Non-classical Modifications of Many-valued Matrices of the Classical Propositional Logic. Part II
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04.05.2017
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Abstract
This paper constitutes the second part of the duology dedicated to many-valued matrices of the classical propositional logic regarded as a tool of construction and analysis of non-classical logics. There are many pairs of three-valued matrices which differ only in classes of designated values present in the literature. However, the majority of them induce non-classical consequence relations with respect to either one and two designated values. At the same time, there are matrices of non-classical logics, obtained from matrices of the classical logic by contraction or expansion of the class of designated values. The principal part of the paper is devoted to the two classes of matrices. The first class consists of matrices which would induce the classical consequence given $D=\{1,2\}$, but are regarded as having $D=\{2\}$. The second class is obtained by assuming $D=\{1,2\}$ in matrices inducing the classical consequence for $D=\{2\}$. For the matrices in question I prove the maximality (in the strong sense) of paraconsistency or paracompleteness of logics they define, as well as analogues of Glivenko or Dual-Glivenko theorems. The matrices in classes under consideration form lattices with respect to functional embeddability relation. Some matrices obtained from matrices of the classical logic through modifications of their classes of designated values are shown to have equivalent formulations as functional extensions of matrices of the classical logic. DOI: 10.21146/2074-1472-2017-23-1-11-47
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Devyatkin L. Y. Non-classical Modifications of Many-valued Matrices of the Classical Propositional Logic. Part II // Logicheskie Issledovaniya / Logical Investigations. 2017. VOL. 23. № 1. C. 11-47.
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References
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Девяткин Л.Ю., Преловский Н.Н., Томова Н.Е. В границах трехзначности. М.: ИФ РАН, 2015. 136 с.
Емельянов Н.Р. О сложности задачи выразимости в многозначных логиках // Доклады Академии Наук СССР. 1985. Т. 282. № 3. С. 525–529.
Карпенко А.С. Развитие многозначной логики. М.: ЛКИ, 2010. 448 с.
Карпенко А.С., Томова Н.Е. Трехзначная логика Бочвара и литеральные паралогики. М.: ИФ РАН, 2016. 110 с.
Раца М.Ф. О классе функций трехзначной логики, соответствующем первой матрице Яськовского // Проблемы кибернетики. 1969. Вып. 21. С. 185–214.
Томова Н.Е. О четырехзначных регулярных логиках // Логические исследования. М.: Наука, 2009. Вып. 15. C. 223–228.
Томова Н.Е. Естественные p-логики // Логические исследования. Вып. 17. Изд-во ЦГИ, 2011. С. 256–268.
Томова Н.Е. Естественные трехзначные логики: функциональные свойства и отношения. М.: ИФ РАН, 2012. 89 с.
Яблонский С.В. Функциональные построения в k-значной логике // Труды математического института им. В.А. Стеклова. Т. 51. М., 1958. С. 5—142.
Arieli O., Avron A. Three-Valued Paraconsistent Propositional Logics // New Directions in Paraconsistent Logic / Ed. by J.-Y. Beziau et al. Springer India, 2015. P. 91–129.
Arieli O., Avron A., Zamansky A. Maximally Paraconsistent ThreeValued Logics // Proceedings of the Twelfth International Conference on the Principles of Knowledge Representation and Reasoning. Toronto, Ontario, Canada, 2010. P. 310–318.
Arieli O., Avron A., Zamansky A. Maximal and Premaximal Paraconsistency in the Framework of Three-Valued Semantics // Studia Logica. 2011. Vol. 97. No. 1. P. 31–60.
Batens D., De Clercq K., Kurtonina N. Embedding and Interpolation for some Paralogics. The Propositional Case // Reports on Mathematical logic. 1999. Vol 33. P. 29–44.
Brunner A.B., Carnielli W.A. Anti-Intuitionism and Paraconsistency // Journal of Applied Logic. 2005. Vol. 3. No. 1. P. 161–184.
Carnielli W., Coniglio M.E., Marcos J. Logics of Formal Inconsistency // Handbook of Philosophical Logic. Vol. 14. Springer Netherlands, 2007. P. 1–93.
Church A. Non-Normal Truth-Tables for the Propositional Calculus // Boletin de la Sociedad Matematica Mexicana. 1953. Vol. 10. P. 41–52.
