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First-order modal logic in the necessary framework of objects

Published online by Cambridge University Press:  01 January 2020

Peter Fritz*
Department of Philosophy, Classics, History of Art and Ideas, University of Oslo, Oslo, Norway.


I consider the first-order modal logic which counts as valid those sentences which are true on every interpretation of the non-logical constants. Based on the assumptions that it is necessary what individuals there are and that it is necessary which propositions are necessary, Timothy Williamson has tentatively suggested an argument for the claim that this logic is determined by a possible world structure consisting of an infinite set of individuals and an infinite set of worlds. He notes that only the cardinalities of these sets matters, and that not all pairs of infinite sets determine the same logic. I use so-called two-cardinal theorems from model theory to investigate the space of logics and consequence relations determined by pairs of infinite sets, and show how to eliminate the assumption that worlds are individuals from Williamson's argument.

Copyright © Canadian Journal of Philosophy 2016

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Barwise, Jon. 1977. “Some Eastern Two Cardinal Theorems.” In Logic, Foundations of Mathematics, and Computability Theory, edited by Butts, Robert E. and Hintikka, Jaakko, 1131. Dordrecht: D. Reidel.CrossRefGoogle Scholar
Bowen, Kenneth A. 1979. Model Theory for Modal Logic: Kripke Models for Modal Predicate Calculi. Dordrecht: D. Reidel.CrossRefGoogle Scholar
Burgess, John P. 2004. “E Pluribus Unum: Plural Logic and Set Theory.” Philosophia Mathematica 12: 193221.CrossRefGoogle Scholar
Chang, C. C. 1965. “A Note on the Two Cardinal Problem.” Proceedings of the American Mathematical Society 16: 11481155.CrossRefGoogle Scholar
Chang, C. C., and Jerome Keisler, H.. 1990. Model Theory. 3rd ed. Amsterdam: North-Holland.Google Scholar
Devlin, Keith J. 1984. Constructibility. Berlin: Springer-Verlag.CrossRefGoogle Scholar
Fine, Kit. 1977. Postscript to Worlds, Times and Selves (with Prior, A. N.). London: Duckworth.Google Scholar
Friedman, Harvey M.. 1999. A Complete Theory of Everything: Satisfiability in the Universal Domain. Scholar
Fritz, Peter. 2013. “Modal Ontology and Generalized Quantifiers.” Journal of Philosophical Logic 42: 643678.Google Scholar
Fuhrken, G. 1964. “Skolem-type Normal Forms for First-order Languages with a Generalized Quantifier.” Fundamenta Mathematicae 54: 291302.CrossRefGoogle Scholar
Hughes, G. E., and Cresswell, M. J.. 1996. A New Introduction to Modal Logic. London: Routledge.CrossRefGoogle Scholar
Jensen, R. Björn 1972. “The Fine Structure of the Constructible Hierarchy.” Annals of Mathematical Logic 4: 229308.CrossRefGoogle Scholar
Keisler, H. Jerome 1966. “First Order Properties of Pairs of Cardinals.” Bulletin of the American Mathematical Society 72: 141144.CrossRefGoogle Scholar
Koellner, Peter. 2009. “On Reflection Principles.” Annals of Pure and Applied Logic 157: 206219.CrossRefGoogle Scholar
Kreisel, Georg. 1967. “Informal Rigour and Completeness Proofs.” In Problems in the Philosophy of Mathematics, edited by Lakatos, Imre, 138186. Amsterdam: North-Holland.CrossRefGoogle Scholar
Kripke, Saul A. 1959. “A Completeness Theorem in Modal Logic.” Journal of Symbolic Logic 24: 114.CrossRefGoogle Scholar
Lewis, David. 1986. On the Plurality of Worlds. Oxford: Basil Blackwell.Google Scholar
Mitchell, William. 1972. “Aronszajn Trees and the Independence of the Transfer Property.” Annals of Mathematical Logic 5: 2146.CrossRefGoogle Scholar
Morley, Michael, and Vaught, Robert. 1962. “Homogeneous Universal Models.” Mathematica Scandinavica 11: 3757.CrossRefGoogle Scholar
Rayo, Agustín, and Williamson, Timothy. 2003. “A Completeness Theorem for Unrestricted First-order Languages.” In Liars and Heaps: New Essays on Paradox, edited by Beall, J. C., 331356. Oxford: Oxford University Press.Google Scholar
Schmerl, James H. 1977. “An Axiomatization for a Class of Two-cardinal Models.” The Journal of Symbolic Logic 42: 174178.CrossRefGoogle Scholar
Shapiro, Stewart. 1987. “Principles of Reflection and Second-order Logic.” Journal of Philosophical Logic 16: 309333.CrossRefGoogle Scholar
Shelah, Saharon. 1979. “On Successors of Singular Cardinals.” In Logic Colloquium 78, edited by Boffa, Maurice, van Dalen, Dirk, and McAloon, Kenneth, 357380. Amsterdam: North-Holland.Google Scholar
Shelah, Saharon. 2005. “The Pair n***0 May Fail 0-compactness.” In Logic Colloquium ’01, edited by Baaz, Matthias, Friedman, Sy-David, and Krajíček, Jan, 402433. Boca Raton, FL: A K Peters/CRC Press.Google Scholar
Silver, Jack. 1971. “The Independence of Kurepa's Conjecture and Two-cardinal Conjectures in Model Theory." In Axiomatic Set Theory. Proceedings of Symposia in Pure Mathematics, Volume XIII, Part I, edited by Scott, Dana S., 383390. Providence, RI: American Mathematical Society.Google Scholar
Stalnaker, Robert. 1976. “Possible Worlds.” Noûs 10: 6575.CrossRefGoogle Scholar
Tarski, Alfred. 2002 [1936]. “On the Concept of Following Logically.” History and Philosophy of Logic 23: 155196. Originally published in Polish and German in 1936.CrossRefGoogle Scholar
Vaught, R. L. 1964. “The Completeness of Logic with the Added Quantifier ‘There are Uncountably Many’.” Fundamenta Mathematicae 54: 303304.CrossRefGoogle Scholar
Vaught, R. L. 1965a. “A Löwenheim--Skolem Theorem for Cardinals for Apart.” In The Theory of Models, edited by Addison, J. W., Henkin, L., and Tarski, A., 390401. Amsterdam: North-Holland.Google Scholar
Vaught, R. L. 1965b. “The Löwenheim--Skolem Theorem.” In Methodology Logic and Philosophy of Science, edited by Bar-Hillel, Yehoshua, 8189. Amsterdam: North-Holland.Google Scholar
Williamson, Timothy. 2000a. “Existence and Contingency.” Proceedings of the Aristotelian Society 100: 321343.CrossRefGoogle Scholar
Williamson, Timothy. 2000b. “The Necessary Framework of Objects.” Topoi 19: 201208.CrossRefGoogle Scholar
Williamson, Timothy. 2003. “Everything.” Philosophical Perspectives 17: 415465.CrossRefGoogle Scholar
Williamson, Timothy. 2013. Modal Logic as Metaphysics. Oxford: Oxford University Press.CrossRefGoogle Scholar
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