Generalization of Kalmar’s Proof of Deducibility in Two Valued Propositional Logic into Many Valued Logic
Pure and Applied Mathematics Journal
Volume 6, Issue 2, April 2017, Pages: 71-75
Received: Feb. 13, 2017; Accepted: Mar. 15, 2017; Published: Mar. 22, 2017
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Chubaryan Anahit, Department of Informatics and Applied Mathematics, Yerevan State University and Russian-Armenian University, Yerevan, Armenia
Khamisyan Artur, Department of Informatics and Applied Mathematics, Yerevan State University, Yerevan, Armenia
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This paper focuses on the problem of constructing of some standard Hilbert style proof systems for any version of many valued propositional logic. The generalization of Kalmar’s proof of deducibility for two valued tautologies inside classical propositional logic gives us a possibility to suggest some method for defining of two types axiomatic systems for any version of 3-valued logic, completeness of which is easy proved direct, without of loading into two valued logic. This method i) can be base for direct proving of completeness for all well-known axiomatic systems of k-valued (k≥3) logics and may be for fuzzy logic also, ii) can be base for constructing of new Hilbert-style axiomatic systems for all mentioned logics.
Many-Valued Logics, Hilbert-Style Axiomatic Systems, Completeness of Formal System
To cite this article
Chubaryan Anahit, Khamisyan Artur, Generalization of Kalmar’s Proof of Deducibility in Two Valued Propositional Logic into Many Valued Logic, Pure and Applied Mathematics Journal. Vol. 6, No. 2, 2017, pp. 71-75. doi: 10.11648/j.pamj.20170602.12
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E.Mendelson, Introduction to Mathematical Logic, Van Nostrand, Princeton, 1975.
An.Chubaryan, Relative efficiency of some proof systems for classical propositional logic, Proceedings of NASA RA, Vol.37, N5, 2002, and Journal of CMA (AAS),Vol.37, N5, 2002, 71-84.
A.A.Chubaryan, A.S.Tshitoyan, A.A.Khamisyan, On some proof systems for many-valued logics and on proof complexities in it, (in Russian) Reports of NASA RA, Vol. 116, N2, 2016, 18-24.
J.Lukasiewicz, O Logice Trojwartosciowej, Ruch filoseficzny (Lwow), Vol.5, 1920, 169-171.
E.Post, Introduction to a general theory of elementary propositions, Amer. Journ. Math., Vol.43, 1921, 163-185.
K.Godel, Zum intuitionistishen Aussagen kalkul, Akademie der Wissenshaften in Wien, Mathematische-naturwissenschaftliche Klasse, Auzeiger, Vol.69, 1932, 65-66.
S.Jaskowski, On the rules of suppositions in formal logic, Studia logica, No.1, Warsaw, 1934, 5-32.
S.C.Kleene, On the interpretation of intuitionistic number theory, Journ. Symb. Log. Vol.10, No.4, 1945, 109-123.
D.A. Bochvar, M. Bergmann, On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus, Journal History and Philosophy of Logic, Volume2, Issue1-2, 1981, 87-112.
N.D. Belnap, Auseful four-valued logic, in: Modern Uses of Multiple-valued Logic, Oriel Press., 1977, 35-56.
I.D. Zaslawskij, Symmetrical Konstructive Logic, Press. ASAR, (in Russian), 1978.
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