Ciuciura J. A Weakly-Intuitionistic Logic I1 // Logical Investigations. 2015. Vol. 21. No. 2. P. 53–60.
Cobreros P. Vagueness: Subvaluationism // Philosophy Compass. 2013. Vol. 8. No. 5. P. 472–485.
Da Costa N.C.A. On the Theory of Inconsistent Formal Systems // Notre Dame Journal of Formal Logic. 1974. Vol. 15. No. 4. P. 497–510.
D’Ottaviano I.M.L. The Completeness and Compactness of a ThreeValued First-Order Logic // Revista Colombiana de Matematicas. 1985. Vol. 19. P. 77–94.
D’Ottaviano I.M.L., da Costa N.C.A. Sur un probleme de Jaskowski // Comptes Rendus de l’Acad_emie de Sciences de Paris. Ser. A. 1970. Vol. 270. P. 1349–1353.
Epstein R.L. The Semantic Foundations of Logic. Vol. 1: Propositional logic. Dordrecht: Kluwer, 1990. 388 p.
Ferguson T.M. Lukasiewicz Negation and Many-Valued Extensions of Constructive Logics // Proceedings of the 44th International Symposium on Multiple-Valued Logic (ISMVL 2014). IEEE Computer Society Press, 2014. P. 121–127.
Finn V.K., Grigolia R. Nonsense Logics and their Algebraic Properties // Theoria. 1993. Vol. 59. No. 1–3. P. 207–273.
Godel K. On the Intuitionistic Propositional Calculus / Godel K. Collected works I: Publications 1929–1936 / Ed. by S. Feferman et al. Oxford University Press, 1986. P. 223–225.
Goodman N.D. The Logic of Contradiction // Mathematical Logic Quarterly. 1981. Vol. 27. No. 8–10. P. 119–126.
Gore R. Dual Intuitionistic Logic Revisited // Automated Reasoning with Analytic Tableaux and Related Methods / Ed. by R. Dyckhoff. SpringerVerlag, 2000. P. 252–267.
Humberstone L. The Connectives. MIT Press, 2011. 1512 p.
Hyde D. From Heaps and Gaps to Heaps of Gluts // Mind. 1997. Vol. 106. No. 424. P. 641–660.
Kalicki J. A Test for the Existence of Tautologies According to ManyValued Truth-Tables // Journal of Symbolic Logic. 1950. Vol. 15(3). P. 182–184.
Karpenko A.S. A Maximal Paraconsistent Logic: the Combination of Two Three-Valued Isomorphs of Classical Propositional Logic // Frontiers of Paraconsistent Logic / Ed. by D. Batens, C. Mortensen, G. Priest, J.-P. van Bendegem. Baldock Research Studies Press, 2000. P. 181–187.
Karpenko A.S., Tomova N.E. Bochvar’s Three-Valued Logic and Literal Paralogics: Their Lattice and Functional Equivalence // Logic and Logical Philosophy. 2016. URL: http://apcz.pl/czasopisma/index.php/LLP/article/view/LLP.2016.029 (дата обращения: 21.10.2016).
Kleene S.C. On Notation for Ordinal Numbers // The Journal of Symbolic Logic. 1938. Vol. 3. No. 4. P. 150–155.
Kubyshkina E., Zaitsev D.V. Rational Agency From a Truth-Functional Perspective // Logic and Logical Philosophy. 2016. Vol. 25. No. 4. P. 499–520.
Loparic A., da Costa N.C.A. Paraconsistency, Paracompleteness, and Valuations // Logique et Analyse. 1984. Vol. 27. No. 106. P. 119–131.
Lewin R.A., Mikenberg I.F. Literal-Paraconsistent and LiteralParacomplete Matrices // Mathematical Logic Quarterly. 2006. Vol. 52. No. 5. P. 478–493.
Marcos J. Nearly Every Normal Modal Logic is Paranormal // Logique et Analyse. 2005. Vol. 48. No. 189–192. P. 279–300.
Marcos J. On a Problem of da Costa // Essays on the Foundations of Mathematics and Logic 2 / Ed. by G. Sica. Polimetrica, 2005. P. 53–69.
McKinsey J.C.C., Tarski A. On Closed Elements in Closure Algebras // Annals of Mathematics. Second Series. 1946. Vol. 47. No. 1. P. 122–162.
