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REALIZABILITY SEMANTICS FOR QUANTIFIED MODAL LOGIC: GENERALIZING FLAGG’S 1985 CONSTRUCTION

  • BENJAMIN G. RIN (a1) and SEAN WALSH (a2)
Abstract

A semantics for quantified modal logic is presented that is based on Kleene’s notion of realizability. This semantics generalizes Flagg’s 1985 construction of a model of a modal version of Church’s Thesis and first-order arithmetic. While the bulk of the paper is devoted to developing the details of the semantics, to illustrate the scope of this approach, we show that the construction produces (i) a model of a modal version of Church’s Thesis and a variant of a modal set theory due to Goodman and Scedrov, (ii) a model of a modal version of Troelstra’s generalized continuity principle together with a fragment of second-order arithmetic, and (iii) a model based on Scott’s graph model (for the untyped lambda calculus) which witnesses the failure of the stability of nonidentity.

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Corresponding author
*FACULTEIT DER NATUURWETENSCHAPPEN, WISKUNDE EN INFORMATICA INSTITUTE FOR LOGIC, LANGUAGE, AND COMPUTATION UNIVERSITEIT VAN AMSTERDAM P.O. BOX 94242 1090 GE AMSTERDAM E-mail: benjamin.rin@gmail.com
DEPARTMENT OF LOGIC AND PHILOSOPHY OF SCIENCE 5100 SOCIAL SCIENCE PLAZA UNIVERSITY OF CALIFORNIA, IRVINE IRVINE, CA 92697-5100, U.S.A. E-mail: swalsh108@gmail.com or walsh108@uci.edu
References
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Barendregt, H. P. (1981). The Lambda Calculus. Studies in Logic and the Foundations of Mathematics, Vol. 103. Amsterdam: North-Holland.
Beeson, M. J. (1985). Foundations of Constructive Mathematics. Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 6. Berlin: Springer.
Bell, J. L. (1985). Boolean-Valued Models and Independence Proofs in Set Theory (second edition). Oxford Logic Guides, Vol. 12. New York: Clarendon.
Corsi, G. (2002). A unified completeness theorem for quantified modal logics. The Journal of Symbolic Logic, 67(4), 14831510.
Fitting, M., & Mendelsohn, R. L. (1998). First-Order Modal Logic. Synthese Library, Vol. 277. Dordrecht: Kluwer.
Flagg, R. C. (1985). Church’s thesis is consistent with epistemic arithmetic. In Shapiro, S., editor, Intensional Mathematics, Studies in Logic and the Foundations of Mathematics, Vol. 113. Amsterdam: North-Holland, pp. 121172.
Goodman, N. D. (1985). A genuinely intensional set theory. In Shapiro, S., editor, Intensional Mathematics. Studies in Logic and the Foundations of Mathematics, Vol. 113. Amsterdam: North-Holland, pp. 6379.
Goodman, N. D. (1986). Flagg realizability in arithmetic. The Journal of Symbolic Logic, 51(2), 387392.
Goodman, N. D. (1990). Topological models of epistemic set theory. Annals of Pure and Applied Logic, 46(2), 147167.
Hájek, P., & Pudlák, P. (1998). Metamathematics of First-Order Arithmetic. Perspectives in Mathematical Logic. Berlin: Springer.
Halbach, V., & Horsten, L. (2000). Two proof-theoretic remarks on EA + ECT. Mathematical Logic Quarterly, 46(4), 461466.
Horsten, L. (1997). Provability in principle and controversial constructive principles. Journal of Philosophical Logic, 26(6), 635660.
Horsten, L. (1998). In defense of epistemic arithmetic. Synthese, 116, 125.
Horsten, L. (2006). Formalizing Church’s thesis. In Olszewski, A., Woleński, J., & Janusz, R., editors, Church’s Thesis After 70 Years. Frankfurt: Ontos, pp. 253267.
