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The point of the theory is to give first-order logic the power of set theory, but without any "existential commitment" to such objects as sets. The classic expositions are Boolos 1984, and Lewis 1991.
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2 Plural Quantification 3 Criticism 4 References 5 Links |
The view is commonly associated with George Boolos, though it is older, and is related to the view of classes defended by John Stuart Mill and other nominalist philosophers. Mill argued that universals or "classes" are not a peculiar kind of thing, having an objective existence distinct from the individual objects that fall under them, but "is neither more nor less than the individual things in the class". (Mill 1904, II. ii. 2,also I. iv. 3).
A similar position was also discussed by Bertrand Russell in chapter VI of Russell (1903), but later dropped in favour of a "no-classes" theory. See also Gottlob Frege 1895 for a critique of an earlier view defended by Ernst Schroeder.
Interest revived in plurals with work in linguistics in the 1970's by Scha, Link, Landman, Schwarzschild and Lasersohn and others, who developed ideas for a semantics of plurals.
It is well known that standard first order logic has difficulties in representing some sentences with plurals. Most well-known is the Geach-Kaplan sentence "some critics admire only one another". It seems implausible to invoke sets to explain these. "Alice, Bob and Carol admire only one another " clearly does not involve sets, being equivalent to
Boolos argued that 2nd-order monadic existential quantification may be systematically intepreted in terms of plural existential quantification, and that, therefore, 2nd-order monadic existential quantification is "ontologically innocent"
Later Tom McKay (2003) and others argued that sentences such as
McKay has argued for a unified account of plural terms that allows for both distributive and non-distributive satisfaction of predicates, which he has defended against the "singularist" assumption that such predicates are predicates of sets of individuals (or of mereological sums)
Philippe de Rouilhan (2000) has argued that Boolos relied on the assumption, never defended in detail, that plural expressions in ordinary language are "manifestly and obviously" free of existential commitment. But when I utter "there are critics who admire only one another" is it manifest and obvious that I am only committing myself with respect to critics? Or is Boolos victim of a "grammatical illusion" (p. 10)? Consider
Background
Plural Quantification
But this seems to be an instantiation of "some people admire only one another", which cannot be interpreted in standard first order logic.
also cannot be interpreted, in standard first order logic. This is because predicates such as "are shipmates", "are meeting together", "are surrounding a building" are not distributive. A predicate F is distributive if, whenever some things are F, each one of them is F. But in standard logic, every predicate is distributive. Yet such sentences also seem innocent of any existential assumptions. If true, they are about individuals who are shipmates, who meet together, lift pianos &c, and nothing else (not sets, or abstract Platonic entities).
McKay, as well as other writers such as Cameron (1999), have suggested that plural logic opens the prospect of simplifying the foundations of mathematics, avoiding the paradoxes of set theory, and "simplifying the complex and unintuitive axiom sets needed in order to avoid them.Criticism
The first case is clearly "innocent". But what about the second? There is an obvious logical difference, since in the first case the plural is distributive, in the second, it is collective, and irreducibly so. How is it obvious that this difference is innocent? Also, the second is equivalent to
But what is a "group" or "collection" in this sense? "That is the whole problem". Perhaps Boolos has accorded a kind of innocence to [the second] that would actually belong only to the first.References
Links