Movatterモバイル変換

1.Second-order logic has quantifiers that bind variablestaking predicate position. These quantifiers are typically understoodas ranging over concepts and relations.Monadic second-orderlogic is the subsystem of second-order logic which admits onlyquantification over concepts, not over (polyadic) relations.

2. See Boolos 1984: 432–3 [1998a: 57] for an ingenious proof,which he attributes to David Kaplan. The basic idea is as follows.Assume for simplicity that the domain consists only of critics. Thenthe sentence can be formalized as

Now reinterpret the predicateAuv as the arithmeticalpredicate \(“u = 0 \vee u = v + 1”.\) The resultingsentence

can be seen to be true in all and only non-standard models ofarithmetic, by letting \(xx\) be all and only the non-standardelements of the model. But it follows from the compactness theorem forfirst-order logic that any consistent first-order theory of arithmetichas both standard and non-standard models. Hence the original sentencecannot be formalized in a first-order language.

3. For further discussion of this and other attempts to analyze plurallocutions in non-plural terms, see, e.g., Oliver and Smiley 2001 andYi 2005.

4. Although most contemporary work on plural quantification appears tohave been inspired, directly or indirectly, by the work of GeorgeBoolos, there is also a substantial pre-Boolosian literature on thetopic. Pluralities are closely related to what the early BertrandRussell called “classes as many,” as opposed to“classes as one”; see Russell 1903, esp. Sections 70, 74,and 104, as well as Klement 2014 for discussion. Simons 1997 arguesthat Stanislaw Leśniewski in the early twentieth centuryanticipated many of the ideas later defended by Boolos. More recentpre-Boolosian writings on plurals include Black 1971; Stenius 1974;Morton 1975; Armstrong 1978: 32–34; and Simons 1982. For morereferences and historical information, see Lewis 1991: 63 and Hazen1993: 133–34.

5. It does not suffice for the predicate \(P\) to be distributive thatthis biconditional is true or even necessarily true. For instance, let\(P\) be “has/have a mereological sum”. Then a believer inarbitrary mereological sums takes the relevant biconditional to thetrue, perhaps even necessarily so, although the predicate \(P\) is notdistributive. (I owe this example to Gabriel Uzquiano.)

6. The phenomenon of non-distributive predication was noted already byFrege, who pondered the sentence “Siemens and Halske have builtthe first major telegraph network” and suggested that in it,

“Siemens and Halske” designates a compound object aboutwhich a statement is being made, and the word “and” isused to help form the sign for this object. (Frege 1914:227–8)

7. Even this extended language will only capture the core features ofthe plural locutions of English and other natural languages. Nosystematic analysis of the syntax or semantics of plural locutions innatural language will be attempted here. For more on the complexitiesof plural locutions in natural languages, the reader is referred toLandman 2000, Link 1998, Lønning 1997, McKay 2006, and Schein1993 and 2006.

8. See Rayo 2002 and McKay 2006 for plural logics that eliminatesingular variables in favor of plural ones.

9. Although Boolos never formalized the theory PFO, he was clearly awareof this interpretation. See his 1984 and 1985a.

10. See Williamson 2003: 456–457, 2010: 699–700, 2013:245–251, and 2016; Rumfitt 2005: sect. VII; Uzquiano 2011; andLinnebo 2016; see Hewitt 2012a for criticism. See Florio and Linnebo2021, ch. 10 for a development and formal proofs of relevant argumentsinvolving plurals and modals, and Roberts forthcoming for a defense ofa definitive modal logic of plurals involving, but going beyond, theprinciples (10) and (11). These two principles are assumed and put tophilosophical use in Bricker 1989: 386–390 and Forbes 1989:93–102.

11. I will, for related reasons, not discuss a proposal due to Taylor andHazen 1992, according to which English contains ordered lists ofreferring expressions, as in “London, Paris, and Berlin are thecapitals of England, France, and Germany” and associatedquantifiers. See Hewitt 2012b for a related proposal.

12. For one useful discussion of the question of logicality, see Rayo2007.

13. However, the “non-trivial mathematical truths” alluded toabove will then have to be expressed by meta-language quantificationover dyadic predicates rather than by object language quantificationover dyadic relations. Besides, since (as A.P. Hazen pointed out tome) the monadic second-order theory of the successor operation isdecidable, the amount of mathematics that can be obtained in this wayis very limited. See Büchi 1962.

14. A further problem is that the neo-logicist construction of thenatural numbers makes essential use of empty concepts. But it isdoubtful that it makes sense to talk about empty pluralities.

15. Some philosophers fail to take this tradition into account or find itunnecessary to do so. They thus move directly from the claim thatcertain locutions incur no controversial ontological commitments ofthe kind incurred by singular first-order quantifiers to the claimthat these locutions incur no controversial ontological commitmentwhatsoever. For some examples, see Boolos 1984 and 1985a, and Rayo andYablo 2001.