Motivating Reductionism about Sets

Abstract

The paper raises some difficulties for the typical motivations behind set reductionism, the view that sets are reducible to entities identified independently of set theory.

Notes

¹For classic discussion of various reductions (e.g. of numbers to sets) see Quine 1960: chap. 7; 1964. Note that I am not using the word ‘domain’ in this paragraph in a specifically set-theoretic sense.

²Armstrong endorses LMT in Armstrong 1997:185 – 8; 1991:189 – 90].

³See also Armstrong 1997:189]. Unit-making properties were called ‘particularising properties’ in Armstrong 1978a; 1978b. Armstrong now feels that the old terminology was infelicitous, presumably because it suggests that these properties somehow turn their instances into particulars.

⁴Armstrong writes: ‘must every member of a class have some unit-making property? I think it must. To be a member of a class something must be a one.…To be a member there must be something about it that makes it a one: that sets the one's boundaries as it were. And what can this something be except something about the nature of the one: some property that does the one-making?’1991: 198]. Note that to avoid the set paradoxes Armstrong has to restrict mereological fusion or argue that some fusions do not have one-making properties.

⁵See Lewis 1991: 56, n. 13].

⁶David Lewis's 1991 metaphysics of sets is not reductionist. The reason is that Lewis takes singletons as primitive rather than reducing them to some other types of entities; hence he accepts unreduced set-theoretic primitives. His position in the book is somewhat complicated by the fact that in its appendix and in the article summarizing it [Lewis 1993, he proposes a further, structuralist (and also nonreductionist) reading of set theory besides the official mereological reduction of subsethood. Nevertheless, the official metaphysics of sets in Lewis 1991 combines LMT, which reduces subsethood to mereological parthood, with a primitive singleton-formation operation. Lewis considers possible metaphysical reductions of the singleton function, including Armstrong's, but rejects them all for various reasons, in the end grudgingly accepting the singleton function as primitive. His 1991 metaphysics of sets was motivated not by anti-platonism but by the thought that the only mode of composition should be mereological.

⁷An example is George Bealer's property-theoretic reductionism 1982: chap. 5, esp. 119], which Bealer construes as an elimination rather than a reduction 1982: 114]; see the next paragraph for more on this point. He solves the property-theoretic paradoxes via an iterative conception of properties parasitic on the iterative conception of sets 1982: 96, 115].

⁸Or rather by at least one, since the kind set may be deemed to have sub-kinds.

⁹A non-vindicatory elimination or just an elimination simpliciter would be one that claims that apparent singular terms such as ‘Ø’, ‘{Ø}’, etc. do not refer to anything (and hence, on the typical views, that set-theoretic statements are either false or truth-valueless because they contain false existential presuppositions). Another kind of elimination is achieved when set terms are shown to be incomplete symbols that appear to denote but disappear under analysis—like, say, ‘the average Welsh family’. On this hypothesis, set theory does not make the existential presuppositions suggested by its surface form, since set talk is construed as a façon de parler. Russell and Whitehead in Principia Mathematica famously construed set terms as incomplete symbols, and indeed the terminology of ‘incomplete symbols’ originates in Russell's work leading up to Principia.

¹⁰Reductionists have also offered specific motivations on behalf of their individual reductions (see e.g. Armstrong 1991: 194 – 5] for why states of affairs might be well suited to play the role of classes). The difficulties I raise for the generic motivations also undermine some reductionists' specific motivations.

¹¹This is Armstrong's 1997 formulation: 41]. Armstrong wields the Eleatic Principle cautiously, calling arguments based on it ‘pragmatic’, i.e. heuristic or less than conclusive 1978: chap. 16, esp. 45; 1989: 7], and he is less certain of it than Naturalism (see below). He also acknowledges that the principle's formulation leaves much to be desired. For instance, read literally, the Eleatic Principle proscribes neither isolated spacetimes nor possible worlds understood realistically à la Lewis 1997: 42]. Since my objections do not depend on the letter of the principle, let us agree to understand it as Armstrong intends it. Critical discussion of the Eleatic Principle can be found in Oddie 1982 and Colyvan 1998. Oddie's complaints about early formulations of the principle were taken on board by Armstrong, resulting in the 1997 formulation presented here.

