Non-classical Modifications of Many-valued Matrices of the Classical Propositional Logic. Part I


L. Y. Devyatkin


This paper constitutes the first 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, and it is for the most part of the survey nature. First, I analyze the three approaches to the question when a many-valued matrix defines the classical propositional logic, which are based on the notions of theory, logical consequence relation with single conclusions and multiple-conclusion consequence relation. Then I deal with the matrices of non-classical logics which are functional extensions of classical matrices. The examples of individual matrices of this kind, as well as some classes of them, are considered, some of them known in the literature, and some completely new. Their functional properties are investigated. Among the examples considered are the matrices of three-valued logics of Post, _ukasiewicz, Bochvar and others. Moreover, I explore a class of matrices which define logics of formal inconsistency $(LFI)$. On the basis of duality between paraconsistent and paracomplete logics, a class of matrices which define logics of formal uncertainty is constructed.Furthermore, I develop a class of four-valued matrices which combine formal inconsistency and formal uncertainty.In the concluding part of the paper I~investigate another class of matrices, defining paraconsistent logics which are not logics of formal inconsistency. DOI: 10.21146/2074-1472-2016-22-2-27-58






Бочвар Д.А. Об одном трехзначном исчислении и его применении к анализу парадоксов классического расширенного функционального исчисления // Математический сборник. 1938. Т. 4. № 2. С. 287–308.
Карпенко А.С. Неклассические логики versus классической // Логикофилософские штудии. 2005. № 3. С. 48–73.
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 c.
Попов В.М. Об одной трехзначной параполной логике // Логические исследования. 2002. № 9. С. 175–178.
Попов В.М. Об одной четырехзначной паранормальной логике // Логика и В.Е.К. К 90-летию со дня рождения профессора Войшвилло Евгения Казимировича. / Под ред. В.И. Маркина. М.: Современные тетради, 2003. С. 192–195.
Томова Н.Е. Естественные трехзначные логики: функциональные свойства и отношения. М.: ИФ РАН, 2012. 89 с.
Финн В.К. О предполноте класса функций, соответствующего трехзначной логике Я. Лукасевича // Научно-техническая информация. Сер. 2. Вып. 10. М., 1969. С. 35–38.
Шестаков В.И. О взаимоотношении некоторых трехзначных логических исчислений // Успехи математических наук. 1964. Т. 19. Вып. 2(116). С. 177—181.
Яблонский С.В. Функциональные построения в k-значной логике // Труды математического института им. В.А. Стеклова. Т. 51. М., 1958. С. 5—142.
Anshakov O., Rychkov S. On Finite-Valued Propositional Logical Calculi // Notre-Dame Journal of Formal Logic. 1995. Vol. 36. No. 4. P. 606–629.
Arieli O., Avron A., Zamansky A. Maximally Paraconsistent ThreeValued Logics // Proceedings of the Twelfth International Conference on the Principles of Knowledge Representation and Reasoning. 2010. P. 310–318.
Avron A. Natural 3-Valued Logics — Characterization and Proof Theory // The Journal of Symbolic Logic. Vol. 56. No. 1. 1991. P. 276–294.
Bergman C., Juedes D., Slutzki G. Computational Complexity of TermEquivalence // International Journal of Algebra and Computation. 1999. Vol. 9. No. 1. P. 113–128.
Brunner A.B., Carnielli W.A.. Anti-Intuitionism and Paraconsistency // Journal of Applied Logic. 2005. Vol. 3. No. 1. P. 161–184.
Carnielli W.A., Marcos J. A Taxonomy of C-systems. 2001. URL: (дата обращения —21.06.2016).