Monteiro A. Sur les Alg`ebres de Heyting Sym_etriques // Portugaliae Mathematica. 1980. Vol. 39. No. 1–4. P. 1–237.
Priest G. Logic of Paradox // Journal of Philosophical Logic. 1979. Vol. 8. P. 219–241.
Priest G. Paraconsistent Logic // Handbook of Philosophical Logic / Ed. by Dov M. Gabbay, F. Guenthner. Springer Netherlands, 2002. P. 287–393.
Rasiowa H., Sikorski R. The Mathematics of Metamathematics. Warszawa, 1963. 520 p.
Rauszer C. Semi-Boolean Algebras and Their Applications to Intuitionistic Logic with Dual Operations // Fundamenta Mathematicae. 1974. Vol. 83. No. 3. P. 219–249.
Rescher N. Many-Valued Logic. New York: McGraw-Hill, 1969. Reprinted: Aldershot: Gregg Revivals, 1993. 349 p.
Ripley D. Sorting out the Sorites // Paraconsistency: Logic and Applications / Ed. by K. Tanaka, F. Berto, E. Mares, F. Paoli. Springer Netherlands, 2013. P. 329–348.
Segerberg K. A Contribution to Nonsense-Logic // Theoria. 1965. Vol. 31. P. 199–217.
Sette A.M. On propositional calculus P1 // Mathematica Japonica. 1973. Vol. 18. P. 173–180.
Sette A.M., Carnielli W.A. Maximal Weakly-Intuitionistic Logics // Studia Logica. 1995. Vol. 55. P. 181–203.
Shoesmith D.J., Smiley T.J. Deducibility and Many-Valuedness // The Journal of Symbolic Logic. 1971. Vol 36. No. 4. P. 610–622.
Shoesmith D.J., Smiley T.J. Multiple-Conclusion Logic. Cambrige University Press, 1978. 409 p.
Shramko Y., Wansing H. Entailment Relations and/as Truth Values // Bulletin of the Section of Logic. 2007. Vol. 36. No. 3–4. P. 131–144.
Tomova N.E. A Lattice of Implicative Extensions of Regular Kleene’s Logics // Reports on Mathematical Logic. 2012. No. 47. P. 173–182.
Tranchini L. Natural Deduction for Dual-Intuitionistic Logic // Studia Logica. 2012. Vol. 100. No. 3. P. 631–648.
Urbas I. Dual-Intuitionistic Logic // Notre Dame Journal of Formal Logic. 1996. Vol. 37. No. 3. P. 440–451.
Wansing H. Constructive Negation, Implication, and Co-implication // Journal of Applied Non-Classical Logics. 2008. Vol. 18. No. 2–3. P. 341–364.
Wojcicki R. Theory of Logical Calculi: Basic Theory of Consequence Operations. Dordrecht: Kluwer, 1988. 474 p.
Девяткин Л.Ю., Карпенко А.С., Попов В.М. Трехзначные характеристические матрицы классической пропозициональной логики // Труды научно-исследовательского семинара Логического центра Института философии РАН. 2007. Т. XVIII. С. 50–62.
Девяткин Л.Ю., Преловский Н.Н., Томова Н.Е. В границах трехзначности. М.: ИФ РАН, 2015. 136 с.
Емельянов Н.Р. О сложности задачи выразимости в многозначных логиках // Доклады Академии Наук СССР. 1985. Т. 282. № 3. С. 525–529.
Карпенко А.С. Развитие многозначной логики. М.: ЛКИ, 2010. 448 с.
Карпенко А.С., Томова Н.Е. Трехзначная логика Бочвара и литеральные паралогики. М.: ИФ РАН, 2016. 110 с.
Раца М.Ф. О классе функций трехзначной логики, соответствующем первой матрице Яськовского // Проблемы кибернетики. 1969. Вып. 21. С. 185–214.
Томова Н.Е. О четырехзначных регулярных логиках // Логические исследования. М.: Наука, 2009. Вып. 15. C. 223–228.
Томова Н.Е. Естественные p-логики // Логические исследования. Вып. 17. Изд-во ЦГИ, 2011. С. 256–268.
Томова Н.Е. Естественные трехзначные логики: функциональные свойства и отношения. М.: ИФ РАН, 2012. 89 с.
Яблонский С.В. Функциональные построения в k-значной логике // Труды математического института им. В.А. Стеклова. Т. 51. М., 1958. С. 5—142.
