Abstract
We present a plural logic that is as expressively strong as it can be without sacrificing axiomatisability, axiomatise it, and use it to chart the expressive limits set by axiomatisability. To the standard apparatus of quantification using singular variables our object-language adds plural variables, a predicate expressing inclusion (is/are/is one of/are among), and a plural definite description operator. Axiomatisability demands that plural variables only occur free, but they have a surprisingly important role. Plural description is not eliminable in favour of quantification; on the contrary, quantification is definable in terms of it. Predicates and functors (function signs) can take plural as well as singular terms as arguments, and both many-valued and single-valued functions are expressible. The system accommodates collective as well as distributive predicates, and the condition for a predicate to be distributive is definable within it; similarly for functors. An essential part of the project is to demonstrate the soundness and completeness of the calculus with respect to a semantics that does without set-theoretic domains and in which the use of set-theoretic extensions of predicates and functors is replaced by the sui generis relations and functions for which the extensions were at best artificial surrogates. Our metalanguage is designed to solve the difficulties involved in talking plurally about individuals and about the semantic values of plural items.
Similar content being viewed by others
References
Boolos, G. (1984): To be is to be a value of a variable (or to be some values of some variables), J. Philos. 81, 430–49; reprinted in his (1998), pp. 54–72.
Boolos, G. (1985): Reading the Begriffsschrift, Mind 94, 331–344; reprinted in his (1998), pp. 155–170.
Boolos, G. (1998): Logic, Logic, and Logic, Harvard University Press, Cambridge, MA.
Burge, T. (1974): Truth and singular terms, Noûs 8, 309–325; reprinted in M. Platts (ed.), Reference, Truth and Reality, Routledge & Kegan Paul, London, 1980, pp. 167–181.
Cantor, G. (1895): Beiträge zur Begründung der transfiniten Mengenlehre 1, in E. Zermelo (ed.), Gesammelte Abhandlungen mathematischen und philosophischen Inhalts Springer, Berlin Heidelberg New York, 1932, pp. 282–311.
Carnap, R. (1953): Meaning postulates, Philos. Stud. 3, 65–73; reprinted in the 2nd edition of his Meaning and Necessity, University of Chicago Press, Chicago, 1957, pp. 222–229.
Cartwright, R. (1994): Speaking of everything, Noûs 28, 1–20.
Church, A. (1956): Introduction to Mathematical Logic, Vol. I, Princeton University Press, Princeton.
Dummett, M. (1973): Frege: Philosophy of Language, Duckworth, London.
Enderton, H. B. (1972): A Mathematical Introduction to Logic, Academic, New York.
Frege, G. (1891): Function and concept, in P. Geach and M. Black (eds.), Translations from the Philosophical Writings of Gottlob Frege, Blackwell, Oxford, 1952, pp. 21–41.
Frege, G. (1919): [Notes for Ludwig Darmstaedter], in H. Hermes, F. Kambartel and F. Kaulbach (eds.), Posthumous Writings, Blackwell, Oxford, 1979, pp. 253–257.
Hardy, G. H. (1925): A Course of Pure Mathematics, 4th edn., Cambridge University Press, Cambridge.
Henkin, L. (1949): The completeness of the first-order functional calculus, J. Symb. Log. 14, 159–166.
Kleene, S. C. (1952): Introduction to Metamathematics, North-Holland, Amsterdam.
Lewis, D. (1991): Parts of Classes, Blackwell, Oxford.
Morton, A. (1975): Complex individuals and multigrade relations, Noûs 9, 309–318.
Neale, S. (1990): Descriptions, MIT Press, Cambridge, MA.
Oliver, A. and Smiley, T. (2001): Strategies for a logic of plurals, Philos. Q. 51, 289–306.
Oliver, A. and Smiley, T. (2004): Multigrade predicates, Mind 113, 609–681.
Oliver, A. and Smiley, T. (2005): Plural descriptions and many-valued functions, Mind 114.
Rayo, A. (2002): Word and objects, Noûs 36, 436–464.
Schock, R. (1968): Logics Without Existence Assumptions, Almqvist & Wiksell, Stockholm.
Smiley, T. (2004): The theory of descriptions, in T. R. Baldwin and T. J. Smiley (eds.), Studies in the Philosophy of Logic and Knowledge, Oxford University Press, Oxford, pp. 131–161.
van Inwagen, P. (1990): Material Beings, Cornell University Press, Ithaca.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Oliver, A., Smiley, T. A Modest Logic of Plurals. J Philos Logic 35, 317–348 (2006). https://doi.org/10.1007/s10992-005-9019-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10992-005-9019-2