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  • ARNON AVRON (a1) and YONI ZOHAR (a1)

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.

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Anderson, A. R. & Belnap, N. D. (1975). Entailment: The Logic of Relevance and Necessity, Vol. I. Princeton, NJ: Princeton University Press.
Arieli, O. & Avron, A. (1998). The value of the four values. Artificial Intelligence, 102(1), 97141.
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. 91129.
Arieli, O., Avron, A., & Zamansky, A. (2011). Maximal and premaximal paraconsistency in the framework of three-valued semantics. Studia Logica, 97(1), 3160.
Asenjo, F. G. (1966). A calculus of antinomies. Notre Dame Journal of Formal Logic, 7, 103106.
Avron, A. (1986). On an implication connective of RM. Notre Dame Journal of Formal Logic, 27, 201209.
Avron, A. (2007). Non-deterministic semantics for logics with a consistency operator. Journal of Approximate Reasoning, 45, 271287.
Avron, A. (2016). Rm and its nice properties. In Bimbó, K., editor. J. Michael Dunn on Information Based Logics. Cham: Springer International Publishing, pp. 1543.
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. 8594.
Avron, A. & Lev, I. (2005). Non-deterministic multi-valued structures. Journal of Logic and Computation, 15, 241261. 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.
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. 227304.
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. 118123.
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), 10311069.
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. 194.
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. 193.
da Costa, N. (1974). On the theory of inconsistent formal systems. Notre Dame Journal of Formal Logic, 15, 497510.
Dummett, M. (1959). A propositional calculus with denumerable matrix. Journal of Symbolic Logic, 24, 97106.
Dunn, M. & Restall, G. (2002). Relevance logic. In Gabbay, D. and Guenthner, F., editors. Handbook of Philosophical Logic, Vol. 6. Dordrecht: Kluwer, pp. 1128.
Ertola, R., Esteva, F., Flaminio, T., Godo, L., & Noguera, C. (2015). Paraconsistency properties in degree-preserving fuzzy logics. Soft Computing, 19(3), 531546.
Hájek, P. (1998). Metamathematics of fuzzy logic, Vol. 4. Dordrecht: Springer Science & Business Media.
Kleene, S. C. (1938). On notation for ordinal numbers. The Journal of Symbolic Logic, 3, 150155.
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. 5363. Available at
Lahav, O. (2013). Studying sequent systems via non-deterministic multiple-valued matrices. In International Symposium on Multiple-Valued Logic, Vol. 9, pp. 575595.
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, 5177.
Marcos, J. (2005). On negation: Pure local rules. Journal of Applied Logic, 3(1), 185219.
Priest, G. (1979). The logic of paradox. Journal of Philosophical Logic, 8(1), 219241.
Urquhart, A. (2001). Many-valued logic. In Gabbay, D. and Guenthner, F., editors. Handbook of Philosophical Logic, Second Edition, Vol. II. Dordrecht: Kluwer, pp. 249295.
Zohar, Y. (2018). Gentzen-type Proof Systems for Non-classical Logics, Ph.D. Thesis, Tel Aviv University.
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The Review of Symbolic Logic
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