Arieli O., Avron A. Three-Valued Paraconsistent Propositional Logics // New Directions in Paraconsistent Logic / Ed. by J.-Y. Beziau et al. Springer India, 2015. P. 91–129.
Arieli O., Avron A., Zamansky A. Maximally Paraconsistent ThreeValued Logics // Proceedings of the Twelfth International Conference on the Principles of Knowledge Representation and Reasoning. Toronto, Ontario, Canada, 2010. P. 310–318.
Arieli O., Avron A., Zamansky A. Maximal and Premaximal Paraconsistency in the Framework of Three-Valued Semantics // Studia Logica. 2011. Vol. 97. No. 1. P. 31–60.
Batens D., De Clercq K., Kurtonina N. Embedding and Interpolation for some Paralogics. The Propositional Case // Reports on Mathematical logic. 1999. Vol 33. P. 29–44.
Brunner A.B., Carnielli W.A. Anti-Intuitionism and Paraconsistency // Journal of Applied Logic. 2005. Vol. 3. No. 1. P. 161–184.
Carnielli W., Coniglio M.E., Marcos J. Logics of Formal Inconsistency // Handbook of Philosophical Logic. Vol. 14. Springer Netherlands, 2007. P. 1–93.
Church A. Non-Normal Truth-Tables for the Propositional Calculus // Boletin de la Sociedad Matematica Mexicana. 1953. Vol. 10. P. 41–52.
Ciuciura J. A Weakly-Intuitionistic Logic I1 // Logical Investigations. 2015. Vol. 21. No. 2. P. 53–60.
Cobreros P. Vagueness: Subvaluationism // Philosophy Compass. 2013. Vol. 8. No. 5. P. 472–485.
Da Costa N.C.A. On the Theory of Inconsistent Formal Systems // Notre Dame Journal of Formal Logic. 1974. Vol. 15. No. 4. P. 497–510.
D’Ottaviano I.M.L. The Completeness and Compactness of a ThreeValued First-Order Logic // Revista Colombiana de Matematicas. 1985. Vol. 19. P. 77–94.
D’Ottaviano I.M.L., da Costa N.C.A. Sur un probleme de Jaskowski // Comptes Rendus de l’Acad_emie de Sciences de Paris. Ser. A. 1970. Vol. 270. P. 1349–1353.
Epstein R.L. The Semantic Foundations of Logic. Vol. 1: Propositional logic. Dordrecht: Kluwer, 1990. 388 p.
Ferguson T.M. Lukasiewicz Negation and Many-Valued Extensions of Constructive Logics // Proceedings of the 44th International Symposium on Multiple-Valued Logic (ISMVL 2014). IEEE Computer Society Press, 2014. P. 121–127.
Finn V.K., Grigolia R. Nonsense Logics and their Algebraic Properties // Theoria. 1993. Vol. 59. No. 1–3. P. 207–273.
Godel K. On the Intuitionistic Propositional Calculus / Godel K. Collected works I: Publications 1929–1936 / Ed. by S. Feferman et al. Oxford University Press, 1986. P. 223–225.
Goodman N.D. The Logic of Contradiction // Mathematical Logic Quarterly. 1981. Vol. 27. No. 8–10. P. 119–126.
Gore R. Dual Intuitionistic Logic Revisited // Automated Reasoning with Analytic Tableaux and Related Methods / Ed. by R. Dyckhoff. SpringerVerlag, 2000. P. 252–267.
Humberstone L. The Connectives. MIT Press, 2011. 1512 p.
Hyde D. From Heaps and Gaps to Heaps of Gluts // Mind. 1997. Vol. 106. No. 424. P. 641–660.
Kalicki J. A Test for the Existence of Tautologies According to ManyValued Truth-Tables // Journal of Symbolic Logic. 1950. Vol. 15(3). P. 182–184.
Karpenko A.S. A Maximal Paraconsistent Logic: the Combination of Two Three-Valued Isomorphs of Classical Propositional Logic // Frontiers of Paraconsistent Logic / Ed. by D. Batens, C. Mortensen, G. Priest, J.-P. van Bendegem. Baldock Research Studies Press, 2000. P. 181–187.
