w'; i.e.,iff[i|>]is trueat all worldsw'. Since [i'f] neednotin generalexistata worldtobe trueatit,itfollowsthata propositioncan be trueat all worldswithout beingtruein all worlds.In particular, of propositional logic are trueat all worlds.Consider,for singulartruths [Lq v ~Lq] thatQuineis eithera logicianornota example,theproposition is trueat a givenworldwjustin case either[Lq] logician.Thisproposition is trueatw or[-Lq] is. SupposeQuineexistsinw.Thenboth[Lq] and[-Lq] existthereas well,andobviouslyone or theotheris true;i.e.,thepropositionis weaklynecessary. However,supposeQuinedoesn'texistin w. Then at all [-Lq] is stilltrueat w, since[Lq] is not.Takingnecessityto be truth worlds,then,itfollowsthat[{Lq v -Lq)] is necessary;i.e.,theproposition neceshavetheoriginalscopeofstrong [UiLq v -Lq)] is true.Wetherefore to ouroriginal to us intact,and this,in turn,signalsa return sityreturned DI (and hencealso suitable modalprinciplesD/0 and fullnecessitation to CQT, forreasonsnotedabove). modifications We can now see clearlywhatlies behindAdams' denialthat"^ltis ^ couldhavebeen that-/? possiblethatp alwaysimpliesthattheproposition thatas a specialcase ofAl , whereP is a proposition true."Notefirst [iļi],we 132 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions = T. This havethattheproposition [[i|>]]is trueat wjustin case extw([''t]) whenwe use ourpseudo-truth readsa bitmoreintuitively predicate:[7ļiļ/]] = T ; i.e.,theproposition is trueat wjustin case extw([''t]) that[ iļi]is trueis itselftrueat w just in case theproposition [iļj]is truein w. Eitherway,the to [i|i]ingeneral) important pointtonoticeis that[[i(/]](incontradistinction is theproposition andhence,likeall expressedby a (0-place)predication, fortheseriousactualist, thecomponents of [[iļi]],hence[[vļj]] predications itself,mustexistin a worldw in orderforittobe trueatw. thatQuinehas essentially; So considernow anyproperty existenceis theobviousexample.ThenbecausethereareworldsinwhichQuinedoesn't exist,theprinciplesabove entailthatit is possiblethatQuinefailto exist, 0~E!#;i.e.,theproposition [0~E!g] thatthisformula expressesis true(i.e., trueat theactualworld):[0~E!g] is trueiff[~E!#]is trueat someworldw (byA3) iff[E!#]is nottrueat someworldw (byA2) iffthereis a worldw suchthatq č extw( theclaim E!), as we've supposed.Considerbycontrast 0[~E!#]thattheproposition [~E!#]couldhavebeentrue.Thisclaimis true ifftheproposition [0[~E!#]]itexpressesis trueiffthereis a worldw such thattheatomicproposition [[~E!g]] is trueatw (byA3) iffextw([~E'q])= T (byAl); i.e.,ifftheproposition [~E!#]is trueinw, andhenceonlyifq e Dw = extw(E'),byourconstruction. Thisofcoursecan'tbe,sincew is a possible stateof affairs. So whileit is possiblethatAdamsfailto exist,0~E!g, the [~E!g] thathe does notcouldnotbe true;i.e., ~0[~E!#].Thus, proposition totighten thetruth above,onthisapproachwe distinguish upAdams'remark ofa proposition totheeffect thata givenproposition P is possiblefromthe ofP's beingtrue. possibility The argument andrelations: We can showin generalizesto properties thesamefashionthat,e.g.,eventhoughitis possiblethatQuinefailtoexist, thathe 0~E!g, it does notfollowthatit is possiblethathe be nonexistent, ofbeingnonexistent, theproperty 0['x ~E!jc]q. Forthisis so iff exemplify theproposition is [0[Xjc~E!jt]g]thatitis possiblethatQuinebe nonexistent trueiffthereis a worldw suchthattheatomicproposition [[Xjc~E'x]q] that is trueat w iffq e extw(['x~E!jc]) andhenceonlyif Quineis nonexistent = g q Dw extjfi^ onceagain. The generallogicallessonhereis thatthelaw of'-conversionappears tobreakdownin modalcontexts. Thatis, in particular, whereasboth (23) [0~E'q] = 0~E'q and (24) [Xjc0-ELx]q = 0~E'q instancesof appear to be valid (as theyought,being straightforward ^-conversion in whichthemodaloperators no essential both role), play (25) 0[~E'q' = 0~E'q 133 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions and (26) 0[Xjc~E'x]q = 0~E'q arefalse,thedifference beingthatbothsidesof (23) and (24) requireonly that[~E!#]be trueat someworld(given0~E!g ), whiletheleftsidesof(25) and (26) requireitto be truein someworld.Currently, however,thelatter twoareprovablein ourrevisedsystemas it stands:In thecase of (25), for andpropositional instance, byX-conversion logicwe have (27) ~[~E 'ql m E 'q% andso byDI we have (28) □(-[-£!