Grim, Patrick

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186 ANALYSISShoemaker and Noonan appealed to indeterminacy of denotationas a way to save LEM in light of vague identity statements. By theShoemaker-Noonan position, if V and '6' determinately denoteobjects, then1̂  it is true that either a = b or a ¥= b; and2) it is either true that a = b or true that d=£b.If this appeal had worked, we could have said that the objectsweren't indeterminately identical, the statements were indeter-minate due to an indeterminacy of denotation. In the presentsection I have tried to save LEM by distinguishing between LEMand PBV and arguing that (1) can be true even though (2) is false.The explanation of vague identity statements lies neither in theLEM-violating indeterminacy of the objects referred to in thesestatements nor in the indeterminacy of the denotation of certaincomponents of these statements, but in our failure to draw a sharpline between what counts as part of the same object and whatdoesn't.1University of Oklahoma, © MONTE COOK 1986455 West Lindsey Street, Norman,Oklahoma 73019, U.S.A.1 I wish to thank Chris Swoyer for many valuable discussions on the topic of vagueidentity.ON SETS AND WORLDS: A REPLY TO MENZELBy PATRICK GRIMIN [4] I offered a Cantorian argument that there can be no set ofall truths, and noted one application: against an approach topossible worlds in terms of maximal consistent sets of propositions.Shortly thereafter Selmer Bringsjord offered a similar argumentagainst set-theoretical worlds in [1].In [7], a recent and important contribution to the discussion,Christopher Menzel has raised a number of critical points regardingBringsjord's argument. But these points also apply against myearlier and more general argument. I want to address them directlyhere.Downloaded from https://academic.oup.com/analysis/article/46/4/186/160951 by guest on 24 February 2021ON SETS AND WORLDS: A REPLY TO MENZEL 187IThere cannot, it appears, be a set of all truths. For essentially thesame Cantorian reasons, it seems, there can be neither world-storiesnor worlds construed as maximal consistent sets of propositions.But perhaps there can be a non-set something-else of all truths,or a non-set something-else appropriate to the propositions of aworld-story or a world.Menzel does ultimately have such a something-else to offer. Hefirst considers and rejects, however, what may seem the obviouscandidate: the proper classes of Von Neumann-Bernays-Godel(VNBG) and similar systems.In [7] as in [8] Menzel deftly distinguishes two conditions onproper classes: excessive size and unbounded rank.1 It is the secondhe takes as the 'true conceptual boundary' between sets and properclasses. If propositions are admitted as urelements, however, aworld-story S will be of bounded rank:since . . . propositions for the world-story theorist are not sets, theythemselves have no members and so have a rank of 0; hence, S itself hasa rank of 1, i.e. S is bounded, ([7], p. 70)There is then no need to consign world-stories (or a totality oftruths) to the realm of proper classes. Or so the argument goes.Note however that Menzel's 'ranks' here are confined exclusivelyto sets: it is sets alone which have ranks, determined exclusively bythe ranks of their members. But this may be a dangerously pro-vincial view of rank. Hierarchies of types have of course often beenthought necessary for propositions, predicates, and properties aswell as sets, and in the end an integrated theory of types may berequired. We may find it necessary to rank sets of propositionsabove those sets mentioned in propositions they include, forexample.2 But given an integrated ranking of this sort Menzel'sargument fails to go through. It is no longer clear, in particular,that world-stories will have a rank either of 1 or of any otherbounded n.Nonetheless there are strong reasons to think that Menzel'sconclusion here is right: that proper classes won't do as a some-thing-else appropriate to world-stories or a totality of truths.1 Within VNBG proper classes are those too large to be sets. Menzel defines the rankof an object recursively as the least ordinal greater than the ranks of all its members. Aclass is unbounded if it contains members of arbitrarily high rank.2 This would seem the obvious way to deal with the following variant of Russell'sparadox, for example:Some sets of propositions contain propositions which mention those sets themselves.Some sets of propositions contain no such propositions. We will refer to the latter as'non-self-mentioning' propositional sets.Consider now the set S of all propositions which mention only non-self-mentioningsets. Is 5 self-mentioning or not?Downloaded from https://academic.oup.com/analysis/article/46/4/186/160951 by guest on 24 February 2021188 ANALYSISSome of the objections to proper classes are technical. WithinVNBG and similar systems, for example, proper classes are keptimmune from Cantorian arguments only at the cost of sacrificinggeneral provability for instances of the induction schema (see forexample [6] p. 198).Intuitive objections are even stronger, however. For in VNBGand similar