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Abstract
The subobject classifier of a topos is its "object ot truth values" and its morphisms yield the propositional connectives of its inner logic. Continuing previous work for Kripke models, we study connectives on sheaves over topological spaces. These are determined by certain operations in open sets. Monadic and Heyting connectives form a functionally complete basis in any espacial topos. In particular, a single new connective generates all connectives implicit in Heyting's intuitionistic trivalent logic. We provide a complete axiomatization for the corresponding intermediate modal logic. We consider also connectives invariant under local homeomorphisms, and the global uniform choosings of connectives on distinct topological spaces. Toe main results generaliz.e readily to sheaves over complete Heyting algebras.
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References
Caicedo, X. 1995a: Lógica de los haces de estructuras. Rev. Acad. Colomb. Cienc. 21(74):569-585.
Caicedo, X. 1995b: Investigaciones sobre los conectivos intuicionistas. Rev. Acad. Colomb. Cienc. 21(75):705-716.
Cigonoli, R., D'Ottaviano, L. & D. Mundici 1995: Álgebras das lógicas de Lukasiewicz. CLE, Unicamp, Brasil.
Fraissé, R. 1954: Sur quelques classifications des systemes de relations. Université d'Alger, Publications Scientifiques, Série A, 1, 35-182.
Freyd, P. 1972: Aspects of Topoi. Bull. Austral. Math. Soc. 7:1-76.
Freyd, P. & A. Scedrov 1990: Categories, Allegories. North Holland, Amsterdam.
Godement, R. 1958: Topologie Algébrique et Théorie des Fais-ceaux. Hermann, Paris.
Goldblatt, R. 1984: Topoi, the Categorical Analysis of Logic. North Holland, Amsterdam.
Goldblatt, R. 1987: Grothendieck topologies as geometric modalities. Zeitschr. f. math. Logik und Grundl. d. Math., d. 27:495-529.
Heyting, A. 1930: Die formalen Regen der intuitionistchen Logik. Sitzungsbericthe de Preussischen Akademie de Wissenschaften, Physikalisch-mathmeatische Klasse, 42-56
Johnstone, P. T. 1977: Topos Theory. Academic Press, New York.
Johnstone, P. T. 1982: Stone Spaces. Cambridge University Press.
Kripke, S. 1965: Semantical Analyisis of Intuitionistic logic I. Formal systems and recursive functions, (Crossley and Dummett, eds.). North Holland, Amsterdam.
Lukasiewicz, J. 1920: O logique Trójwarkościowej. Ruch Filozoficzny 6:170-171.
Makkai, M. & G. E. Reyes 1977: First order categorical Logic. Lecture Notes in Math. 611, Springer Verlag.
MacLane, S. 1971: Categories for the Working Mathematician. Springer Verlag.
MacLane, S. & I. Moerdijk 1992: Sheaves in Geometry and Logic, Springer Verlag.
Monteiro, L. 1964: Sur les Algèbres de Heyting trivalentes. Notas de Lógica Matemática No. 19, Universidad Nacional del Sur, Bahia Blanca, Argentina.
Monteiro, L. 1970: Les algebres de Heyting et the Lukasie- vicz trivalentes. Notre Dame Journal of Formal Logic, XI, 4: 453-466.
Reyes, G. E. 1974: From sheaves to logic. MAA Studies in Algebraic Logic.
Sette, A. M. & X. Caicedo 1993: Equivalencia elementar entre feixes. Proceedings of the IX Latin American Symposium on Mathematical Logic, Notas de Lógica Matemática, 38:129-141.
Stone, M. H. 1937. Topological representation of distributive lattices and Brouwerian logics. Cas. Mat. Phys. 67:1-25
Tennison, B. R. 1975:Sheaf Theory. London Math. Soc. Lect. Notes 20. Cambridge University Press.
van Dalen, D. 1983: Logic and Structure. Springer Verlag.
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