*] = E 'q) whichcreatesthevexatiousmodalcontext, and(25) followsbyK, D/0, and contraposition. The latterthreeprinciplesare unimpeachable in thecurrent context, so theproblemappearsto lie witheitherX-conversion or necessitation. thantheprinX-conversion, however,seemsscarcelymorecontroversial In above: in the case used in what else could ciples (27), particular, 0-place itbe forcptoholdthanfortheproposition itexpressestobe true?In fact,we shallaffirm thevalidity ofX-conversion, rather thanan thoughas a theorem withnecessitation. axiom;theproblemlies moredirectly To getatthenatureoftheproblemanditssolution, letus return to the of in modal The is that E!x recall, CQT problem quantified logic. problem, is a theorem ofCQT (letting E'x =df3y(y = jc)),and so bynecessitation it followsthatŒ'x is a theorem as well; i.e., it becomesa truth of logic thateverything is a necessary A common around this being. way difficulty, as noted,is tomovetoa freequantification inparticular, toreplace theory;28 theusualuniversal instantiation axiom z) (pí,foranyterm UI: Vxcp t thatis freeforjcincp withitsfreecounterpart FUI: Vjtxp t thatis freeforx in9. d(E!td 9Í), foranyterm FromFUI, together withthereflexivity axiomforidentity Id: jc = jc, itis possibleonlytoproveEbe =) 3y (y = jc),i.e.,E!xd E!jc,notE!jc simand so insteadof necessitarianism we deriveonlytheinnocuous pliciter, 3 □(Ebe E!*); i.e., it is trueat everyworldw thatifx existsin w, then x existsin w; i.e., it is trueat everyworldw thateitherx existsthereorx doesn'texistthere;i.e.,that[E!jc]e w or [E !jc]g w. So in thisrespectFUI appearstobejustwhatwe need.However,thissolutionis in one important behindfreelogic is to have a logical wayunlovely.The chiefmotivation 134 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions ofnondenoting terms. Thatis systemcapableofdealingwiththepossibility all well and but as a solution to the of (perhaps) good; problem CQT in modallogicitfoistsundesirable us. baggageupon FollowingPrior,we are an actualist modal beings, and hencewe constructing logic of contingent wouldliketo have at ourdisposalat leastthepossibility of signalingthis factbyhavingE'x falloutas a theorem. thus us FUI gives therightresult in modalcontexts, butis undulyrestrictive in nonmodalcontexts. in thiscase at least,we can haveourcake andeat it too. Fortunately, ConsiderId once again.Ifwe take'= ' tobe a predicate thatdenotesa fullrelation I we then we do not want itto follow (as, think, ought), fledged - that□(* = jc); i.e., thattheproposition as it does givennecessitation true,foranyx. Forotherwise, [x = jc]is necessarily by(ii) inAl (andthe definition of extw),it followsthatx existsin everyworld.Equallyclearly, we don'twantto lose Id as a logicaltruth; 'x = jc'is trueunder however, So this a natural restriction onnecessitation; viz., anyinterpretation. suggests □I': h cp=> h □ cp,so longas cpprovable without instance of Id. any This of coursepreventstheinference fromId to itsnecessitation. At the withSA, Id yieldsE'x and hence,in consame time,however,together junctionwithFUI, fullUI. Butsincea proofofE'x requiresId, itsnecessitationis notprovable.Thus,by a well-motivated (foractualists, anyway) we avoid thenecessitarian restriction on necessitation problemsof CQT without it. abandoning We can framea preciselyanalogoussolutionto theproblemof Xthattheusualaxiomcanbe brokendownintotwo conversion. Considerfirst conditionals: 'R: [Kx <p]y=><Py and cp?z>[Ä.X<p]y, wherex andy arenonrepeating sequencesx{ . . . xnand . . . yn, respectively, andforall i < n,y.is freeforx. in cp.Applying necessitation to instances of'R corresponding to (25) and(26), we have (29) D([~E 'q] ZD~E 'q) and (30) D([Xjc ~E!jt'q => ~E!$), whichareclearlyunproblematic: (29) is trueifftheproposition [[~E!#] =) ~E 'q ] is trueateveryworldw iffeither[[~E!#]]is falseatw or[~E!#]is true at w iffextw([~E'q])= F or [E!#]is nottrueat w iffextw('E'q')= T orq ë q g extJE') or q £ extw(El).(30) followsin extw(El)iff(by construction) 135 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions instancesof muchthesameway.Notso therelevant viz., (31) D(~E 'q 3 [~E'q') and (32) D(~E!^ =) ['x~E'x]q). doesnotfolFromthefactthat[~E!g] is trueata givenworldw, itcertainly itcannotfollow)thateither low (andindeed,when'E!' denotesexistence, [[~E!#]]is trueatw; i.e.,thatextJ'~E'q') = T, orthat[[Xjc ~E!jc]#]is true atw; i.e.,thatq e extw([kx-E!*]). Forbothwouldrequireq's existencein The problemin general,ofcourse,is thata negated w byourconstruction. atomicproposition cfs existing [~Pq] can be trueat a worldeitherthrough inw butfailingtoexemplify P , inwhichcase both[[~P#]]and[[Kx ~Px'q] aretrueatw as well,orthrough q's failingtoexistthere,in whichcase neitherproposition is trueat w. of analoWhattheseobservations suggest,then,is a qualification of UI thatyieldsFUI: An instanceof thecondigous to thequalification referred to tionalX^Lholdsin a givenworldw so longas all theindividuals in theantecedent existinw. Thatis, we replace with Tcontains where thevariables cpjd(E!td [Xxcp]y), y¿and freein<py, andwhere allother terms occurring noncomplex foralli < n,y. is freeforx. incp. □r cannowbe appliedunproblematically toVRL,butbecauseoftherestricilkwillno tionon itsgeneralapplicability, (31), (32), andtheirtroublesome - thoughnot fullX-conversion longerfollow.ButbecauseE'x is a theorem, - is provablefrom and itsnecessitation XLR ''RL, as desired. This thenis thekeyto thesolutionof theRussellianpuzzle on our reconstruction ofAdams'approach.29 As notedabove,themovefrom(1) to ofnecessitation to~E!g 3 [~E!g] toyield(31) theapplication (2) requires of~E'q 3 [~E!