systems proper classes are introduced precisely as classeswhich are not members of further (even finite) classes. That wouldclearly not seem to hold for either a 'class of all truths' or forclasses appropriate to world-stories. For surely a class of all truthswould be a member of further classes — of various classes of prop-ositional classes and various classes of classes of truths. Surely aclass of all truths would form a pair with a class of all falsehoods,for example. Worlds and their world-stories would seem if anythingless amenable to treatment as proper classes. For consider the classof worlds in which I exist and the class of worlds in which thisvase is hit with a hammer. Current philosophical use quite generaUyseems to demand classes of possible worlds.IIThe 'something-else' that Menzel ultimately suggests involves arecourse not to amplified but to amputated set theories. Perhapsthe world-story theorist, though clearly a profligate realist regardingpropositions, can follow an abstemiously constructivist line when itcomes to sets:By adopting ZF-Power (or some similarly conservative set theory), then,and adjusting it appropriately to allow for the existence of large sets (andurelements), the world-story theorist is free to postulate the existence ofhis world-stories without fear of paradox, at least by way of [theCantorian] argument; for that argument depends essentially on therebeing a power set of the world-story S; but there is simply no way ofgenerating the full classical power set of an infinite set in ZF-Power . . .([7],P. 71)It is far from clear, however, that large sets within ZF-Po-wer willprove any more satisfactory here than proper classes.Not too surprisingly, sacrifice of the power set axiom results inquite major technical limitations; at one blow we are effectivelyexiled from Cantor's paradise (some surprising technical limitationsof ZF-Power appear in [10]). Nor does ZF-Power applied as Menzelenvisages it really seem to satisfy constructivist scruples. Menzelrelies on the fact that 'there is simply no way of generating the fullclassical power set of an infinite set in ZF-power . . . ' But of coursethere is no way of generating basic infinite sets within ZF-Power atall; they must be added by fiat of special axiom. The set of zero andits successors appears by axiom within ZF, and a set of all ordinalsis added by special axiom in Menzel's system in [8]. If we are toDownloaded from https://academic.oup.com/analysis/article/46/4/186/160951 by guest on 24 February 2021ON SETS AND WORLDS: A REPLY TO MENZEL 189include the reals, and world-stories, and infinite sets of world-stories within ZF-Power we will need special axiomatic provisionfor these as well — a prodigal postulation of sets far from con-structivist in spirit.It is not clear, moreover, that sacrifice of the power set axiomalone will be enough to escape Cantorian difficulties regarding aset of all truths. For consider the following argument.Let us suppose we did have a set T of all truths within ZF-Power.Then by the axiom schema of separation, carried over directly fromZF, we would have as a subset of T a set of all truths satisfying aparticular condition B(x). As long as our basic language is richenough to express the notion that a truth t is about a topic c, itappears, one such condition will be lx is about a set of truths.' Byseparation, then, we would have a set C of all truths about sets oftruths.More formally, using 'Axy' to indicate that x is about y, ourcondition *x is about a set of truths' is:3yVz(z eyDzET-8c Axy).The axiom schema of separation, taken directly from ZF, isx Sy = x <=a8cB(x)),where zx, ..., zn are the free variables of B(x) other than x, and theonly restriction on B(x) is that it does not contain y as a freevariable (see for example [6] p. 175). Using T fora in this schemaand our condition above for B(x) would give us a set C of all truthsabout sets of truths:z(z EyDzET- 8c Axy)).The existence of a set C, however, would give us a reductio.C would, in particular, be larger than T. For consider any one-to-one function / from T into C, mapping truths onto truths con-cerning sets of truths, and consider further a set c of all truthswhich do not belong to the sets their assigned element is about.Here using 'a(/(x))' to indicate the set (or union of sets) f(x) isabout,xGc' =x£T8cx<£a(f(x)).Clearly C will contain some truth about c just as it contains sometruth regarding any set of truths. But by familiar reasoning / canassign no element of T to any truth about c . C is larger than T.One branch of the reductio, then, is this: given a set T of alltruths, it appears, there would be a subset C larger than the set Tof which it is a subset.33 Note that this is not a problem for sets in ZF-Power in general; the argument aboverelies essentially on the supposition that T is a set of all truths.Here and elsewhere I am obliged to an anonymous referee for helpful comments.Downloaded from https://academic.oup.com/analysis/article/46/4/186/160951 by guest on 24 February 2021190 ANALYSISAnother branch is this. Each element of C is a truth. There arethus more truths than elements of