On the ‘classical’ operations in three-valued logics
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28.09.2015
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Abstract
The general aim of the present paper is to provide the analysis of the connection between proof-theoretical and functional properties of certain logical matrices. To be more precise, we consider the class of three-valued matrices that induce the classical consequence relation and show that their operations always constitute a subset of one of the maximal classes of functions, which preserve non-trivial equivalence relations. We use a matrix with the single designated value as a sample for in-depth analysis, and generalize the results to suit other cases. Furthermore, on the basis of obtained results we conclude the paper with methodological considerations concerning the nature and interpretation of the truth-values in logical matrices.
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How to Cite
Devyatkin L. Y. On the ‘classical’ operations in three-valued logics // Logicheskie Issledovaniya / Logical Investigations. 2015. VOL. 21. № 2. C. 61-69.
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Papers
References
Devyatkin, L.Yu. “Three-valued matrices with classical consequence relation for an arbitrary propositional language”, Logical Investigations, 2014, vol. 20, pp. 248–254.
Bochvar, D.A. “Ob odnom trekhznachnom ischislenii i ego primenenii k analizu paradoksov klassicheskogo rasshirennogo funktsional’nogo ischisleniya” [On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus], Matematiceskij Sbornik [Sbornik: Mathematics], 1938, vol. 4, no 46, pp. 287–308. (In Russian)
Church, A. Introduction to mathematical logic. Princeton, 1956, p. 117.
Jablonskij, S.V. “Funktsional’nye postroeniya v k-znachnoi logike” [Functional constructions in many-valued logics], Sbornik statei po matematicheskoi logike i ee prilozheniyam k nekotorym voprosam kibernetiki [Collected papers on mathematical logic and its applications to some problems of cybernetics], Tr. MIAN USSR, 1958, vol. 51, pp. 5-–142. (In Russian)
Jablonskij, S.V., Gavrilov, G.P., Kudryavtsev, V.B. Funktsii algebry logiki i klassy Posta [Functions of the algebra of logic and post classes]. Moscow: Nauka, 1966, 90 pp. (In Russian)
Lau, D. Function Algebras on Finite Sets: Basic Course on Many-Valued Logic and Clone Theory. Springer Science & Business Media, 2006, p. 228.
Malinowski, G. Many-Valued Logics. Oxford: Oxford University Press, 1993, p. 30.
Mal’tsev, A.I. “Iterativnye algebry i mnogoobraziya Posta” [Iterative algebras and Post’s varieties], Algebra i logika [Algebra and Logic], 1966, vol. 5, no 2, pp. 5–24.
Mal’tsev, A.I. Algebraicheskie sistemy [Algebraic systems]. Moscow: Nauka, 1970, p. 62. (In Russian)
Post, E.L. The two-valued iterative systems of mathematical logic, London; H. Milford: Oxford University Press, 1941, 122 pp.
Prelovskiy, N.N. “Cardinality of sets of closed functional classes in weak 3-valued logics”. Logical Investigations, 2013, vol. 19, pp. 334–343.
Raca, M.F. “O klasse funktsii trekhznachnoi logiki, sootvetstvuyushchem pervoi matritse Yas’kovskogo” [On the class of functions of three-valued logic corresponding to Ja_skowski’s first matrix], Problemy kibernetiki [Problems of cybernetics], 1969, no 21, pp. 185—214. (In Russian)
Rescher, N. Many-Valued Logic. New York (McGraw-Hill). 1969, p. 116.
Tomova, N.E. Estestvennye trekhznachnye logiki: Funktsional’nye svoistva i otnosheniya [Natural three-valued logics: functional properties and relations]. Moscow. IFRAN Publ. 2012. 89 pp. (In Russian)
Wojcicki, R. Lectures on Propositional Calculi. Pub. House of the Polish Academy of Sciences. 1984, p. 99.
Bochvar, D.A. “Ob odnom trekhznachnom ischislenii i ego primenenii k analizu paradoksov klassicheskogo rasshirennogo funktsional’nogo ischisleniya” [On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus], Matematiceskij Sbornik [Sbornik: Mathematics], 1938, vol. 4, no 46, pp. 287–308. (In Russian)
Church, A. Introduction to mathematical logic. Princeton, 1956, p. 117.
Jablonskij, S.V. “Funktsional’nye postroeniya v k-znachnoi logike” [Functional constructions in many-valued logics], Sbornik statei po matematicheskoi logike i ee prilozheniyam k nekotorym voprosam kibernetiki [Collected papers on mathematical logic and its applications to some problems of cybernetics], Tr. MIAN USSR, 1958, vol. 51, pp. 5-–142. (In Russian)
Jablonskij, S.V., Gavrilov, G.P., Kudryavtsev, V.B. Funktsii algebry logiki i klassy Posta [Functions of the algebra of logic and post classes]. Moscow: Nauka, 1966, 90 pp. (In Russian)
Lau, D. Function Algebras on Finite Sets: Basic Course on Many-Valued Logic and Clone Theory. Springer Science & Business Media, 2006, p. 228.
Malinowski, G. Many-Valued Logics. Oxford: Oxford University Press, 1993, p. 30.
Mal’tsev, A.I. “Iterativnye algebry i mnogoobraziya Posta” [Iterative algebras and Post’s varieties], Algebra i logika [Algebra and Logic], 1966, vol. 5, no 2, pp. 5–24.
Mal’tsev, A.I. Algebraicheskie sistemy [Algebraic systems]. Moscow: Nauka, 1970, p. 62. (In Russian)
Post, E.L. The two-valued iterative systems of mathematical logic, London; H. Milford: Oxford University Press, 1941, 122 pp.
Prelovskiy, N.N. “Cardinality of sets of closed functional classes in weak 3-valued logics”. Logical Investigations, 2013, vol. 19, pp. 334–343.
Raca, M.F. “O klasse funktsii trekhznachnoi logiki, sootvetstvuyushchem pervoi matritse Yas’kovskogo” [On the class of functions of three-valued logic corresponding to Ja_skowski’s first matrix], Problemy kibernetiki [Problems of cybernetics], 1969, no 21, pp. 185—214. (In Russian)
Rescher, N. Many-Valued Logic. New York (McGraw-Hill). 1969, p. 116.
Tomova, N.E. Estestvennye trekhznachnye logiki: Funktsional’nye svoistva i otnosheniya [Natural three-valued logics: functional properties and relations]. Moscow. IFRAN Publ. 2012. 89 pp. (In Russian)
Wojcicki, R. Lectures on Propositional Calculi. Pub. House of the Polish Academy of Sciences. 1984, p. 99.