#]inthe andthence0~E!g Z) 0[~E!#].Butthededuction above logic requiresVRL,whichrequiresE!#,whichin turnrequiresId. is blocked. Thus,CU'cannotbe invokedto yield(31), andso theargument We getonlytheinnocuous, indeedpropositionally trivial, [H(~E!g =) (E!g z> [~E !<?])). SOLUTION 3: FULL PERSPECTIVALISM TRUTHGENERALIZING AT,I: RELATIONS itis still Preferable as thislogicis tothePriorean logicabove,however, To see this,fora givennotionofextension toomuchinthelatter' s clutches. P ispositivewithrespecttoa given ofa proposition e, saythattheextension worldjust in case e(P , w) = T. Whattheconceptof truth-at givesus, in 136 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions on whicha proposicontrast to thePrioreanview,is a notionof extension tionhas a positiveextension withrespecttoworldsinwhichitdoesn'texist. as we've seen,givesus a muchmorepalatablequanUsingthisdistinction, tifiedmodallogicthatnonetheless remainsfaithful to theintuitions behind arelimiting cases of theRussellianviewof propositions. Butpropositions forrelations. andtruthvalueslimiting casesofextensions n-placerelations, we can the above definition: For notion ofextenThus, anygiven generalize R is positiveat a worldwjustin sione, theextensionofan n-placerelation Thenon ourreconcase e(R, w) = T ifn = 0, and e(R, w) ž 0 otherwise. can havea struction ofAdams' approachas it stands,a relationgenerally in which it doesn't exist if it is a proposiextension at a world only positive restriction. For thenotionof truth-at tion.But thisis an unwarranted by whichpositiveextensions are assignedto propositions at worldsin which to a notionof holding-at forntheydon'texistgeneralizesstraightaway a in which Adams for all n. For consider world w instance, place relations, existsbutQuinedoesnot.Justas itmakesgoodsenseto saythatthepropofromQuineis trueatw,itis equally sition[Xa ž q] thatAdamsis distinct fromQuine the jc sensibleto say that property [Xjc ^ q] of beingdistinct holdsofAdamsat w as well. Thisobservation requiresthatwe definea generalnotionofa relation's theextension functions extension atworldsthatcomports with,butextends, to haveextensions usedto defineworldsso as to allowrelationsgenerally = atworldsinwhichtheydon'texist.So let*Rw >0D.andlet<R= KJWG W^RW whereWis theclassofall possibleworlds;% thatis,is theclassof"allpossible" PRPs (and hence, given our definitionof worlds,will include PRPs"involving "surrogate surrogate possibiliaamongtheirconstituents).30 be a function thatmapstheeleThenfora givenworldw,we letexit 3 extw ifR = [Xjc~<p], extensions in Dw.In particular, mentsof*Rintoappropriate = theproperty thenextw(R [Xjcx ï q] andlet extwiVvc ) cp]).So consider wbe theworldjustnotedinwhichAdamsexistsandQuinedoesnot.Thenwe have extw([kxx ž q]) = T)w-extt{['x x = q]) = Dw- { b e w I (b, q) e x = y])} = D^- { b e w I (b, q) e extj[hxyx = y])} = Dw- 0 = exíÜ;([ÁJ9> x ï q]). Adamsg extwiV^x Dw.So inparticular PRPs are neededexplicitlybecausetheabove Note that"surrogate" thatforcesus toconsider theextenina manner canbe generalized argument Consideragaina sionsof PRPs thatwouldexistifthingsweredifferent. worldw in whichJohnPaul II- though,God forbid,notas pope- has a andhence worldu in whichthatgrandson, andconsideranother grandson, theproperty fromhim,does notexist,butAdamsdoesofbeingdistinct theactualworlditselfwilldo. Then"lookingat" bothw andu we can see fromJohnPaulII's ofw,Adamsis no lessdistinct that,fromtheperspective 137 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions as fromQuine;i.e.,figuratively, fromourperspective onceagain grandson andassignan extension to we can "project"ourselvesintow's perspective theproperty ofbeingdistinct fromJohnPaulIPs grandson atu thatincludes tocomeouttrueevenifwe Adams.Moregenerally, we wantthefollowing believetherecouldhavebeenobjectsotherthanthosethathappenactually to exist: (33) DVx(a * Xz>□((~E!jcAE!û) d [Kyx * y]a)). ourrepresentation ofthemodal Thus,inthepossibleworldsthatconstitute oftheproperties therewouldbe ifthings were facts,we needrepresentations as indicatedby modallyand quantificationally embeddedterms different, like6[Xyx* yYin (33). affects thewaywe evaluate Thisgeneralization oftruth-at significantly withrespectto a givenpossibleworld.The keydifferatomicpropositions encebetweentruth-in andtruth-at as definedaboveshowedup in theevaluationofnegatedpropositions: UnlikeP2,A2- thata negatedproposition [~i|/]is trueat wjustin case [vļi]is nottrueat w- permitted negatedsingularpropositions like [~E!