T, and thus T cannot, asassumed, be a set of all truths. A similar argument would of courseapply to world-stories.In order to escape the Cantorian difficulties of C, it appears,something more would have to be sacrificed — something aboveand beyond the power set axiom. Perhaps additional restrictionson the axiom schema of separation are called for. Perhaps thelanguage of the system must exclude or restrict the notion of truthsabout sets of truths — though since there clearly are truths aboutany sets of truths this would seem far from satisfactory. Or perhapshere again we should simply conclude that there is no set T of alltruths.IllVariations on the Cantorian argument can be used to show thatthere is no set of all necessary truths, no set of all falsehoods, noset of all propositions, and no set of all things known by anomniscient being (see also [3] and [5]). If extensions are construedstandardly in terms of sets, 'true', 'false', 'necessary', and 'is aproposition' have no extensions. Since the conjunction of a neces-sary and a contingent proposition is itself contingent, moreover,the argument can also be adapted to show there is no set of allcontingent truths or falsehoods.4 Nor is there any need to interpretthe argument as restricted to linguistic entities in any sense; theargument would be the same against a set of all facts or all states ofaffairs.In a concluding caveat, however, Menzel appeals to two outlinesof possible worlds which appear to escape Cantorian arguments.One of these is Plantinga's:a state of affairs S is complete or maximal if for every state of affairs S',S includes S or precludes S . And a possible world is simply a possiblestate of affairs that is maximal. ([9], p. 45)The other is Menzel's own:A world-story [could] be a set S that is maximal in the sense that for anyproposition q, S entails (but not necessarily contains) either q or itsnegation, but not both . . . ( [7 ] , p. 72)4 In the standard argument (as in [4]) we envisage a necessary truth corresponding toeach element of the power set Here we envisage the conjunction of each of these with achosen contingent proposition.It appears that even a set of all atomic propositions will fall victim. For suppose a setA of all atomic propositions and a set C of all connectives. Within classical atomismcompounds from these will give us all propositions. If A and C are sets, however, bystandard principles of set theory all permutations of elements of A sprinkled withpermutations of elements of C will also form a set. We know independently that therecan be no set of all propositions, however, and thus by modus tollens it appears there canbe no set of all classical atomic propositions.Downloaded from https://academic.oup.com/analysis/article/46/4/186/160951 by guest on 24 February 2021ON SETS AND WORLDS: A REPLY TO MENZEL 1 9 1Do these approaches offer an escape from Cantorian difficultiesof sets and worlds?That is unclear. Menzel's outline demands universal quantifica-tion over propositions, Plantinga's calls for quantification over allstates of affairs, and it appears that any genuine explication of thenotion of 'maximality' crucial to possible worlds must be similarlyquantificational in form. The lingering difficulty here is the follow-ing. By our Cantorian arguments there can be neither a set of allpropositions nor a set of all states of affairs. But the only realsemantics for quantification we have is in terms of sets.5State University of New York, ©PATRICK GRIM 1986Stony Brook, NY 11794, U.S.A.REFERENCES[1] Selmer Bringsjord, 'Are There Set-Theoretic Possible Worlds?', Analysis 45.1,January 1985, p. 64.[2] Martin Davies, Meaning, Quantification, Necessity, London: Routledge and KeganPaul, 1981.[3] Patrick Grim, 'Some Neglected Problems of Omniscience', American PhilosophicalQuarterly, 20 (1983), 265-76.[4] Patrick Grim, 'There Is No Set Of All Truths', Analysis 44.4. October 1984,pp. 206-8.[5] Patrick Grim, 'Against Omniscience: The Case from Essential Indexicals', Nous,19(1985), pp. 151-80.[6] Azriel Levy, 'The Role of Classes in Set Theory', in Gert H. Miiller, ed., Sets andClasses, Amsterdam: North-Holland, 1976, pp. 173-215.[7] Christopher Menzel, 'On Set Theoretic Possible Worlds', Analysis, 46.2, March1986, pp. 68-72.[8] Christopher Menzel, 'On The Iterative Explanation of the Paradoxes', PhilosophicalStudies, 49(1986), pp. 37.[9] Alvin Plantinga, The Nature of Necessity, Oxford: Clarendon Press, 1974.[10] Andrzej Zarach, 'Unions of ZF--models which are themselves ZF~-models\ inD. Van Dalen, D. Lascar, and T. J. Smiley, eds., Logic Colloquium '80, Amsterdam:North-Holland, 1982.5 In [2] , p. 262, Martin Davies suggests further difficulties: a Cantorian argumentagainst a set of all worlds, which would seem to apply however worlds are conceived.Crucial premises of Davies' argument, however, involve principles both messy andintriguing regarding philosophy of mind and the conceivability of propositions.Downloaded from https://academic.oup.com/analysis/article/46/4/186/160951 by guest on 24 February 2021

On Sets and Worlds: A Reply to Menzel

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