Von Wright’s truth-logic and around
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09.04.2013
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Abstract
In this paper von Wright’s truth-logic T__ is considered. It seems that it is a De Morgan four-valued logic DM4 (or Belnap’s four-valued logic) with endomorphism e2. In connection with this many other issues are discussed: twin truth operators, a truth-logic with endomorphism g (or logic Tr), the lattice of extensions of DM4, modal logic V2, Craig interpolation property, von Wright–Segerberg’s tense logic W, and so on.
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How to Cite
Karpenko A. Von Wright’s truth-logic and around // Logicheskie Issledovaniya / Logical Investigations. 2013. VOL. 19. C. 39-50.
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References
Anshakov, O.M., and S.V. Rychkov, On the axiomatization of finite-valued logical calculi, Math. USSR Sbornik 51:473–491, 1985.
Anshakov, O.M., and S.V. Rychkov, On finite-valued propositional logical calculi, Notre Dame Journal of Formal Logic 36(4):606–629. 1995.
Belnap, N.D., A useful four-valued logic, in G. Epstein, and J.M. Dunn (eds.), Modern Uses of Multiple-Valued Logic, Reidel, Dordrecht, 1977, pp. 7–37.
Bochvar, D.A., and V.K. Finn, On many-valued logics that permit the formalization of analysis of antinomies, I, in D.A. Bochvar (ed.), Researches on Mathematical Linguistics, Mathematical Logic and Information Languages, NAUKA Publishers, Moscow, 1972, pp. 238–295 (in Russian).
Ermolaeva, N.M., and A.A. Muchnik, A modal extensions of Hao Wang-type of logical calculi, in D.A. Bochvar (ed.), Investigations on Formalized Language and Non-Classical Logics, NAUKA Publishers, Moscow, 1974, pp. 172–193 (in Russian).
Ermolaeva, N.M., and A.A. Muchnik, Modal logics defined by endomorpismus in distributive lattices, in D.A. Bochvar and V.N. Grishin (eds.), Investigations on the Set Theory and Non-classical Logics, NAUKA Publishers, Moscow, 1976, pp. 229–246 (in Russian).
Ermolaeva N.M., and A.A. Muchnik, A pretabular tense logic, in A. I. Mihailov (ed.), Investigations on Non-classical Logics and the Set Theory, NAUKA Publishers, Moscow, 1979, pp. 288–297 (in Russian).
Fitting, M., Bilattices and the theory of truth, Journal of Philosophical Logic 18:225-–256, 1989.
Font, J.M., Belnap’s four-valued logic and De Morgan lattices, Logic Journal of the IGPL 5(3):413–440, 1997.
Font, J.M., and M. Rius, An abstract algebraic logic approach to tetravalent modal logics, The Journal of Symbolic logic 65(2):481–518, 2000.
Ginsberg, M.L., Multivalued logics: A uniform approach to inference in artificial intelligence, Computational Intelligence 4(3):265—315, 1988.
Halbach, V., Axiomatic theories of truth, in Stanford Encyclopedia of Philosophy, ed. E.N. Zalta, 2009, http://plato.stanford.edu/entries/truth-axiomatic/.
Karpenko, A.S., Development of Many-Valued Logic, URSS, Moscow, 2010.
Lewis, C.I., and C.H. Langford, Symbolic Logic, N. Y., 1932 (2nd ed. in 1959).
Lukasiewicz, J., Interpretacja liczbowa teorii zda_n, Ruch Filozoficzny 7:92–93, 1922/1923. (English trans.: A numerical interpretation of theory of propositions, in Lukasiewicz, J., Selected Works, North-Holland & PWN, Amsterdam & Warszawa, 1970, pp. 129–130).
Lukasiewicz, J., A system of modal logic, The Journal of Computing Systems 1:111–149, 1953.
Maksimova, L.L., Interpolation theorems in modal logics and amalgamable varieties of topoboolean algebras, Algebra i Logica 18(5): 556–586. 1979 (in Russian).
Meshi, V.Y., Kripke’s semantics for modal systems including S4.3, Mathematical Notes 15(6):875–884, 1974 (in Russian).
Pavlov, S.A., Logic of falsehood FL4, in Proceedings of the Reseach Logical Seminar of Institute of Philosophy of Russian Academy of Sciences, IFRAN, Moscow, 1994, pp. 14-35 (in Russian).
Post, E.L., Introduction to a general theory of elementary propositions, American Journal of Mathematics 43(3):163–185, 1921. (Repr. in J. van Heijenoort (ed.), From Frege to Godel: A Source Book in Mathematical Logic, 1879—1931, Harvard Univ. Press, Cambridge, 1967, pp. 264–283).
Pynko, A.P., Functional completeness and axiomatizability within Belnap’s four-valued logic and its expansion, Journal of Applied NonClassical Logics 9(1):61–105, 1999.
Scroggs, S.J., Extensions of the Lewis system S5, The Journal of Symbolic Logic 16:112–120, 1951.
Segerberg, K., On the logic of ‘To-morrow’, Theoria 33(1):46–52, 1967.
Segerberg, K., Von Wright’s tense logic, in V.A. Smirnov (ed.), Logical Deduction, NAUKA Publishers, Moscow, 1979, pp. 173–205 (in Russian). (English version in P.A. Schilpp, and L.E. Hahn (eds.), The Philosophy of Georg Henrik von Wright, Open Court, Illinois, 1989, pp. 602–635).
Sobochinski, B., Modal system S4.4, Notre Dame Journal of Formal Logic 5(4):305–312, 1964.
