Truth Values in t-norm based Systems Many-valued FUZZY Logic

Usó-Doménech J.L.¹, Nescolarde-Selva J.^1, and Perez-Gonzaga S.¹

¹Department of Applied Mathematics, University of Alicante, Alicante, Spain

Cite this paper:
Usó-Doménech J.L., Nescolarde-Selva J. and Perez-Gonzaga S.. Truth Values in t-norm based Systems Many-valued FUZZY Logic. American Journal of Systems and Software. 2014; 2(6):139-145. doi: 10.12691/ajss-2-6-1

Abstract

In t-norm based systems many-valued logic, valuations of propositions form a non-countable set: interval [0,1]. In addition, we are given a set E of truth values p, subject to certain conditions, the valuation v is v=V(p), V reciprocal application of E on [0,1]. The general propositional algebra of t-norm based many-valued logic is then constructed from seven axioms. It contains classical logic (not many-valued) as a special case. It is first applied to the case where E=[0,1] and V is the identity. The result is a t-norm based many-valued logic in which contradiction can have a nonzero degree of truth but cannot be true; for this reason, this logic is called quasi-paraconsistent.

Keywords:
contradiction denier logic coordinations propositions truth value

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References:

[1]	Arruda A. I., Chuaqui R., da Costa N. C. A. (eds.) 1980 . Mathematical Logic in Latin America. North-Holland Publishing Company, Amsterdam, New York, Oxford.
[2]	Asenjo, F.G. 1966. A calculus of antinomies. Notre Dame Journal of Formal Logic 7. Pp 103-105.
[3]	Avron, A. 2005. Combining classical logic, paraconsistency and relevance. Journal of Applied Logic. 3(1). pp 133-160.
[4]	Belnap, N.D. 1977. How a computer should think. In Contemporary aspects of philosophy, ed. G. Ryle, 30-55. Oriel Press. Boston.
[5]	Béziau J.Y. 1997. What is many-valued logic? Proceedings of the 27th International Symposium on Multiple-Valued Logic, IEEE Computer Society, Los Alamitos, pp. 117-121.
[6]	Bueno, O. 1999. True, Quasi-true and paraconsistency. Contemporary mathematics. 39. pp 275-293.
[7]	Bueno, O. 2010. Philosophy of Logic. In Fritz Allhoff. Philosophies of the Sciences: A Guide. John Wiley & Sons. p. 55.
[8]	Carnielli, W.A., Coniglio and M. Lof D'ottaviano, I.M. 2002. Paraconsistency: The Logical Way to the Inconsistent. Marcel Dekker, Inc. New York.
[9]	Carnielli, W. and Marcos, J. 2001. Ex contradictione non sequitur quodlibet. Proc. 2nd Conf. on Reasoning and Logic (Bucharest, July 2000).
[10]	Castiglioni, J. L. and Ertola Biraben, R. C. 2013. Strict paraconsistency of truth-degree preserving intuitionistic logic with dual negation. Logic Journal of the IGPL. Published online August 11, 2013.
[11]	Cignoli, R. L. O., D'Ottaviano, I, M. L. and Mundici, D., 2000. Algebraic Foundations of Many-valued Reasoning. Kluwer.
[12]	Da Costa, N., Nuccetelli, S., Schutte, O. and Bueno, O. 2010. “Paraconsistent Logic” (with (eds.), A Companion to Latin American Philosophy (Oxford: Wiley-Blackwell), pp. 217-229.
[13]	Dunn, J.M. 1976. Intuitive semantics for first-degree entailments and coupled trees. Philosophical Studies 29. pp 149-168.
[14]	Fisher, J. 2007. On the Philosophy of Logic. Cengage Learning. pp. 132-134.
[15]	Malinowski, G. 2001. Many-Valued Logics, in Goble, Lou. Ed., The Blackwell Guide to Philosophical Logic. Blackwell.
[16]	Miller, D, M. and Thornton, M. A. 2008. Multiple valued logic: concepts and representations. Synthesis lectures on digital circuits and systems 12. Morgan & Claypool Publishers.
[17]	Nescolarde-Selva, J. and Usó-Domènech, J. L. 2013a. Semiotic vision of ideologies. Foundations of Science.
[18]	Nescolarde-Selva, J. and Usó-Domènech, J. L. 2013b. Reality, Systems and Impure Systems. Foundations of Science.
[19]	Nescolarde-Selva, J. and Usó-Doménech, J. 2013c. Topological Structures of Complex Belief Systems. Complexity. pp 46-62.
[20]	Nescolarde-Selva, J. and Usó-Doménech, J. 2013d. Topological Structures of Complex Belief Systems (II): Textual materialization.
[21]	Priest G. and Woods, J. 2007. Paraconsistency and Dialetheism. The Many Valued and Nonmonotonic Turn in Logic. Elsevier.
[22]	Usó-Domènech, J.L. and Nescolarde-Selva, J.A. 2012. Mathematic and Semiotic Theory of Ideological Systems. A systemic vision of the Beliefs. LAP LAMBERT Academic Publishing. Saarbrücken. Germany.