#]to be truewithrespectto worldsin whichthey - viz.,exit- thatallowsrelations do notexist.Givena notionofextension withrespecttoworldsinwhichtheydon'texist, tobe exemplified generally we mustnowmodify Al, which,beingdefinedin termsof extw, prevents this.Accordingly all we needto do is replaceextw inAl withexit AI* [Paļ . . . an]is trueatwjustincase(i) n = 0 andext*(P)= T, or(ii)n> 0 and(av . . . , an)e ext*(P). A2 andA3 thenremainas before.If we nowthinkin termsofext„ rather thanextw thegeneralized seriousactualismprinciple GS A no longerholds; moreexactly, we can no longerinfer[E'R] fromthetruth of [Rax. . . an]at a givenworld.However,sinceext„ stillmapsextensions intoD^, serious actualismstillholds.Thatis, whilewe lose (12), tttz> E!tt,we retainSA, 7tT3 E!t. Furthermore, we havetorelaxtherestriction on X-conversion in allow for true at worlds in which the relato accordingly predications tionpredicated does notexist;so now,itappears,we haveinstead: =) (E!y=>[Xx<p]y), where forall i < n, y.is freeforx. '£l: cp* in(p. in thatthedenotations of However,thiswon'tquitedo. The condition incp*existintheworldofevaluationguartermsoccurring all noncomplex relationin thatworld.Our anteedthat'[Xx<p]'wouldindicatea legitimate observation thatthemetaphor ofperspective enablesus toassignextensions inworldsinwhichtheydon'tthemselves existcausedus torelax torelations thisrestriction in this relaxation thattheembedHowever, presupposes fromsomeperspective, a "possible" dedterm'[Xxcp]'willalwaysrepresent, 138 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions relation;i.e., a relationthatexistsin someworld.Thereis no longerany itseemsthattherecouldbe "incomthatthiswillbe so. Intuitively, guarantee itseemstobe possiblethattherebe some Moreexactly, possible"objects.31 x objectX andpossibletherebe someotherobjecty suchthat,necessarily, existsonlyify doesn'tand(hence)viceversa;i.e., (34) Q3x03y[J(E'xz)~E'y). Thenthereis no worldw inwhichbothobjectsexist,andhenceitis notposas thefolbothobjectsas constituents, siblethattherebe a PRP containing a theorem of our is logic: lowing (35) DVjOtyQE!* 3 ~Ely) 3 D~E!['z 9]), V and cy'32Ifwe nowcounta predicawhere9 is anyformula containing - as falseatworldsinwhich 7r°,inparticular tionitt- a 0-placepredication certaininstancesof it has no denotation, yieldinvalidconsequences. Consider,forexample, (36) (E'x 3 ~E!y)3 [E!jc3 ~E 'yl □I andK yield (37) D(E!jc3 ~E 'y)3 D[E'x 3 ~E 'y], fromwhichwe derive (38) DVjOVyQE!* 3 ~E 'y)3 U[E'x 3 ~E'y]) and DI. But iftherecouldbe incompossible via universalgeneralization if is then true, i.e., (38) is false,as therecouldnotbe a propo(34) objects, sitionof theform[E!jc3 ~E!j] involvingthemas constituents. Thinking ofpossibleworlds,if'ť and y intermsofourrepresentation semantically andmodalopervalueswhenwe unpackthequantifiers takeincompossible for'[Ebe3 ~E!;y]' no worldconatorsin (38), thenthereis no denotation ofthe involvingthosevalues,and so theantecedent taininga proposition false. conditional wouldbe trueandtheconsequent of Now we couldtake(36)-(38) as a logicalproofoftheimpossibility decide itself shouldn't but that seems untoward; logic objects, incompossible into condition suchissues.A moreseemlysolutionis tobuildan additional GivenEx, thefollowofincompossibles. to accountforthepossibility ingwilldo: whereforall i < n, (p£3 ((E!yA 0E!['x 9]) 3 ['x cp]y), y.is freeforx. in9; then i.e.,if9y holds,thenifall they.existand ['x 9] is a possiblerelation, [Xx 9]y.33Since 0E!['x 9] is provable,(36) now followsas a theorem. - in particular, theproof However,šincetheproofof0E!['x 9] in general of(36) is notprovable of0E![E!x 3 ~E!y|- requiresId,34thenecessitation 139 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions in general,and so theabove proofof (38) is prevented. Of course,if one desirestoruleoutincompossibles so so to say,anyn possible explicitly that, are then becomes Id, and objects compossible, 0E!['x <p] provablewithout theaddedcondition in becomes simply superfluous. WhatthenofRP in thisrevisedlogic?Notethatas a limiting case of we have (39) ~E 'q 3 (0E![~E'q' 3 [~E'q]' whichbypropositional logic,DI', andK yields (40) D0E![~E!^] z>D(~E!^ Z>[~E!#]). But0>E![~E!#] is provablefromId, SA, theaxiomE!=, Ex, T, andtheS5 axiom0E![~E!#]z> D0E![~E!^]. Hence,unlikeon ourearlierpicture, (31) n(~E'qz>[~E'q]) andthus (41) Q~E'qz>0[~E'q] are provable,and so themovefrom(1) to (2) is now valid.This is as it shouldbe onourgeneralized view,sinceitis nolongerrequired perspectivai thata PRP existin a worldtohavea positiveextension there.However,the premise (40 □(~E![~E!tf]3~7T~E!<?]) in thepuzzlenowfails,sinceit requiresthediscredited (12), specifically, theinstance7ļ~E!#] z> E![~E!