Plurals and modals

Abstract

Consider one of several things. Is the one thing necessarily one of the several? This key question in the modal logic of plurals is clarified. Some defenses of an affirmative answer are developed and compared. Various remarks are made about the broader philosophical significance of the question.

Keywords:

• Plural logic
• modality
• extensionality
• set theory
• Timothy Williamson

Notes

1 See Rumfitt (2005, Section VII), Uzquiano (2011), Williamson (2003, 456–457), Williamson (2010, 699–700), and Williamson (2013). The view is challenged in Hewitt (2012).

2 Here and in what follows, a displayed open formula will be short for the necessitation of its universal closure. It is important to realize that this convention differs from another one in discussions of modal logic, which lets a formula be short for its modal closure, where this is defined as the result of prefixing the formula with any string of universal quantifiers and necessity operators.

3 Here and in what follows, I use the word ‘plurality’ as a convenient shorthand to convey claims whose proper expression eschews this word in favor of some plural construction.

4 See e.g. Landman (1989) and Uzquiano (2004).

5 A similar example is attributed to Dorothy Edgington in Rumfitt (2005). Further examples are found in Hewitt (2012).

6 Sets – understood as on the iterative conception – come close but have the additional and complicating factor that their members are ‘bound together’ into a single object.

7 What about reducing plural logic to monadic second-order logic by translating a plural quantifier by means of a monadic second-order quantifier restricted to concepts that are non-empty and rigid? Technically, this should work. But philosophically, the proposal seems strained and lacking in motivation.

8 See Williamson (2003, Section IX).

9 See Williamson (2010).

10 See Linnebo (2010) and Linnebo (2013). While this approach draws inspiration from Parsons (1983) and to some extent also Putnam (1967) and Hellman (1989), these earlier views do not rely in the same way on the rigidity of pluralities.

11 In fact, as Williamson (1996) has pointed out, () can also be derived without use of the Brouwerian axiom by invoking suitable principles of actuality.

12 Since the necessitation of (Leibniz) ensures □(x = y → x = x), the existential presupposition `x = x', presentin (□ =), would be redundant in (□ ≠). For were x ≠ y to fail, thementioned presupposition would anyway be satised. (Thanks toWilliamson for this observation.)

13 Here, and in the paragraphs that follow, I leave implicit the proviso that the sets still exist.

14 This is ‘the basic thought’ from the end of Section 1.

15 For sets, the former option is unattractive. As Boolos (1971, 229–230) reminds us, if ever there was an example of an analytic truth, then the extensionality of sets is one.

16 This is significant for Fregean and neo-Fregean approaches to collections (or extensions, or Wertverläufe). These approaches regard extensionality as a criterion of identity and are thus committed to soft extensionalism. But they also view a collection as in some way ‘obtained from’ its defining (Fregean) concept and are thus potentially on collision course with hard extensionalism. See Parsons (2012) for a discussion of Frege’s concept of extension.

17 Thanks to Jeremy Goodman for articulating this objection.

18 Abstract objects would be a limiting case where the material contribution is nil.

19 The proper analogue of the set theoretic principle of extensionality would be the principle that sameness of material parts ensures identity. Now the analogy that seemed to cause trouble for my plausibility argument does work: any reason to accept this principle is also a reason to accept rigidity of material parthood. Of course, anyone attracted to non-rigid parts should respond to this observation by denying that sameness of material parts ensures identity.

20 See Stalnaker (1997) and essays 1–3 of Fine (2005) for some closely related considerations.

21 Notice that this enables us to drop the existential assumptions Ex and Eyy from (Rgd) on p. 1.

22 Here it is essential to observe that (Indisc) is short for the necessitation of its universal closure; cf. footnote 2. Had (Indisc) instead been short for its modal closure, we could have derived , which has contingentist S5 countermodels. (Let xx and yy be distinct pluralities at , neither of which exists at . Then holds at . Since the antecedent is vacuously true at , so is the consequent. But this contradicts the distinctness of xx and yy at .) Thanks to Tim Williamson for questions that prompted this clarification.

23 I am assuming that the statutes are partially constitutive of the committees, in the sense that, were one to change the statutes, the original committees would cease to exist and be replaced by new ones. If necessary, this persistence condition for the committees can be written into the statutes.

24 Notice that this enables us to drop the existential assumptions Ex from (Rgd) on p. 1.

25 A closely related concern is expressed in Uzquiano (2014).

26 Again, it is instructive to consider the mereological analogue, which in the case of (3) is the principle that for any objects x and y there is a sum. A natural formalization of this principle would be:(4)

Assume a molecule m is part of a cat c. For convenience, write for their sum. We easily derive and , and hence by anti-symmetry, . Since necessarily has m as part, by Leibniz’s law, so does c. However, unlike its plural analogue, this argument involves controversial steps. If requires merely that the matter of x be included in that of y, then anti-symmetry can plausibly be denied. On the other hand, if is sensitive also to the formal aspects of x and y, then the relevant instance of (4) can plausibly be denied. True, there is a sum which necessarily has both m and c as parts. But this sum has a formal aspect which falsifies the third conjunct of the relevant instance of (4). Although , we do not have : for this sum has a formal aspect, manifested in its modal profile, that goes beyond anything found in c.

27 In fact, as Jeremy Goodman observed, if a singleton plurality is uniformly traversed by its sole member, then Uniform Adjunction allows us to prove that any finite plurality is uniformly traversed by its members.

28 In fact, the entailment of covariance goes through on the contingentist version as well.

29 If desired, one can tweak (UniTrav-C) so as to ensure that (Dep) too follows, namely by adding the following as a third (and perfectly sensible) conjunct: .

30 Recall the claim from p. that (Indisc) can be ‘factorized’ into (Sup) and (Cov).

31 Of course, uniform traversability is another matter, as demonstrated by the argument from Section 5.4.

32 There is a more indirect connection, however. In mathematics, the prevailing view has come to be that quasi-combinatorial reasoning should be extrapolated ‘as far as possible.’ How far is that? Given (Sup), Rigidity opens for the possibility of extrapolating such reasoning very far, namely to any plurality. The reason is that such reasoning ensures traversability, and thus by (Sup) also uniform traversability, whence by the argument developed in Section 5.4, also Rigidity. Thus, without Rigidity, it would not be permissible to extend quasi-combinatorial reasoning to any plurality.

33 Consider e.g. the breezy arguments for the status of the axioms of plural as ‘genuine logical truths’ found in Boolos (1985, 342) (corresponding to Boolos (1998, 167)) and Hossack (2000, 422).

34 See Feferman (2005) for a survey of debates concerning the legitimacy of impredicative reasoning in mathematics.

35 Our example from Section 6 of the Hiring Committee and the Graduate Admissions Committee will do.

36 See Linnebo (2010).

37 See Linnebo (2013). This approach uses a non-metaphysical modality to identify the legitimate forms of plural comprehension. This alternative modality allows us to represent the stages of ‘the process of set formation’. Plural comprehension is permissible on any condition whose instances can be exhausted by one of these stages.

38 I am grateful for comments from Salvatore Florio, Jeremy Goodman, Simon Hewitt, Timothy Williamson, Gabriel Uzquiano, as well as participants at a London research seminar and a Montreal workshop where this work was presented. Much of the research was undertaken while benefiting from an ERC Starting Grant.