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TOPOLOGY AND MODALITY: THE TOPOLOGICAL INTERPRETATION OF FIRST-ORDER MODAL LOGIC

  • STEVE AWODEY (a1) and KOHEI KISHIDA (a2)

Abstract

As McKinsey and Tarski showed, the Stone representation theorem for Boolean algebras extends to algebras with operators to give topological semantics for (classical) propositional modal logic, in which the “necessity” operation is modeled by taking the interior of an arbitrary subset of a topological space. In this article, the topological interpretation is extended in a natural way to arbitrary theories of full first-order logic. The resulting system of S4 first-order modal logic is complete with respect to such topological semantics.

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Corresponding author

*PHILOSOPHY DEPARTMENT, CARNEGIE MELLON UNIVERSITY E-mail:awodey@cmu.edu
PHILOSOPHY DEPARTMENT, UNIVERSITY OF PITTSBURGH E-mail:kok6@pitt.edu

References

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Awodey, S. (2006). Category Theory, Oxford Logic Guides 49. New York: Oxford University Press.
Awodey, S., & Kishida, K. (2005). Topological semantics for first-order modal logic. Paper read (by Kishida) at Topos Theory Summer School in Haute Bodeux, Belgium, on June 3.
Awodey, S., & Kishida, K. (in preparation). Topological completeness of first-order modal logic.
Dragalin, A. G. (1979). Mathematical Intuitionism: Introduction to Proof Theory. Moscow: Nauka (in Russian); English translation (1988) by Mendelson, E., Providence, RI: American Mathematical Society.
Fourman, M. P., & Scott, D. S. (1979). Sheaves and logic. In Fourman, M. P., Mulvey, C. J., and Scott, D. S., editors. Applications of Sheaves: Proceedings of the Research Symposium on Applications of Sheaf Theory to Logic, Algebra, and Analysis, Durham, July 9–21, 1977. Berlin: Springer-Verlag, pp. 302401.
Ghilardi, S. (1989). Presheaf semantics and independence results for some non classical first order logics. Archive for Mathematical Logic, 29, 125136.
Ghilardi, S. (1990). Modalità e categorie. PhD Thesis, Universite à degli studi di Milano (in Italian).
Ghilardi, S. (1991). Incompleteness results in Kripke semantics. Journal of Symbolic Logic, 56, 517538.
Ghilardi, S., & Meloni, G. (1988). Modal and tense predicate logic: models in presheaves and categorical conceptualization. In Borceux, F., editor. Categorical Algebra and Its Applications: Proceedings of a Conference, Held in Louvain-La-Neuve, Belgium, July 26–August 1, 1987. Berlin: Springer-Verlag, pp. 130142.
Ghilardi, S., & Meloni, G. (1991). Relational and topological semantics for modal and temporal first order predicative logic. In Costantini, D., and Galavotti, M. C., editors, Nuovi Problemi della Logica e della Filosofia della Scienza, vol. 2. Bologna: CLUEB, pp. 5977.
Goldblatt, R. (1979). Topoi: The Categorial Analysis of Logic. Amsterdam: North-Holland, (revised ed. 1984).
Hilken, B., & Rydeheard, D. (1999). A first order modal logic and its sheaf models. In Fairtlough, M., Mendler, M., and Moggi, E., editors. FLoC Satellite Workshop on Intuitionistic Modal Logics and Applications (IMLA’99), Trento, Italy, on July 6.
Kishida, K. (2007). Topological semantics for first-order modal logic. MSc Thesis, Carnegie Mellon University.
Lawvere, F. W. (1969). Adjointness in foundations. Dialectica, 23, 281296. Reprinted in Reprints in Theory and Applications of Categories, 16, 116.
Lawvere, F. W. (1970). Quantifiers and sheaves. Actes du Congrès International des Mathématiciens (Nice, 1970), 1, 329334.
Mac Lane, S., & Moerdijk, I. (1992). Sheaves in Geometry and Logic: A First Introduction to Topos Theory. New York: Springer-Verlag.
Makkai, M., & Reyes, G. E. (1995). Completeness results for intuitionistic and modal logic in a categorical setting. Annals of Pure and Applied Logic, 72, 25101.
McKinsey, J. C. C., & Tarski, A. (1944). The algebra of topology. Annals of Mathematics, 45, 141191.
Rasiowa, H., & Sikorski, R. (1963). The Mathematics of Metamathematics. Warsaw: Państwowe Wydawnictwo Naukowe.
Reyes, G. E. (1991). A topos-theoretic approach to reference and modality. Notre Dame Journal of Formal Logic, 32, 359391.
Reyes, G. E., & Zolfaghari, H. (1991). Topos-theoretic approaches to modality. In Carboni, A., Pedicchio, M. C., and Rosolini, G., editors. Category Theory: Proceedings of the International Conference Held in Como, Italy, July 22–28, 1990. Berlin: Springer-Verlag, pp. 359378.
Shehtman, V., & Skvortsov, D. (1990). Semantics of non-classical first-order predicate logics. In Petkov, P. P., editor. Mathematical Logic. New York: Plenum Press, pp. 105116.

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