#]; i.e., [~E!g] 3 E![~E!g]. So thelogic remainssafefromRP. II: EXTENSIONS GENERALIZING TRUTH-AT, The movefromPI first toAl thentoAl has provideda broadersense oftruth forpropositions withrespecttoa givenpossibleworldthatpreserves a muchmorestandard modallogicwithout thebasic quantified abandoning Russellianmetaphysics. The modifications thatled to Al and Al* were drivenbythemetaphor ofperspective, whichprovidedan intuitive underof positiveextensionsto PRPs withrespectto pinningto theassignment worldsin whichtheydon'texistwithout violatingactualistscruples.Given thatmetaphor, itseemswe cango further still.Forifwe do indeed however, have something like a perspective on otherworlds,can we not,fromour in theactualworld,considera property's extensionat a given perspective worldto encompassnotjust objectsin thatworld,butin otherworldsas well?Consider, onceagain,a worldw inwhichQuinedoes notexist.Then, fromourperspective, itseemsquitepossibleto considerQuinetohavethe nonexistenceat- thoughnotin,to be sure- w. As theexample property tonegative relations. There shows,onceagainthepointappliesmostdirectly twowaysin whichI canbe includedin theextension ofa are,inparticular, 140 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions [Kx ~<p]at a worldw: I can existin w andbe amongthe negativeproperty that have [Xjc<p],orI can simplyfailtoexistinw, andhencefromthe things the perspective of actualworldcanbe countedamongthethingsthatexemplify[Xjc~Pjc]at w.35 To incorporate thisidea intoourapparatus ofpossibleworldsrequires onlythatwe allow extt to map n-placerelationsintothenthCartesian of "possibleindividuals". productof the entiredomainT> = in continue to be evaluated at worlds accordancewiththeprinPropositions Al in the are First,sinceobjects ciples ,A2, andA3. Changes logic minimal. canbe intheextensions atworldsinwhichtheydon'texist,SA ofrelations nowfails.Andsecond,we candropthecondition on 'y' inA.^ thatthedenoin the world of tationsofall theyf. exist evaluation;thus,we havesimply where foralli < n, y.is free (p£z>(0E!['x 9] =>[kxcp]y) forX.in(p. RP remainsblockedas above,as (12) is stillinvalid.However,without SA, itis no longerpossibletoprovesuchdesirabletheorems as (31). Thiscanbe remediedbysimplyaddingtherelevantspecialcase ofSA: term t SA=: T = T d E!t,foranynoncomplex totheeffect thatanything intheidentity relation withitselfexists.36 standing All proofsintheprevioussystem on the Id/SA combination nowgo relying as before. through just NotewellthattoabandontheaxiomSA is nottoabandonseriousactualismas a metaphysical anymorethanabandoningPrior'sII principle, '~ for DI' is to abandon themetaphysics ofRussellianproposi=> (l-cp Hep) aretheresultant oftwovectors:ourmetaphysics tions.Ourlogicalprinciples we use to evaluate(thepropositions and thesemanticprinciples expressed sentences of our Russellianmetaby) logical language.Our underlying in thedefinition of a possibleworld(Dw, extw)insofar physicsis reflected as everypossiblestateofaffairs mustsatisfy theprinciple Ex. Ml reflects modalsemantics. botha commitment to themetaphysics and an internalist themovetodi' reflected As is evidentinthediscussion ofAdams'approach, the no changein thebasic metaphysics; thedefinition of worldsremained itremained thecase thatall andonlythoseproposame,and,in particular, sitionsexistwhoseconstituents alsoexist.Rather, itreflected onlytheswitch to a perspectivai a changein thewaypropositions areevaluated semantics, withrespectto worlds,as indicatedby theswitchfromthesemanticprinciplesP1-P4 toA1-A3. of SA (and the Preciselythesamepointappliesto ourabandonment a in our semanticprinmovefrom to X.^). Bothreflect only change we are abletoderive not in our Given the metaphysics, ciples, metaphysics. on a broadernotionofexemplification intermsofthenotionofperspective, Al*.37 thebasisofwhichwe arriveat ournewsemanticprinciple 141 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions Of course,another wayto abandonSA is to abandonthemetaphysics the and relationsin a givenworldto takevalues by allowing properties outsidethatworld;i.e., in termsof our apparatus,to allow extwto take andrelations toextensions thatincludeobjectsthatdon'texistin properties w. Butwhynot?We defineextļ in sucha manner, whynotextji How do we showourselvesmorecommitted toactualismbyrestricting therangeof and then not without such restriction? exit defining Why justallow any extw the itself to take values at other worlds from outset? extw The reasonforthisis thateach worldstructure w = (Dw, extj in our framework in itselfrepresents means of surrogates) a way (in general,by All the would have been if that world had been actual: things objectsthere wouldhave been- theroleof Dw- and theway thatthoseobjectswould - therole of ex An actualistwill therefore have been configured not tw. includeanything in Dwthatis notin theextension ofexistenceE! (thisis a condition onworldstructures, willnotinclude recall),anda seriousactualist but such the extensions of and anything existingthingsamong properties relations. since to be in the extension of a or relation is to (Indeed, property be in somesense,it is hardto see how an actualistcouldnotbe a serious itprovidesa fixed, actualist.)Oncewe aregivenan arrayofsuchstructures, of the modal facts in terms of whichto cash the objectiverepresentation notionofperspective, andthereby the semantic foundations forour develop alternative modallogics.Again,though, as byrepresenting