Sobochinski, B., Certain extensions of modal system S4, Notre Dame Journal of Formal Logic 11(3):347–367, 1970.
Von Wright, G.H., And next, Acta Philosopica Fennica 18:293–304, 1965.
Von Wright, G.H., Truth-logics, Logique et Analyse 30(120):311–334, 1987; repr. in Acta Philosopica Fennica 60:71–91, 1996.
Anshakov, O.M., and S.V. Rychkov, On finite-valued propositional logical calculi, Notre Dame Journal of Formal Logic 36(4):606–629. 1995.
Belnap, N.D., A useful four-valued logic, in G. Epstein, and J.M. Dunn (eds.), Modern Uses of Multiple-Valued Logic, Reidel, Dordrecht, 1977, pp. 7–37.
Bochvar, D.A., and V.K. Finn, On many-valued logics that permit the formalization of analysis of antinomies, I, in D.A. Bochvar (ed.), Researches on Mathematical Linguistics, Mathematical Logic and Information Languages, NAUKA Publishers, Moscow, 1972, pp. 238–295 (in Russian).
Ermolaeva, N.M., and A.A. Muchnik, A modal extensions of Hao Wang-type of logical calculi, in D.A. Bochvar (ed.), Investigations on Formalized Language and Non-Classical Logics, NAUKA Publishers, Moscow, 1974, pp. 172–193 (in Russian).
Ermolaeva, N.M., and A.A. Muchnik, Modal logics defined by endomorpismus in distributive lattices, in D.A. Bochvar and V.N. Grishin (eds.), Investigations on the Set Theory and Non-classical Logics, NAUKA Publishers, Moscow, 1976, pp. 229–246 (in Russian).
Ermolaeva N.M., and A.A. Muchnik, A pretabular tense logic, in A. I. Mihailov (ed.), Investigations on Non-classical Logics and the Set Theory, NAUKA Publishers, Moscow, 1979, pp. 288–297 (in Russian).
Fitting, M., Bilattices and the theory of truth, Journal of Philosophical Logic 18:225-–256, 1989.
Font, J.M., Belnap’s four-valued logic and De Morgan lattices, Logic Journal of the IGPL 5(3):413–440, 1997.
Font, J.M., and M. Rius, An abstract algebraic logic approach to tetravalent modal logics, The Journal of Symbolic logic 65(2):481–518, 2000.
Ginsberg, M.L., Multivalued logics: A uniform approach to inference in artificial intelligence, Computational Intelligence 4(3):265—315, 1988.
Halbach, V., Axiomatic theories of truth, in Stanford Encyclopedia of Philosophy, ed. E.N. Zalta, 2009, http://plato.stanford.edu/entries/truth-axiomatic/.
Karpenko, A.S., Development of Many-Valued Logic, URSS, Moscow, 2010.
Lewis, C.I., and C.H. Langford, Symbolic Logic, N. Y., 1932 (2nd ed. in 1959).
Lukasiewicz, J., Interpretacja liczbowa teorii zda_n, Ruch Filozoficzny 7:92–93, 1922/1923. (English trans.: A numerical interpretation of theory of propositions, in Lukasiewicz, J., Selected Works, North-Holland & PWN, Amsterdam & Warszawa, 1970, pp. 129–130).
Lukasiewicz, J., A system of modal logic, The Journal of Computing Systems 1:111–149, 1953.
Maksimova, L.L., Interpolation theorems in modal logics and amalgamable varieties of topoboolean algebras, Algebra i Logica 18(5): 556–586. 1979 (in Russian).
Meshi, V.Y., Kripke’s semantics for modal systems including S4.3, Mathematical Notes 15(6):875–884, 1974 (in Russian).
Pavlov, S.A., Logic of falsehood FL4, in Proceedings of the Reseach Logical Seminar of Institute of Philosophy of Russian Academy of Sciences, IFRAN, Moscow, 1994, pp. 14-35 (in Russian).
Post, E.L., Introduction to a general theory of elementary propositions, American Journal of Mathematics 43(3):163–185, 1921. (Repr. in J. van Heijenoort (ed.), From Frege to Godel: A Source Book in Mathematical Logic, 1879—1931, Harvard Univ. Press, Cambridge, 1967, pp. 264–283).
Pynko, A.P., Functional completeness and axiomatizability within Belnap’s four-valued logic and its expansion, Journal of Applied NonClassical Logics 9(1):61–105, 1999.
Scroggs, S.J., Extensions of the Lewis system S5, The Journal of Symbolic Logic 16:112–120, 1951.
Segerberg, K., On the logic of ‘To-morrow’, Theoria 33(1):46–52, 1967.
Segerberg, K., Von Wright’s tense logic, in V.A. Smirnov (ed.), Logical Deduction, NAUKA Publishers, Moscow, 1979, pp. 173–205 (in Russian). (English version in P.A. Schilpp, and L.E. Hahn (eds.), The Philosophy of Georg Henrik von Wright, Open Court, Illinois, 1989, pp. 602–635).
Sobochinski, B., Modal system S4.4, Notre Dame Journal of Formal Logic 5(4):305–312, 1964.
Sobochinski, B., Certain extensions of modal system S4, Notre Dame Journal of Formal Logic 11(3):347–367, 1970.
Von Wright, G.H., And next, Acta Philosopica Fennica 18:293–304, 1965.
Von Wright, G.H., Truth-logics, Logique et Analyse 30(120):311–334, 1987; repr. in Acta Philosopica Fennica 60:71–91, 1996.