Correspondence Analysis for First Degree Entailment
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03.03.2016
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In this paper natural deduction systems for four-valued logic $FDE$ (first degree entailment) and its extensions are constructed. At that B. Kooi and A. Tamminga’s method of correspondence analysis is used. All possible four-valued unary $\star$ and binary $\circ $ propositional connectives which could be added to $FDE$ are considered. Then $FDE$ is extended by Boolean negation $\sim$and every entry (line) of truth tables for $\star$ and $\circ $is characterized by inference scheme. By adding all inference schemes characterizing truth tables for $\star$ and $\circ $as rules of inference to the natural deduction for $FDE$, natural deduction for extension of $FDE$ is obtained. In addition, applying of correspondence analysis gives axiomatizations of implicative extensions of $FDE$ including $BN4$ and some extensions by classical implications.
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Петрухин Я. И. Correspondence Analysis for First Degree Entailment // Логические исследования / Logical Investigations. 2016. Т. 22. № 1. C. 108-124.
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Литература
Anderson, A.R., Belnap, N.D. “Tautological entailments”, Philosophical Studies, 1962, vol. 13(1-2), pp. 9-24.
Avron, A. “Natural 3-valued logics — characterization and proof theory”, The Journal of Symbolic Logic, 1991, vol. 56(1), pp. 276-294.
Belnap, N.D. “A useful four-valued logic”, Modern Uses of Multiple-Valued Logic, ed. by J.M. Dunn, G. Epstein. Boston: Reidel Publishing Company, 1977, pp. 7-37.
Belnap, N.D. “How a computer should think”, Contemporary Aspects of Philosophy, ed. by G. Rule. Stocksfield: Oriel Press, 1977, pp. 30-56.
Belnap, N.D. “Tautological entailments”, The Journal of Symbolic Logic, 1959, vol. 24(4), p. 316.
Brady, R.T. “Completeness proofs for the systems RM3 and BN4”, Logique et Analyse, 1982, vol. 25(97), pp. 51-61.
De, M., Omori, H. “Classical Negation and Expansions of Belnap-Dunn Logic”, Studia Logica, 2015, vol. 103(4), pp. 825-851.
Dunn, J.M. “Intuitive semantics for first-degree entailment and coupled trees”, Philosophical Studies, 1976, vol. 29(3), pp. 149-168.
Font, J.M. “Belnap’s Four-Valued Logic and De Morgan Lattices”, Logic Journal of the IGPL, 1997, vol. 5(3), pp. 1-29.
Karpenko, A.S. Razvitie mnogoznachnoi logiki [The development of manyvalued logic]. Moscow: LKI Publ., 2010 (3d edition). 448 p. (In Russian)
Kleene, S.C. Introduction to metamathematics. Amsterdam: WoltersNoordhoff Publishing and North-Holland Publishing Company, 1971 (6th reprint). 560 pp.
Kleene, S.C. “On a notation for ordinal numbers”, The Journal of Symbolic Logic, 1938, vol. 3(1), pp. 150-155.
Kooi, B., Tamminga, A. “Completeness via correspondence for extensions of the logic of paradox”, The Review of Symbolic Logic, 2012, vol. 5(4), pp. 720-730.
Popov, V.M. “Sekventsial’nye formulirovki paraneprotivorechivykh logicheskikh sistem” [Sequent formulations of paraconsistent logical systems], Semanticheskie i sintaksicheskie issledovaniya neekstensional’nykh logik [Semantic and syntactic investigations of non-extensional logics], ed. by V.A. Smirnov. Moscow: Nauka Publ., 1989, pp. 285-289. (In Russian)
Priest, G. “Paraconsistent logic”, Handbook of philosophical logic. 2nd edition. Vol.6, ed. by M. Gabbay, F. Guenthner. Dordrecht: Kluwer, 2002, pp. 287-393.
Priest, G. “The logic of paradox”, Journal of Philosophical Logic, 1979, vol. 8(1), pp. 219-241.
Pynko, A.P. “Functional completeness and axiomatizability within Belnap’s four-valued logic and and its expansion”, Journal of Applied Non-Classical Logics, 1999, vol. 9(1), pp. 61-105.
Shramko, Y., Wansing, H. “Entailment relations and/as truth values”, Bulletin of the Section of Logic, 2007, vol. 36(3-4), pp. 131-143.
Slaney, J.K. “The implications of paraconsistency”, Proceedings of the 12th IJCAI, vol. 2, ed. by J. Mylopoulos and R. Reiter. Sydney: Morgan Kaufmann Publ., 1991, pp. 1052-1057.
Tamminga, A. “Correspondence analysis for strong three-valued logic”, Logical Investigations, 2014, vol. 20, pp. 255-268.
