Keywords
lntuitionism
Sentential connectives
lntermediate logics
How to Cite
Societal impact
Abstract
We propose and study a general notion of intuitionistic sentential connective in the semantical context of Kripke models. This includes the ordinary connectives of the Heyting calculus as well as the connectives introduced by Gabbay. The resulting extensions of the intuitionistic propositional calculus preserve the paradigms of intuitionism, for example, the disjunction property. We study in detail several specific connectives, providing for them complete axiomatizations that extend the Heyting calculus conservatively. These axiomatizations may be utilized to give new proofs of known facts about intermediate logics, such as Dummett’s logic for Kripke models over linear orders.
References
Bowen, K. A. (1971). An extension of the intuitionistic propositional calculus. Indagationes Mathematicae, 33, 287–294.
Calcedo, X. (1990). Elementos de lógica y calculabilidad. Segunda edición. Una Empresa Docente.
Van Dalen, D. (1983). Logic and structure. Second edition. Springer-Verlag.
Van Dalen, D. (1986). Intuitionistic logic. En M. Gabbay & F. Guenthner (eds.), Handbook of Philosophical Logic, cap. 4. Reidel.
Dummett, M. A. (1959). A propositional calculus with a denumerable matrix. Journal of Symbolic Logic, 24, 96–107.
Fitting, M. C. (1969). Intuitionistic logic, model theory and forcing. North-Holland.
Freyd, P. (1972). Aspects of topoi. Bulletin of the Australian Mathematical Society, 7, 1–76.
Gabbay, D. M. (1977). On some new intuitionistic propositional connectives. Studia Logica, 36, 127–139.
Gabbay, D. M. (1981). Semantical investigations in Heyting intuitionistic logic. Reidel.
Goldblatt, R. (1984). Topoi: the categorical analysis of logic. North-Holland.
Iovino, J. (1988). Conectivos intuicionistas y lógica proposicional intuicionista de segundo orden. Preprint, Universidad de los Andes.
Kaminski, M. (1988). Nonstandard connectives of intuitionistic propositional logic. Notre Dame Journal of Formal Logic, 29(3), 309–331.
Kirk, R. E. (1980). A characterization of the classes of finite frames that are adequate for intuitionistic logic. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 26(6), 197–501.
Kripke, S. (1965). Semantical analysis of intuitionistic logic. En J. Crossley & M. Dummett (eds.), Formal systems and recursive functions. North-Holland, Amsterdam, 92–130.
Lindström, P. (1969). First-order predicate logic with generalized quantifiers. Theoria, 32, 186–195.
López-Escobar, E. G. K. (1985). On intuitionistic sentential connectives. Revista Colombiana de Matemáticas, 19, 117–130.
Maksimova, L. (1972). Pretabular superintuitionistic logics. Algebra i Logika, 11, 558–570.
McCullough, D. P. (1971). Logical connectives of intuitionistic propositional logic. Journal of Symbolic Logic, 36, 5–20.
Rauzer, C. (1974). A formalization of propositional calculus of HB logic. Studia Logica, 33, 22–33.
Troelstra, A. S. (1981). On a second-order propositional operation in intuitionistic logic. Studia Logica, 40, 113–140.
Zalamea, F. (1986). Conectivos intuicionistas. Preprint, Universidad de los Andes.
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