things theycould havebeenwithactuallyexistingobjects,we capturethemodalfactswithoutanyontological commitment and toindividuals, propositions, properties, relations thatdon'texistinfactbutwouldhaveexistedhadthingsbeenotherwise.In thiswayactualism, seriousactualism,and theRussellianmetaof andpropositions areall preserved relations, physics singular properties, a robust modal alongside quantified logic. APPENDIX: THE LOGICS In thisappendixwe assemblethefouractualistlogicsin one place. Let E! = jc)], and let x and y be arbitrary abbreviate[Xjc nonrepeating sequencesofvariablesofanylengthn > 0. WhereT = Tj . . . Tmis a nonwe writeE!t to abbreviate repeating sequenceofterms, E!tj A . . . A EÎTm. THEPRIOREAN SYSTEMQ The following modallogic,foris Prior'sfullsystemQ of quantified mulatedin termsofourricherlanguage.Let M<p=df~0~cp. Taut: Propositional tautologies. PK: ■ (9 =>iļj)3 (ŪE!t =>(■ cpz>■ iļ/)), T contains all where theterms freein<pbutnoti|>. occurring 142 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions PT: P5: cpz>O9. O93 MO(p. □/■: D<ps lep A DE! [9]. if* is notfreein9. Qu: Vx((p3 iļj)3 (9 3 Vjciļi), t is freefor* in(p. 3 cpi,where UI: Vxxp where GSA: 9 3 E![cp], 9 is atomic. = all andonlythenonEx: E!['x 9] E!t, whereT contains thanthex.thatoccurfreein9. terms other complex T is as inEx. □Ex: DE!['x 9] = DE!t, where □E!=: DE!=. x = x. Id: and9' isjustlike9 x = y3 (9 3 9'), where 9 is atomic, ofx oneormore(free)occurrences exceptthatyreplaces in9, andwhere yis freeforx in9. forall i < n,y. is freeforx. in9. '-con: 9* = [Xx 9] y,where LL: axiomsforPRPs.38 PRP: Fine-grainedness RulesofInference ■I: ^ <p=> Iq m<p. PMP: Iq<P,ÍqIP3>|i=>^I|>. PGen: ^ 9 => ^ '/xņ. THESYSTEMAl It is easiestto definethissystemandtheonesbelowbasedonAdams' We start approachbymeansoftwologics,one a freesublogicoftheother. ofthe called which consists a sublogicofbothsystems, first SI, bydefining following. Taut: Propositional tautologies. K: T: 5: FUI: Qui: 3 (Ū9 3 Q (/). D(9 3 vļi) Ū9 3 9. O93 [HO9. t thatis freeforx in9. Vx93 (E!t 3 9?),foranyterm Vx(93 iļi)3 (V*93 x is notfreein9. Qu2: 9 3 V*9,where GSA: 9 3 E! [9],where 9 is atomic. all andonlythenonEx: E!['x 9] = E!t, whereT contains that terms other than the complex xř occurfreein9. 143 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions E!=: NI: Ind: A^: E!=. T = t'z)D(E!T=) T = T'). and9' isjustlike t = t' d (cpd (pO,where 9 is atomic, of oneormore(free)occurrences that t' replaces 9 except t in9, andt' is freefort in9. forall i < n,y.is freeforx. in9. ['x 9] y z>9* where Tcontains thevariables yřand X'RL: 9Jz>(E!t z>[Xx9]y),where freein9J,andwhere terms allother occurring noncomplex foralli < n,y.is freeforx. in9. axiomsforPRPs. PRP: Fine-grainedness Let Gl be thesystemthatresultsfromaddingto SI thefollowingaxioms andrulesofinference: GId: 'fx(x = x). term t. 0E!t,foranynoncomplex GMP: ^ļ9» »GÍ9=> ^GI^- OB!: GGen: ^9 =>^*9. GNec: ^9 ^I^DcpBy Al we willmeanthesystemthatresultsfromaddingto SI thefollowingaxiomsandrulesofinference: t. term t = t,foranynoncomplex Id: Gen: => inGl is Nec: Iqi9 => ^Ckp. (N.B.:Anything provable inAl.) necessary THESYSTEMA2 Let S2 be thesystemthatresultsfromSI whenwe replaceGSA with SA: ttis anypredicate, tp . . . , inany iTTļ. . . Tnz>E!t.,where and1 < i < n, terms, is replacedwith andX.'RL < n, '™L: 9jz>((E!y A0E![Xx9])z>[Xx9]y),whereforall/ y.is freeforxiin9. thatresultsfromaddingtoS2 theaxiomsandrulesof Let G2 be thesystem inference GId, 0E!, GMP, GGen, and GNec above,exceptwiththesub'Gl' script replacedby 'G2' . thatresultsbyaddingtoS2 theaxioms A2 By we willmeanthesystem 'Gl' and andrulesofinference Nec,exceptwiththesubscript Id, MP, Gen, 'A2' 'Al' and the subscripts replacedby replacedby 'G2' 144 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions THESYSTEMA3 Let S3 be thesystemthatresultsfromS2 whenwe replaceSA with term t. SA=: T = tdE!t, foranynoncomplex and with X*L: z>(OE![Xxcp]=) [Xx(p]y), where foralli < n,y.isfree <p* forX.in(p. Let G3 be thesystemthatresultsfromaddingtoS3 theaxiomsandrulesof inference GId, OE!, GMP, GGen, and GNec, exceptwiththesubscript 'Gl' replacedby»G3' By A3 we willmeanthesystemthatresultsbyaddingtoS3 theaxioms 'Gl ' andrulesofinference Id, MP, Gen,andNec,exceptwiththesubscript 4 39,40 'Al' replacedby A3'. replacedby 'G3' andthesubscript NOTES andH.Wettstein, inJ.Almog, J.Perry, 1.See,e.g.,D. Kaplan, eds., "Demonstratives," Themes Oxford Press, 1989);N.Salmon, (NewYork: Frege's University from Kaplan "Direct Mass.:MITPress/Bradford Puzzle Books, 1986);andS. Soames, (Cambridge, inN.Salmon andS.Soames, andSemantic Attitudes, Content," Reference, Prepositional Oxford 197-239. andAttitudes Press, 1988), eds.,Propositions (NewYork: University 49(1981):18. 