¹²See Field 1989: 68]. This general issue (emphasizing knowledge rather than reliability of belief) is famously broached in Benacerraf 1973. Armstrong's ‘Causal argument’ (see the long quotation below) infers from the fact that we have no causal contact with platonic entities that we have no good reason to postulate them. The distinction between an entity having distinctive causal powers and its potentially having distinctive causal effects on us will not matter here.

¹³The transitive closure of a set X is the set of entities that are related to X by the ancestral of the set-membership relation. The individual A is thus the only individual in the transitive closure of each of {A}, {{A}}, {{{A}}}, etc.

¹⁴As also noted by Rosen 1995: 623]. Another philosopher perplexed by Armstrong's notion of supervenience is Sider 2005: 690 – 1].

¹⁵Do not confuse states of affairs (worldly) with true propositions (linguistic). It may look as if I am assuming the falsity of a view—such as that attributed to Meinong by Russell—that distinguishes between subsistence and existence. But any ‘Meinongian’ who is a reductionist in the sense considered here must maintain that sets exist rather than subsist. (A metaphysician of this kind may of course be a reductionist within the realm of subsisting but not existing entities; but reductionists' starting point is that sets exist, not merely subsist.)

¹⁶I follow Armstrong in simply calling this Naturalism 1997: chap. 1]. His definition of Naturalism has changed over the years—he has in the past taken it to include (a version of) the Eleatic Principle. For John Bigelow's spatiotemporal motivation, see 1988: 1]; for his preference for a causal theory of knowledge, see 1988: 75]; and for his general physicalist motivation, see 1990: 291 – 5; 1993: 77 – 80, 95].

¹⁷An inaccessible—often, ‘strongly inaccessible’—cardinal is one that is uncountable, is regular, and has the property that the cardinal of the power set of a cardinal smaller than it is also smaller than it. (An infinite cardinal β is regular iff there is no ordinal α smaller than β and function f from α to β such that the range of f is unbounded in β.) The initial segments of the hierarchy, the V(α), are defined by ordinal recursion as follows: V(0) = Ø, V(α + 1) = P(V(α)), and V(λ) = U_ξ<λ V(ξ) for λ a limit. The set V(κ) for κ the first inaccessible is the domain of the first natural model of the ZFC axioms. The majority of set theorists view this domain as an initial segment of the full set hierarchy and hence implicitly accept the axiom of inaccessibles.

¹⁸Usually an ‘impure’ set is one with some individual (i.e. a non-set, be it concrete or abstract) in its transitive closure.

¹⁹Maddy 1980 also defends the view that some impure sets are spatiotemporal. For compelling criticism of Maddy, see e.g. Chihara 1990: chap. 10].

²⁰This structural interpretation is supplemented with the claim that there are inaccessibly many mereological atoms to avoid vacuous satisfaction of the universal clauses.

²¹Of course the challenge applies just as much to reductions that take the reducing entities to constitute spacetime (e.g. Armstrong's state of affairs metaphysics—see Armstrong 1991: 199]) as to those that take them to be in it. The difficulties apply more generally to ontologies in which the entities in question are derived entities. For instance, the difficulties of conceiving of inaccessibly many universals in spacetime are not overcome by thinking of them as derived by some kind of abstraction from states of affairs, as in Armstrong 1997.

²²I follow the contemporary literature in calling the principle ‘Humean’. Whether Hume truly intended it to have this implication is another matter.

²³For Lewis's version of the argument, see Lewis 1991: 34 ff.].

²⁴Bigelow himself may understand haecceities in a non-standard way, since in one passage 1988: 109] he claims not to be sure whether there is any real distinction between A's haecceity and A itself. Yet clearly the two cannot be the same, since if {A} = A for any entity A (and not just any individual A) that would undermine his proposed reduction.

²⁵I am grateful to the following for comments on early versions of this paper and/or the sections of my PhD dissertation chapter it grew out of: Alex Oliver, the late David Lewis, Gideon Rosen, John Burgess, Michael Potter, Paul Benacerraf, and Tim Williamson. I have also benefited from more recent discussion of these ideas with Antony Eagle, Brian Leftow, David Charles, Oliver Pooley, and Ralph Wedgwood. Thanks finally to two AJP referees for helpful suggestions.