Abstract
Take a formula of first-order logic which is a logical consequence of some other formulae according to model theory, and in all those formulae replace schematic letters with English expressions. Is the argument resulting from the replacement valid in the sense that the premisses could not have been true without the conclusion also being true? Can we reason from the model-theoretic concept of logical consequence to the modal concept of validity? Yes, if the model theory is the standard one for sentential logic; no, if it is the standard one for the predicate calculus; and yes, if it is a certain model theory for free logic. These conclusions rely inter alia on some assumptions about possible worlds, which are mapped into the models of model theory. Plural quantification is used in the last section, while part of the reasoning is relegated to an appendix that includes a proof of completeness for a version of free logic.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Armstrong, D. M.: 1989, A Combinatorial Theory of Possibility, Cambridge University Press, Cambridge.
Boolos, G.: 1984, 'To Be Is to Be a Value of a Variable (or to Be Some Values of Some Variables)', The Journal of Philosophy 81, 430–49.
Cartwright, R.: 1987, 'Implications and Entailments', in Philosophical Essays, MIT Press, Cambridge, MA, pp. 237–56.
Chang, C. C., and H. J. Keisler: 1990, Model Theory, 3rd edn. North-Holland, Amsterdam.
Church, A.: 1958, Introduction to Mathematical Logic, Vol. 1, 2nd printing of the 2nd edn. with corrections, Princeton University Press, Princeton.
Deutsch, H.: 1990, 'Contingency and Modal Logic', Philosophical Studies 60, 89–102.
Dummett, M. A. E.: 1981, Frege: Philosophy of Language, 2nd edn., Duckworth, London.
Dummett, M. A. E.: 1991, The Logical Basis of Metaphysics, Duckworth, London.
Etchemendy, J.: 1990, The Concept of Logical Consequence, Harvard University Press, Cambridge, MA.
Hughes, G. E., and M. J. Cresswell: 1996, A New Introduction to Modal Logic, Routledge, London.
Kreisel, G.: 1967, 'Informal Rigor and Completeness Proofs', in Imre Lakatos (ed.), Problems in the Philosophy of Mathematics: Proceedings of the International Colloquium in the Philosophy of Science, London 1965, volume 1, North-Holland, Amsterdam, pp. 138–171.
Lambert, K.: 1991, 'The Nature of Free Logic', in K. Lambert (ed.), Philosophical Applications of Free Logic, Oxford University Press, New York, pp. 3–14.
Lewis, D.: 1986, On the Plurality of Worlds, Blackwell, Oxford.
Mendelson, E.: 1987, Introduction to Mathematical Logic, 3rd ed., Wadsworth & Brooks/Cole, Pacific Grove, CA.
Quirk, R., S. Greenbaum, G. Leech and J. Svartvik: 1985, A Comprehensive Grammar of the English Language, Longman, London.
Sainsbury, R. M.: 1991, Logical Forms: An Introduction to Philosophical Logic, Blackwell, Oxford.
Stephanou, Y.: 1994, Aspects of Meaning and Justification in Logic, Ph.D. diss., University of London.
Strawson, P. F.: 1952, Introduction to Logical Theory, Methuen & Co, London.
Suppes, P.: 1972, Axiomatic Set Theory, 2nd edn., Dover Publications, New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Stephanou, Y. Model Theory and Validity. Synthese 123, 165–193 (2000). https://doi.org/10.1023/A:1005217809302
Issue Date:
DOI: https://doi.org/10.1023/A:1005217809302