2. R.Adams, "Actualism andThisness," Synthese ofterms metavariables toindicate I willoften useboldface 3. Asillustrated here, sequences ofthecorresponding type. in inR.L.Martin, "Truth andParadox," 4. See,for A.Gupta, ed.,Recent Essays example, H. Oxford Truth andtheLiarParadox 175-235; Press, 1984), (NewYork: University andR. inR.L. Martin, "Notes onNaiveSemantics," op.cit.,133-174; Herzberger, 455-472. "ATheory ofProperties," Journal Turner, ofSymbolic Logic52(1987): I find their andR.Turner, 5. Noteveryone G.Chierchia arguments though agrees, notably andPhilosophy andProperty seetheir "Semantics Linguistics Theory," paper unpersuasive; "TheProper Treatment of seeC. Menzel, 11(1988):261-302, andfora response, inJ.Tomberlin, Intensional Predication inFine-Grained ed.,Philosophical Logic," Calif.: andLogic(Atascadero, Vol.7:Language Press, 1993). Ridgeview Perspectives, atlength. value and ofsuch a language where I for the 6. Cf.Menzel, propriety argue op.cit., E. Zalta, Oxford 7. SeeG.Bealer, andConcept Press, 1982); (Oxford, University Quality and"AComplete C.Menzel, Abstract Reidel, 1983); (Dordrecht: Typeop.cit., Objects no.CSLICSLIReport FreeSecond-Order' Foundations," LogicandItsPhilosophical Stanford andInformation, fortheStudy ofLanguage Calif.: Center 85-40(Stanford, 1985). University, distinction between that theintuitive ofdual-role 8. Itisanimportant syntax advantage inprediIfX-terms cannot occur canbeexpressed these twopropositions syntactically. andthemodalization both thepredication then oneistempted toexpress cateposition, inparticular, tosort out oneunable, ofthesameabstract '[X0~E!g]'leaving bymeans some ofthedetails oftheRussellian puzzle. ButasChris for ofsemantics 9. Intheusual Lewis/Stalnaker counterfactuals, anyway. types necwith for some counterfactuals isnotintuitive reminded me,theequivalence Swoyer Athat false consider thenecessarily false antecedents. Forexample, proposition essarily axiomatization. Ontheusual arithmetic hasa complete recursive semantics, anycounterif false that likeAistrue. Butitisintuitively antecedent factual with a necessarily false 145 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions still would havebeenableto recursive axiomatization Godei hada complete arithmetic A'snecessary falsehood hissecond However, theorem, notwithstanding. important prove inthe ofalltheconditionals astheantecedents asitis,thisissueisnotrelevant here, areintuitively Russellian contingent. puzzle 44(1983):1-20.1should Studies "OnExistentialism," 10.SeeA.Plantinga, Philosophical alsoprovide solutions toRPinthepresent that thethree non-Priorean note replies paper on9-10ofthis article. constructs totheanti-existentialist important argument Plantinga = is = y)].ByExwehaveE!=z>E],andsobyDI wehaveD(E!= term theonly occurring noncomplex = y)]),and that z>E![Xjc3j(jc = y)]),andbyK itfollows DE!= => IZIE!['*3;y(jt = y)]). ofE!= wehaveDE![Xjc3;y(;c sobythenecessitation nr.By inthecaseof(12)(hence 12.Toseethis case),suppose (11)aswellinthe0-place Ifitisa primitive then GSAwehaveE![ttt]. byEx'#ehaveE!irimmedipredicate, t' . Then from itis['x cp] Sosuppose E![['x <p]T], byExwehaveE!t',where ately. occur free in[[Xx<p] terms that allandonly thenoncomplex contains t],andsobyprepoterms t" contains thenoncomplex that allandonly sitional logicwehaveE!t",where wehaveE!['x cp] SAfollows in[Xx9] alone. SobyExagain occur free ,asrequired. toterms. with respect bysimilar reasoning sec.6; A. Nous8 (1974):211-231, "Theories ofActuality," 13.See,e.g.,R. Adams, ch.4; and Oxford TheNature Press, 1974), (Oxford: ofNecessity University Plantinga, Princeton TheFoundations Semantics J.Pollock, (Princeton: University ofPhilosophical ch.3. Press, 1984), matters willofcourse havetoqualify careful account 14.Amore somewhat; e.g.,D , can't bea set.Wecould take ittobea class, ¿ill elseitwouldn't itself tocontain beexpected sets, berevised ofaffairs asanordered thedefinition ofa state butthen (asclasses pairmust worlds Another istodefine ofsetsinstandard settheories). cannot bemembers option true ofalltheelements ofD p relative tosome states ofaffairs tobemaximal property "TheTrueModalLogic," Journal cf.C. Menzel, Logic20(1991): ofPhilosophical of wewillnothavetheluxury asitis,however, n.27.Important 331-374, esp.371-2, this issuehere. pursuing ofaffairs, itmight ofpossible states canthemselves beelements 15.Sincesuchsurrogates tomapevery settosomerepresentational ingeneral wellbenecessary systematically thesetperse.That a setinitsrepresentational rolewith soasnottoconfuse counterpart wecanthen thepure setsbeoursurrogate is,forinstance, represent possibilia, letting anordered butnecessarily setsthemselves counterpart, say, pair existing bysome impure relanumbers aresets, theidentity first element isthenumber 0 (or,ifyoubelieve whose inturn berepresented itself Sucha pair(0,s) would tion). bythepair(0, (0,s)),andso mentioned inthedefinition of themapping from toexisting on.Thus, surrogates objects forsetsalways bemapped thecondition that must alsomeet states ofaffairs surrogates carehastobetaken when conAdditional