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REXPANSIONS OF NONDETERMINISTIC MATRICES AND THEIR APPLICATIONS IN NONCLASSICAL LOGICS

Published online by Cambridge University Press:  26 October 2018

ARNON AVRON  and
YONI ZOHAR
ARNON AVRON*
Affiliation:
School of Computer Science, Tel Aviv University
YONI ZOHAR*
Affiliation:
School of Computer Science, Tel Aviv University
*
*SCHOOL OF COMPUTER SCIENCE TEL AVIV UNIVERSITY TEL AVIV, ISRAEL E-mail: aa@cs.tau.ac.il
†SCHOOL OF COMPUTER SCIENCE TEL AVIV UNIVERSITY TEL AVIV, ISRAEL E-mail: yoni.zohar@cs.tau.ac.il
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Abstract

The operations of expansion and refinement on nondeterministic matrices (Nmatrices) are composed to form a new operation called rexpansion. Properties of this operation are investigated, together with their effects on the induced consequence relations. Using rexpansions, a semantic method for obtaining conservative extensions of (N)matrix-defined logics is introduced and applied to fragments of the classical two-valued matrix, as well as to other many-valued matrices and Nmatrices. The main application of this method is the construction and investigation of truth-preserving ¬-paraconsistent conservative extensions of Gödel fuzzy logic, in which ¬ has several desired properties. This is followed by some results regarding the relations between the constructed logics.

Keywords

03B5003B5203B5303B60non-deterministic matricesnon-classical logicsfuzzy logicparaconsistent logics
Type
Research Article
Information
The Review of Symbolic Logic , Volume 12 , Issue 1 , March 2019 , pp. 173 - 200
Copyright
Copyright © Association for Symbolic Logic 2018 