Karpenko A.S., Tomova N.E. Bochvar’s Three-Valued Logic and Literal Paralogics: Their Lattice and Functional Equivalence // Logic and Logical Philosophy. 2016. URL: http://apcz.pl/czasopisma/index.php/LLP/article/view/LLP.2016.029 (дата обращения: 21.10.2016).
Kleene S.C. On Notation for Ordinal Numbers // The Journal of Symbolic Logic. 1938. Vol. 3. No. 4. P. 150–155.
Kubyshkina E., Zaitsev D.V. Rational Agency From a Truth-Functional Perspective // Logic and Logical Philosophy. 2016. Vol. 25. No. 4. P. 499–520.
Loparic A., da Costa N.C.A. Paraconsistency, Paracompleteness, and Valuations // Logique et Analyse. 1984. Vol. 27. No. 106. P. 119–131.
Lewin R.A., Mikenberg I.F. Literal-Paraconsistent and LiteralParacomplete Matrices // Mathematical Logic Quarterly. 2006. Vol. 52. No. 5. P. 478–493.
Marcos J. Nearly Every Normal Modal Logic is Paranormal // Logique et Analyse. 2005. Vol. 48. No. 189–192. P. 279–300.
Marcos J. On a Problem of da Costa // Essays on the Foundations of Mathematics and Logic 2 / Ed. by G. Sica. Polimetrica, 2005. P. 53–69.
McKinsey J.C.C., Tarski A. On Closed Elements in Closure Algebras // Annals of Mathematics. Second Series. 1946. Vol. 47. No. 1. P. 122–162.
Monteiro A. Sur les Alg`ebres de Heyting Sym_etriques // Portugaliae Mathematica. 1980. Vol. 39. No. 1–4. P. 1–237.
Priest G. Logic of Paradox // Journal of Philosophical Logic. 1979. Vol. 8. P. 219–241.
Priest G. Paraconsistent Logic // Handbook of Philosophical Logic / Ed. by Dov M. Gabbay, F. Guenthner. Springer Netherlands, 2002. P. 287–393.
Rasiowa H., Sikorski R. The Mathematics of Metamathematics. Warszawa, 1963. 520 p.
Rauszer C. Semi-Boolean Algebras and Their Applications to Intuitionistic Logic with Dual Operations // Fundamenta Mathematicae. 1974. Vol. 83. No. 3. P. 219–249.
Rescher N. Many-Valued Logic. New York: McGraw-Hill, 1969. Reprinted: Aldershot: Gregg Revivals, 1993. 349 p.
Ripley D. Sorting out the Sorites // Paraconsistency: Logic and Applications / Ed. by K. Tanaka, F. Berto, E. Mares, F. Paoli. Springer Netherlands, 2013. P. 329–348.
Segerberg K. A Contribution to Nonsense-Logic // Theoria. 1965. Vol. 31. P. 199–217.
Sette A.M. On propositional calculus P1 // Mathematica Japonica. 1973. Vol. 18. P. 173–180.
Sette A.M., Carnielli W.A. Maximal Weakly-Intuitionistic Logics // Studia Logica. 1995. Vol. 55. P. 181–203.
Shoesmith D.J., Smiley T.J. Deducibility and Many-Valuedness // The Journal of Symbolic Logic. 1971. Vol 36. No. 4. P. 610–622.
Shoesmith D.J., Smiley T.J. Multiple-Conclusion Logic. Cambrige University Press, 1978. 409 p.
Shramko Y., Wansing H. Entailment Relations and/as Truth Values // Bulletin of the Section of Logic. 2007. Vol. 36. No. 3–4. P. 131–144.
Tomova N.E. A Lattice of Implicative Extensions of Regular Kleene’s Logics // Reports on Mathematical Logic. 2012. No. 47. P. 173–182.
Tranchini L. Natural Deduction for Dual-Intuitionistic Logic // Studia Logica. 2012. Vol. 100. No. 3. P. 631–648.
Urbas I. Dual-Intuitionistic Logic // Notre Dame Journal of Formal Logic. 1996. Vol. 37. No. 3. P. 440–451.
Wansing H. Constructive Negation, Implication, and Co-implication // Journal of Applied Non-Classical Logics. 2008. Vol. 18. No. 2–3. P. 341–364.
Wojcicki R. Theory of Logical Calculi: Basic Theory of Consequence Operations. Dordrecht: Kluwer, 1988. 474 p.