Abstract
The main goal of this paper is to explain the link between the algebraic models and the Kripke-style models for certain classes of propositional non-classical logics. We consider logics that are sound and complete with respect to varieties of distributive lattices with certain classes of well-behaved operators for which a Priestley-style duality holds, and present a way of constructing topological and non-topological Kripke-style models for these types of logics. Moreover, we show that, under certain additional assumptions on the variety of the algerabic models of the given logics, soundness and completeness with respect to these classes of Kripke-style models follows by using entirely algebraical arguments from the soundness and completeness of the logic with respect to its algebraic models.
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References
A. R. Anderson and N. D. Belnap, Entailment. The Logic of Relevance and Necessity, volume I, Princeton University Press, 1975.
C. Brink and I. Rewitzky, Power Structures and Program Semantics. Studies of Logic, Language and Information (to appear).
S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Graduate Texts in Mathematics, Springer, 1981.
E. W. Dijkstra, ‘Guarded commands, nondeterminacy and formal derivation of programs’ Communications of the ACM 18 (1975), 8, 453-458.
E. W. Dijkstra, A Discipline of Programming, Prentice Halls, Englewood Cliffs, New Jersey, 1976.
B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, Cambridge University Press, 1990.
M. Gehrke and B. JÓnsson, ‘Bounded distributive lattices with operators’ Math. Japonica 40 (1994), 2, 207-215.
R. Goldblatt, ‘Varieties of complex algebras’ Annals of Pure and Applied Logic 44 (1989), 3, 153-301.
L. Iturrioz and E. Orlowska, ‘A Kripke-style and relational semantics for logics based on Lukasiewicz algebras’ Conference in honour of J. Lukasiewicz, Dublin, 1996.
B. JÓnsson and A. Tarski, ‘Boolean algebras with operators. Part I’ American Journal of Mathematics 73 (1951), 891-939.
B. JÓnsson and A. Tarski, ‘Boolean algebras with operators. Part II’ American Journal of Mathematics 74 (1952), 127-162.
E. J. Lemmon, ‘Algebraic semantics for modal logics I’ Journal of Symbolic Logic 31 (1966), 1, 46-65.
E. J. Lemmon, ‘Algebraic semantics for modal logics II’ Journal of Symbolic Logic 31 (1966), 2, 191-218.
H. A. Priestley, ‘Representation of distributive lattices by means of ordered Stone spaces’ Bull. London Math. Soc. 2 (1970), 186-190.
H. A. Priestley, ‘Ordered topological spaces and the representation of distributive lattices’ Proc. London Math. Soc. 3 (1972), 507-530.
I. Rewitzky and C. Brink, ‘Unification of four versions of program semantics’ Formal Aspects of Computing (to appear).
V. Sofronie-Stokkermans, Fibered Structures and Applications to Automated Theorem Proving in Certain Classes of Finitely-Valued Logics and to Modeling Interacting Systems, PhD thesis, RISC-Linz, J. Kepler University Linz, Austria, 1997.
V. Sofronie-Stokkermans, ‘Automated theorem proving by resolution for finitely-valued logics based on distributive lattices with operators’ Multiple-Valued Logic — An International Journal (to appear).
V. Sofronie-Stokkermans, ‘Duality and canonical extensions of bounded distributive lattices with operators, and applications to the semantics of non-classical logics I’ Studia Logica 64 (2000), 1, 93-132.
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Sofronie-Stokkermans, V. Duality and Canonical Extensions of Bounded Distributive Lattices with Operators, and Applications to the Semantics of Non-Classical Logics II. Studia Logica 64, 151–172 (2000). https://doi.org/10.1023/A:1005228629540
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DOI: https://doi.org/10.1023/A:1005228629540