Hughes, G. E., & Cresswell, M. J. (1996). A New Introduction to Modal Logic. London: Routledge.
Kechris, A. S. (1995). Classical Descriptive Set Theory. Graduate Texts in Mathematics, Vol. 156. New York: Springer.
Kleene, S. C. (1943). Recursive predicates and quantifiers. Transactions of the American Mathematical Society, 53, 4173.
Kleene, S. C. (1945). On the interpretation of intuitionistic number theory. The Journal of Symbolic Logic, 10, 109124.
Martin, D. A. (2001). Multiple universes of sets and indeterminate truth values. Topoi, 20(1), 516.
McCarty, D. C. (1984). Realizability and Recursive Mathematics. Technical Report CMU-CS-84-131. Department of Computer Science, Carnegie-Mellon University. Reprint of author’s PhD Thesis, Oxford University 1983.
McCarty, D. C. (1986). Realizability and recursive set theory. Annals of Pure and Applied Logic, 32(2), 153183.
Odifreddi, P. (1999). Classical Recursion Theory, Vol. II. Studies in Logic and the Foundations of Mathematics, Vol. 143. Amsterdam: North-Holland.
Rathjen, M. (2006). Realizability for constructive Zermelo-Fraenkel set theory. In Logic Colloquium ’03. Lecture Notes in Logic, Vol. 24. La Jolla: Association for Symbolic Logic, pp. 282314.
Reinhardt, W. N. (1985). The consistency of a variant of Church’s thesis with an axiomatic theory of an epistemic notion. In Caicedo, X., N. C. A. d. C., & Chuaqui, R., editors, Proceedings of the Fifth Latin American Symposium on Mathematical Logic, Vol. 19. Bogotá: Sociedad Colombiana de Matemáticas, pp. 177200.
Rogers, H Jr.. (1987). Theory of Recursive Functions and Effective Computability (second edtion). Cambridge: MIT Press.
Scedrov, A. (1986). Some properties of epistemic set theory with collection. The Journal of Symbolic Logic, 51(3), 748754.
Scott, D. (1975). Lambda calculus and recursion theory. In Proceedings of the Third Scandinavian Logic Symposium. Studies in Logic and the Foundations of Mathematics, Vol. 82. Amsterdam: North-Holland, pp. 154193.
Shapiro, S. (1985a). Epistemic and intuitionistic arithmetic. In Intensional Mathematics. Studies in Logic and the Foundations of Mathematics, Vol. 113. Amsterdam: North-Holland, pp. 1146.
Shapiro, S. (1985b). Intensional Mathematics. Studies in Logic and the Foundations of Mathematics, Vol. 113. Amsterdam: North-Holland.
Shapiro, S. (1993). Understanding Church’s thesis, again. Acta Analytica, 11, 5977.
Troelstra, A. S. (1977). Aspects of constructive mathematics. In Barwise, J., editor, Handbook of Mathematical Logic. Studies in Logic and the Foundations of Mathematics, Vol. 90. Amsterdam: North-Holland, pp. 9731052.
Troelstra, A. S. (1998). Realizability. In Buss, S. R., editor, Handbook of Proof Theory. Studies in Logic and the Foundations of Mathematics, Vol. 137. Amsterdam: North-Holland, pp. 407474.
Troelstra, A. S., & Schwichtenberg, H. (2000). Basic Proof Theory (second edition). Cambridge Tracts in Theoretical Computer Science, Vol. 43. Cambridge: Cambridge University Press.
Troelstra, A. S., & van Dalen, D. (1988). Constructivism in Mathematics. An Introduction. Amsterdam: Elsevier. Two volumes.
van Oosten, J. (2002). Realizability: A historical essay. Mathematical Structures in Computer Science, 12(3), 239263.
van Oosten, J. (2008). Realizability: An Introduction to its Categorical Side. Studies in Logic and the Foundations of Mathematics, Vol. 152. Amsterdam: Elsevier.
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