Abstract
In this paper we describe a procedure for developing models and associated proof systems for two styles of paraconsistent logic. We first give an Urquhart-style representation of bounded not necessarily discrete lattices using (grill, cogrill) pairs. From this we develop Kripke semantics for a logic permitting 3 truth values: true, false and both true and false. We then enrich the lattice by adding a unary operation of negation that is involutive and antimonotone and show that the representation may be extended to these lattices. This yields Kripke semantics for a nonexplosive 3-valued logic with negation.
This work was performed within the framework of the COST Action 274, entitled: ”Theory and Applications of Relational Structures as Knowledge Instruments” (www.tarski.org). Both authors were supported by a NATO Science and Technology Collaborative Linkages Grant.
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References
Allwein, G., Dunn, M.: Kripke models for linear logic. Journal of Symbolic Logic 58, 514–545 (1993)
Allwein, G., MacCaull, W.: A Kripke semantics for the logic of Gelfand quantales. Studia Logica 61, 1–56 (2001)
Arieli, O., Avron, A.: A model-theoretic approach for recovering consistent data from inconsistent knowledge-bases. Journal of Automated Reasoning 22, 263–309 (1999)
Belnap, N.: A useful four-valued logic. In: Epstein, G., Dunn, J.M. (eds.) Modern Uses of Multiple-Valued Logic, pp. 7–37. Reidel, Dordrecht (1977)
da Costa, N.: On the theory of inconsistent formal systems. Notre Dame Journal of Formal Logic 15, 497–510 (1974)
Dunn, M.: Gaggle Theory: An abstraction of Galois connections and residuation, with applications to negation, implication, and various logical operators. In: van Eijck, J. (ed.) JELIA 1990. LNCS (LNAI), vol. 478, pp. 31–51. Springer, Heidelberg (1991)
Fitting, M.: Bilattices and the semantics of logic programming. Journal of Logic Programming 11, 91–116 (1991)
Ginsberg, M.: Multivalued logics: A uniform approach to reasoning in AI. Computer Intelligence 4, 256–316 (1988)
MacCaull, W., Orłowska, E.: Correspondence results for relational proof systems with application to the Lambek Calculus. Studia Logica 71, 389–414 (2002)
MacCaull, W., Vakarelov, D.: Lattice-based paraconsistent logic. In: Düntsch, I., Winter, M. (eds.) Proceedings of RelMiCS 8, the 8th International Seminar in Relational Methods in Computer Science, pp. 155–162 (2005)
Orłowska, E., Vakarelov, D.: Lattices with modal operators and lattice based modal logics (to appear)
Rauszer, C.: Semi-Boolean algebras and their applications to intuitionistic logic with dual operations. Fundamenta Informaticae 83, 219–250 (1974)
Urquhart, A.: A topological representation for lattices. Algebra Universalis 8, 45–58 (1978)
Vakarelov, D.: Semi-Boolean algebras and semantics for HB-predicate logic. Bull. Acad. Polon. Sci. Ser. Math. Phys. 22, 1087–1095 (1974)
Vakarelov, D.: Consistency, completeness and negation. In: Priest, G., Routley, R., Norman, J. (eds.) Paraconsistent Logic, Essays on the Inconsistent. Philosophia Verlag (1989)
Vakarelov, D.: Intuitive semantics for some three-valued logics connected with information, contrariety and subcontrariety. Studia Logica 48, 565–575 (1989)
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MacCaull, W., Vakarelov, D. (2006). Lattice-Based Paraconsistent Logic. In: MacCaull, W., Winter, M., Düntsch, I. (eds) Relational Methods in Computer Science. RelMiCS 2005. Lecture Notes in Computer Science, vol 3929. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11734673_14
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DOI: https://doi.org/10.1007/11734673_14
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