Zaitsev, D.V. Obobshchennaya relevantnaya logika i modeli rassuzhdenii [Generalized relevant logic and models of reasoning]. Moscow State Lomonosov University, doctoral (Doctor of Science) dissertation, 2012. 284 pp. (In Russian)
Zaitsev, D.V., Shramko, Ya.V. “Logicheskoe sledovanie i vydelennye znacheniya” [Logical entailment and designated values], Logicheskie issledovaniya [Logical investigations], 2004, vol. 11, pp. 126-137. (In Russian)
Avron, A. “Natural 3-valued logics — characterization and proof theory”, The Journal of Symbolic Logic, 1991, vol. 56(1), pp. 276-294.
Belnap, N.D. “A useful four-valued logic”, Modern Uses of Multiple-Valued Logic, ed. by J.M. Dunn, G. Epstein. Boston: Reidel Publishing Company, 1977, pp. 7-37.
Belnap, N.D. “How a computer should think”, Contemporary Aspects of Philosophy, ed. by G. Rule. Stocksfield: Oriel Press, 1977, pp. 30-56.
Belnap, N.D. “Tautological entailments”, The Journal of Symbolic Logic, 1959, vol. 24(4), p. 316.
Brady, R.T. “Completeness proofs for the systems RM3 and BN4”, Logique et Analyse, 1982, vol. 25(97), pp. 51-61.
De, M., Omori, H. “Classical Negation and Expansions of Belnap-Dunn Logic”, Studia Logica, 2015, vol. 103(4), pp. 825-851.
Dunn, J.M. “Intuitive semantics for first-degree entailment and coupled trees”, Philosophical Studies, 1976, vol. 29(3), pp. 149-168.
Font, J.M. “Belnap’s Four-Valued Logic and De Morgan Lattices”, Logic Journal of the IGPL, 1997, vol. 5(3), pp. 1-29.
Karpenko, A.S. Razvitie mnogoznachnoi logiki [The development of manyvalued logic]. Moscow: LKI Publ., 2010 (3d edition). 448 p. (In Russian)
Kleene, S.C. Introduction to metamathematics. Amsterdam: WoltersNoordhoff Publishing and North-Holland Publishing Company, 1971 (6th reprint). 560 pp.
Kleene, S.C. “On a notation for ordinal numbers”, The Journal of Symbolic Logic, 1938, vol. 3(1), pp. 150-155.
Kooi, B., Tamminga, A. “Completeness via correspondence for extensions of the logic of paradox”, The Review of Symbolic Logic, 2012, vol. 5(4), pp. 720-730.
Popov, V.M. “Sekventsial’nye formulirovki paraneprotivorechivykh logicheskikh sistem” [Sequent formulations of paraconsistent logical systems], Semanticheskie i sintaksicheskie issledovaniya neekstensional’nykh logik [Semantic and syntactic investigations of non-extensional logics], ed. by V.A. Smirnov. Moscow: Nauka Publ., 1989, pp. 285-289. (In Russian)
Priest, G. “Paraconsistent logic”, Handbook of philosophical logic. 2nd edition. Vol.6, ed. by M. Gabbay, F. Guenthner. Dordrecht: Kluwer, 2002, pp. 287-393.
Priest, G. “The logic of paradox”, Journal of Philosophical Logic, 1979, vol. 8(1), pp. 219-241.
Pynko, A.P. “Functional completeness and axiomatizability within Belnap’s four-valued logic and and its expansion”, Journal of Applied Non-Classical Logics, 1999, vol. 9(1), pp. 61-105.
Shramko, Y., Wansing, H. “Entailment relations and/as truth values”, Bulletin of the Section of Logic, 2007, vol. 36(3-4), pp. 131-143.
Slaney, J.K. “The implications of paraconsistency”, Proceedings of the 12th IJCAI, vol. 2, ed. by J. Mylopoulos and R. Reiter. Sydney: Morgan Kaufmann Publ., 1991, pp. 1052-1057.
Tamminga, A. “Correspondence analysis for strong three-valued logic”, Logical Investigations, 2014, vol. 20, pp. 255-268.
Zaitsev, D.V. Obobshchennaya relevantnaya logika i modeli rassuzhdenii [Generalized relevant logic and models of reasoning]. Moscow State Lomonosov University, doctoral (Doctor of Science) dissertation, 2012. 284 pp. (In Russian)
Zaitsev, D.V., Shramko, Ya.V. “Logicheskoe sledovanie i vydelennye znacheniya” [Logical entailment and designated values], Logicheskie issledovaniya [Logical investigations], 2004, vol. 11, pp. 126-137. (In Russian)