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POLYNOMIAL RING CALCULUS FOR MODAL LOGICS: A NEW SEMANTICS AND PROOF METHOD FOR MODALITIES

Published online by Cambridge University Press:  14 September 2010

JUAN C. AGUDELO*
Affiliation:
State University of Campinas—UNICAMP, and Eafit University
WALTER CARNIELLI*
Affiliation:
State University of Campinas—UNICAMP, and SQIG—IT
*
*PH.D. PROGRAM IN PHILOSOPHY, AREA OF LOGIC, IFCH AND GROUP FOR APPLIED AND THEORETICAL LOGIC—CLE, STATE UNIVERSITY OF CAMPINAS—UNICAMP, BRAZIL AND LOGIC AND COMPUTATION RESEARCH GROUP, EAFIT UNIVERSITY, COLOMBIA. E-mail: juancarlos@cle.unicamp.br
DEPARTMENT OF PHILOSOPHY AND GROUP FOR APPLIED AND THEORETICAL LOGIC, CENTRE FOR LOGIC, EPISTEMOLOGY AND THE HISTORY OF SCIENCE—CLE, STATE UNIVERSITY OF CAMPINAS—UNICAMP, BRAZIL AND SQIG—INSTITUTE OF TECHNOLOGY, LISBON, PORTUGAL. E-mail:walter.carnielli@cle.unicamp.br

Abstract

A new (sound and complete) proof style adequate for modal logics is defined from the polynomial ring calculus (PRC). The new semantics not only expresses truth conditions of modal formulas by means of polynomials, but also permits to perform deductions through polynomial handling. This paper also investigates relationships among the PRC here defined, the algebraic semantics for modal logics, equational logics, the Dijkstra–Scholten equational-proof style, and rewriting systems. The method proposed is throughly exemplified for S5, and can be easily extended to other modal logics.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2010

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References

BIBLIOGRAPHY

Avron, A., & Zamansky, A. (2007). Generalized non-deterministic matrices and (n,k)-ary quantifiers. In Artemov, S., and Nerode, A., editors. Proceedings of the Symposium on Logical Foundations of Computer Science, LNCS 4514. Berlin/Heidelberg: Springer, pages 2640.Google Scholar
Blackburn, P., & Benthem, J. v. (2006). Modal logic: A semantic perspective. In Blackburn, P., van Benthem, J., and Wolter, F., editors. Handbook of Modal Logic. Amsterdam: Elsevier North-Holland, pp. 182.Google Scholar
Blackburn, P., de Rijke, M., & Venema, Y. (2002). Modal Logic. Cambridge, UK: Cambridge University Press.Google Scholar
Bohórquez, J. A. (2008). Intuitionistic logic according to dijkstra’s calculus of equational deduction. Notre Dame Journal of Formal Logic, 49(4), 361384.CrossRefGoogle Scholar
Carnielli, W., & Pizzi, C. (2008). Modalities and Multimodalities. Amsterdam: Springer.CrossRefGoogle Scholar
Carnielli, W. A. (2005). Polynomial ring calculus for many-valued logics. In Werner, B., editor. Proceedings of the 35th International Symposium on Multiple-Valued Logic. Los Alamitos: IEEE Computer Society, 2005, pp. 2025. Preprint available at CLE e-Prints vol 5, n. 3: www.cle.unicamp.br/e-prints/vol_5,n_3,2005.html.Google Scholar
Carnielli, W. A. (2007). Polynomizing: Logic inference in polynomial format and the legacy of Boole. In Magnani, L., and Li, P., editors. Model-Based Reasoning in Science, Technology, and Medicine, Volume 64 of Studies in Computational Intelligence. Berlin/Heidelberg: Springer, pp. 349364.Google Scholar
Carnielli, W. A., & Coniglio, M. E. (2005). Splitting logics. In We Will Show Them! Essays in Honour of Dov Gabbay. London: College Publications, pp. 389414.Google Scholar
Carnielli, W. A., Coniglio, M. E., & Marcos, J. (2007). Logics of formal inconsistency. In Gabbay, D., and Guenthner, F., editors. Handbook of Philosophical Logic (second edition), Vol. 14. Berlin/Heidelberg: Springer, pp. 15107. Preprint available at CLE e-Prints vol 5, n. 1. www.cle.unicamp.br/e-prints/vol_5,n_1,2005.html.Google Scholar
Dershowitz, N., Hsiang, J., Huang, G. S., & Kaiss, D. (2004). Boolean ring satisfiability. In Hoos, Holger H., & Mitchell, David C., editors. Proceedings of the Seventh International Conference on Theory and Applications of Satisfiability Testing (SAT 2004). Berlin/Heidelberg: Springer, pp. 281286.Google Scholar
Dijkstra, E. W., & Scholten, C. S. (1990). Predicate Calculus and Program Semantics. New York: Springer-Verlag.CrossRefGoogle Scholar
Fagin, R., & Vardi, M. Y. (1985). An internal semantics for modal logic. In Sedgewick, Robert, editor. Proceedings of the Seventeenth Annual ACM Symposium on Theory of Computing (1985). New York: Association for Computing Machinery, pp. 305315.Google Scholar
Foret, A. (1988). Rewrite rule systems for modal propositional logic. In Grabowski, J., Lescanne, P., & Wechler, W., editors. Proceedings of the International Workshop on Algebraic and Logic Programming, Volume 343 of Lecture Notes in Computer Science. Berlin/Heidelberg: Springer Verlag, pp. 147156.Google Scholar
Goldblatt, R. (2005). Mathematical modal logic: A view of its evolution. In Gabbay, D. M., and Woods, J., editors. Handbook of the History of Logic, Vol. 6. Amsterdam: Elsevier, pp. 198.Google Scholar
Gries, D., & Schneider, F. B. (1995). Equational propositional logic. Information Processing Letters, 53, 145152.CrossRefGoogle Scholar
Hsiang, J. (1985). Refutational theorem proving using term-rewriting systems. Artificial Intelligence, 25, 255300.CrossRefGoogle Scholar
Hsiang, J., & Huang, G. S. (1997). Some fundamental properties of Boolean ring normal forms. In Du, D., Gu, J., and Pardalos, P. M., editors. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 35. Providence, RI, USA: American Mathematical Society, pp. 587602.Google Scholar
Kuczynski, J.-M. (2007). Does possible world semantics turn all propositions into necessary ones? Journal of Pragmatics, 39(5), 872916.CrossRefGoogle Scholar
Lemmon, E. J. (1966a). Algebraic semantics for modal logics I. Journal of Symbolic Logic, 31(1), 4665.CrossRefGoogle Scholar
Lemmon, E. J. (1966b). Algebraic semantics for modal logics II. Journal of Symbolic Logic, 31(2), 191218.CrossRefGoogle Scholar
Quine, W. V. O. (2006). From a logical point of view: Nine logico-philosophical essays.In Two Dogmas of Empiricism. Harvard University Press, pp. 2046.Google Scholar
Tarski, A. (1968). Equational logic and equational theories of algebra. In Schmidt, H. A., Schütte, K., & Thiele, H. J. editors. Contributions to Mathematical Logic. Amsterdam: North Holland, pp. 275288.Google Scholar
Taylor, W. (1979). Equational logic. Houston Journal of Mathematics, 5, 151.Google Scholar