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References

BIBLIOGRAPHY

Anderson, A. R. & Belnap, N. D. (1975). Entailment: The Logic of Relevance and Necessity, Vol. I. Princeton, NJ: Princeton University Press.Google Scholar
Arieli, O. & Avron, A. (1998). The value of the four values. Artificial Intelligence, 102(1), 97–141.CrossRefGoogle Scholar
Arieli, O. & Avron, A. (2015). Three-valued paraconsistent propositional logics. In Beziau, J.-Y., Chakraborty, M., and Dutta, S., editors. New Directions in Paraconsistent Logic: 5th WCP, Kolkata, India, February 2014. Springer, New Delhi, India, pp. 91–129.CrossRefGoogle Scholar
Arieli, O., Avron, A., & Zamansky, A. (2011). Maximal and premaximal paraconsistency in the framework of three-valued semantics. Studia Logica, 97(1), 31–60.CrossRefGoogle Scholar
Asenjo, F. G. (1966). A calculus of antinomies. Notre Dame Journal of Formal Logic, 7, 103–106.CrossRefGoogle Scholar
Avron, A. (1986). On an implication connective of RM. Notre Dame Journal of Formal Logic, 27, 201–209.CrossRefGoogle Scholar
Avron, A. (2007). Non-deterministic semantics for logics with a consistency operator. Journal of Approximate Reasoning, 45, 271–287.CrossRefGoogle Scholar
Avron, A. (2016). Rm and its nice properties. In Bimbó, K., editor. J. Michael Dunn on Information Based Logics. Cham: Springer International Publishing, pp. 15–43.CrossRefGoogle Scholar
Avron, A., Konikowska, B., & Zamansky, A. (2012). Modular construction of cut-free sequent calculi for paraconsistent logics. In 2012 27th Annual IEEE Symposium on Logic in Computer Science. Washington, DC: IEEE Computer Society, pp. 85–94.CrossRefGoogle Scholar
Avron, A. & Lev, I. (2005). Non-deterministic multi-valued structures. Journal of Logic and Computation, 15, 241–261. Conference version: Avron, A. & Lev. I. (2001). Canonical propositional Gentzen-type systems. In International Joint Conference on Automated Reasoning, IJCAR 2001. Proceedings, LNAI 2083. Springer, pp. 529–544.CrossRefGoogle Scholar
Avron, A. & Zamansky, A. (2011). Non-deterministic semantics for logical systems: A survey. In Gabbay, D. and Guenther, F., editors. Handbook of Philosophical Logic, Vol. 16. Dordrecht: Springer, pp. 227–304.CrossRefGoogle Scholar
Avron, A. & Zohar, Y. (2017). Non-deterministic matrices in action: Expansions, refinements, and rexpansions. In 2017 IEEE 47th International Symposium on Multiple-Valued Logic (ISMVL). Washington, DC: IEEE Computer Society, pp. 118–123.CrossRefGoogle Scholar
Bou, F., Esteva, F., Font, J. M., Gil, A. J., Godo, L., Torrens, A., & Verdú, V. (2009). Logics preserving degrees of truth from varieties of residuated lattices. Journal of Logic and Computation, 19(6), 1031–1069.CrossRefGoogle Scholar
Carnielli, W. A. & Marcos, J. (2002). A taxonomy of C-systems. In Carnielli, W. A., Coniglio, M. E., and D’Ottaviano, I. M. L., editors. Paraconsistency: The logical way to the inconsistent. Lecture Notes in Pure and Applied Mathematics, Vol. 228. New York: Marcel Dekker, pp. 1–94.CrossRefGoogle Scholar
Carnielli, W., Coniglio, M., & Marcos, J. (2007). Logics of formal inconsistency. In Gabbay, D. and Guenthner, F., editors. Handbook of Philosophical Logic, Second Edition, Vol. 14. Dordrecht: Springer, pp. 1–93.Google Scholar
da Costa, N. (1974). On the theory of inconsistent formal systems. Notre Dame Journal of Formal Logic, 15, 497–510.CrossRefGoogle Scholar
Dummett, M. (1959). A propositional calculus with denumerable matrix. Journal of Symbolic Logic, 24, 97–106.CrossRefGoogle Scholar
Dunn, M. & Restall, G. (2002). Relevance logic. In Gabbay, D. and Guenthner, F., editors. Handbook of Philosophical Logic, Vol. 6. Dordrecht: Kluwer, pp. 1–128.Google Scholar
Ertola, R., Esteva, F., Flaminio, T., Godo, L., & Noguera, C. (2015). Paraconsistency properties in degree-preserving fuzzy logics. Soft Computing, 19(3), 531–546.CrossRefGoogle Scholar
Hájek, P. (1998). Metamathematics of fuzzy logic, Vol. 4. Dordrecht: Springer Science & Business Media.CrossRefGoogle Scholar
Kleene, S. C. (1938). On notation for ordinal numbers. The Journal of Symbolic Logic, 3, 150–155.CrossRefGoogle Scholar
Kulicki, P. & Trypuz, R. (2012). Doing the right things: Trivalence in deontic action logic. In Egre, P. and Ripley, D., editors. Trivalent Logics and their Applications, Proceedings of ESSLLI 2012 Workshop, pp. 53–63. Available at http://paulegre.free.fr/TrivalentESSLLI/esslli_trivalent_proceedings1.pdf.Google Scholar
Lahav, O. (2013). Studying sequent systems via non-deterministic multiple-valued matrices. In International Symposium on Multiple-Valued Logic, Vol. 9, pp. 575–595.Google Scholar
Łukasiewicz, J. (1930). Philosophische bemerkungen zu mehrwertigen systemen der aussagenlogik. Comptes Rendus de la Siciete des Sciences et des Letters de Varsovie, ct.iii 23, 51–77.Google Scholar
Marcos, J. (2005). On negation: Pure local rules. Journal of Applied Logic, 3(1), 185–219.CrossRefGoogle Scholar
Priest, G. (1979). The logic of paradox. Journal of Philosophical Logic, 8(1), 219–241.CrossRefGoogle Scholar
Urquhart, A. (2001). Many-valued logic. In Gabbay, D. and Guenthner, F., editors. Handbook of Philosophical Logic, Second Edition, Vol. II. Dordrecht: Kluwer, pp. 249–295.CrossRefGoogle Scholar
Zohar, Y. (2018). Gentzen-type Proof Systems for Non-classical Logics, Ph.D. Thesis, Tel Aviv University.Google Scholar