totheir correct non-surrogate counterparts. soastoensure that the forsemantical worlds setsofsuchsurrogate purposes sidering are ofthese considerations Thedetails thesame roleineachworld. same surrogate plays inC. Menzel, fashion worked outinrather "Actualism, Ontological excruciating 85(1990):355-389. World andPossible Semantics," Commitment, Ray Synthese Greg Modal in"Ontology-Free these ideashandsomely hastightened upandformalized Arelated, more formal intheJournal Semantics," ofPhilosophical Logic. forthcoming isfound in"TheTrue Modal Logic," esp.350-2. approach hasnarrow 'thesetofexisting that theterm tonote 16.Itisimportant scopehere; objects' weare within thescopeofthepossibility that then, is,itoccurs Vulgarly put, operator. a setofobjects that sethere, butabout about anactually notnecessarily existing talking hadthings beendifferent. would haveexisted modal ownquantified follows isvery closetoPrior's 17.Thelogicthat contingent logicfor inwhich itis duetothemore butismuch more expressive language explicit beings, andprovide a more Prior's I reconstruct andcriticize couched. appealing logicindetail, Modal in"TheTrue alternative, Logic." 146 This content downloaded from 165.91.74.118 on Sun, 10 May 2015 16:00:43 UTC All use subject to JSTOR Terms and Conditions 48.1havemodTime andModality Oxford 18.A.N.Prior, Press, 1957), (Oxford: University ified thequote tocomport with slightly mynotation. 19.Ibid. inaddition toPRPinlogics a genuine truth 20.Admittedly, however, containing predicate with ofother intuthis isproblematic, a couple terms, since, together denoting principle should atleasthold itleadstotheliarparadox. theprinciple itive However, principles, don't themselves involve thetruth insuchlogics forsentences like(14)that predicate. "Toward Useful I," Theories, See,e.g.,R.Turner, Type-Free op.cit.;alsoS. Feferman, inR.L. Martin, 237-287. op.cit., 21.Prior, Time andModality ,48-9. 150. 22.A.N.Prior, Clarendon Past,Present , andFuture Press, 1967), (Oxford: in individuals isweakly 23.Moregenerally, involving contingent possible anyproposition *a' wehave a name that ourPriorean ~DE!a is,where 9 isanysentence involving logic, I- ~ū~9. Toseethis, let9 besucha sentence andassume ~DE!û.By□/■wehave □~<p= ~0<pADE![-9]andhence logic~D~cp= O9v -□E![~cp]. bypropositional wehave~DE![~(p]3 ~ū~9. ByCExwehaveDEÍf-íp]z> DE 'a,andthus, Thus, sincebyassumption -DE!a, bypropositional logicagainwehave-DE![-9]andso -□-9;i.e.,^9; i.e.,[9]isweakly possible. in A. Plantinga, "On 24.Thisobjection wasinspired found argument bya similar seeesp.18-19. Existentialism," tosugandThisness," continues oninthis 25.Robert "Actualism 19.Adams Adams, passage ofmodal should bethought ofinterms that thetruth ofmodal gest singular propositions Butbymylights he ofthose bythesubjects propositions. properties actually possessed instead totheperspectivai never cashes thisidea,turning developed really metaphor below. 22. 26.Ibid., 27.Ibid., 24. for Modal Part IalsoK.Fine, "Model 28.Cf.,e.g.,R.Adams, 24ff; Theory Logic, op.cit., 125-156. TheDERE/DE Journal 1 DICTODistinction," (1978): ofPhilosophical Logic inthis section seems tobetheone tobenoted that while thesystem 29.Itought developed itismarkedly different from theoneheactubehind Adams' lurking intuitively approach, likethenonstandard Priorean endsuplooking much more at,which logic allyarrives I believe, lies Thechief reason forthis, aboveinwhich □ and0 arenotinterdefinable. hislogic. In intheinadequate ofthelanguage Adams usedtoframe expressive power oftheform lackofcomplex ledhimtotakestatements 09(a)to particular, predicates thefine distinctions about a. Thisleft himunable tomake express singular propositions outthe inlogical andwhich arecrucial tosorting form that ourricher language permits, ina more 28ff. attractive fashion. SeeAdams, op.cit., paradox - the - andstudiously fornow butI acknowledge 30.Classtalkisconvenient here, ignore, are factthatsuchtalkis problematic, sinceworlds domains, themselves, i.e.,their I assume with classtalkthat involves classes themselves. wecoulddispense arguably talkofclasses ofclasses ifneed be. in"Existence," 31.See,e.g.,N.Salmon's discussion ofNothan , Philosophical Perspectives Vol.1:Metaphysics Calif.: Co.,1987), 49-108, (Atascadero, Publishing esp. Ridgeview 95ff. - inthecontext - easily but 32.(35)hasinparticular several unusual understood, looking, z>~E!y)z>EHE!*z>-Ely]). instances; e.g.,[ZIV;y(IIII(E!;t into hasbite inmodal contexts inwhich wequantify 33.Note that theadded clause here only intheactual and allthesimple terms in'['x 9]'take values otherwise world, '['x 9]',for hence thepredicate anactual, hence relation. denotes possible, wehaveEUAElyAE!=, E!=that 34.Specifically, exists, identity byId,SA,andtheaxiom = *)])wehaveE![E!;t =>~E!/|,andsobythe andsobyEx(recall that E! = J[kxByCy T axiom =)-Ely]. schema wehave0E
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