Carnielli W., Coniglio M.E., Marcos, J. Logics of Formal Inconsistency // Handbook of Philosophical Logic. Vol. 14. Springer Netherlands, 2007. P. 1–93.
Carnielli W., Marcos J., de Amo S. Formal Inconsistency and Evolutionary Databases // Logic and Logical Philosophy. 2004. Vol. 8. P. 115–52.
Cobreros P. Vagueness: Subvaluationism // Philosophy Compass. 2013. Vol. 8. No. 5. P. 472–485.
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 probl`eme de Jaskowski // Comptes Rendus de l’Acad_emie de Sciences de Paris. Ser. A. 1970. Vol. 270. P. 1349–1353.
D’Ottaviano I.M.L., de Araujo Feitosa H. Paraconsistent Logics and Translations // Synthese. 2000. Vol. 125. No. 1–2. P. 77–95.
Ebbinghaus H.D. Uber eine Pr_adikatenlogik mit partiell definierten Pradikaten und Funktionen // Archive for Mathematical Logic. 1969. Vol. 12. No. 1. P. 39–53.
Epstein R.L. The Semantic Foundations of Logic. Vol. 1: Propositional logic. Dordrecht: Kluwer, 1990. 388 p.
Feitosa H.A., D’Ottaviano I.M.L. Conservative Translations // Annals of Pure and Applied Logic. 2001. Vol. 108. No. 1. P. 205–227.
Finn V.K., Grigolia R. Nonsense Logics and their Algebraic Properties // Theoria. 1993. Vol. 59. No. 1–3. P. 207–273.
Gottwald S. A Treatise on Many-Valued Logics. Baldock: Research Studies Press, 2001. 600 p.
Halkowska K. A Note on Matrices for Systems of Nonsense-Logics // Studia Logica. 1989. Vol. 48. No. 4. P. 461–464.
Hyde D. From Heaps and Gaps to Heaps of Gluts // Mind. 1997. Vol. 106. No. 424. P. 641–660.
Lewin R.A., Mikenberg I.F. Literal-Paraconsistent and LiteralParacomplete Matrices // Mathematical Logic Quarterly. 2006. Vol. 52. No. 5. P. 478–493.
Lukasiewicz J. On Three-Valued Logic // Jan Lukasiewicz. Selected Works / Ed. by L. Borkowski. Amsterdam: North-Holland, 1970. P. 87–88.
Malinowski G. Towards the Concept of Logical Many-Valuedness // Acta Universitatis Lodziensis. Folia Philosophica. 1990. Vol. 7. P. 97–103.
Malinowski G. Many-Valued Logics. Oxford University Press. 1993. 144 p.
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.
Popov V.M. On the Logics Pelated to A. Arruda’s System V1 // Logic and Logical Philosophy. 1999. Vol. 7. P. 87–90.
Post E.L. Introduction to a General Theory of Elementary Propositions // American Journal of Mathematics. 1921. Vol. 43. No. 3. P. 163–185.
Priest G. Logic of Paradox // Journal of Philosophical Logic. 1979. Vol. 8. P. 219–241.
Puga L.Z., da Costa N.C.A. On the Imaginary Logic of N.A. Vasiliev // Mathematical Logic Quarterly. 1988. Vol. 34. P. 205–211.
Rescher N. Many-Valued Logic. New York: McGraw-Hill, 1969. Reprinted: Aldershot: Gregg Revivals, 1993. 349 p.
Rosser J.B., Turquette A.R. Many-Valued Logics. Amsterdam: NorthHolland. 1952. 124 p.
Rozonoer L.I. Proving Contradictions in Formal Theories. I // Automation and Remote Control. 1983. Vol. 44. No. 6. P. 781–790.
Rozonoer L.I. Proving Contradictions in Formal Theories. II // Automation and Remote Control. 1983. Vol. 44. No. 7. P. 908–914.
Rozonoer L.I. On Interpretation of Inconsistent Theories // Information Sciences. 1989. Vol. 47. No. 3. P. 243–266.
Segerberg K. A Contribution to Nonsense-Logic // Theoria. 1965. Vol. 31. P. 199–217.
Sette A.M., Carnielli W.A. Maximal Weakly-Intuitionistic Logics // Studia Logica. 1995. Vol. 55. P. 181–203.
Shoesmith D.J., Smiley T.J. Multiple-Conclusion Logic. Cambrige University Press, 1978. 409 p.
Steinberger F. Why Conclusions Should Remain Single // Journal of Philosophical Logic. 2011. Vol. 40. No. 3. P. 333–355.
Tomova N.E. A Lattice of Implicative Extensions of Regular Kleene’s Logics // Reports on Mathematical Logic. 2012. No. 47. P. 173–182.
Wojcicki R. Lectures on Propositional Calculi. Wroclaw: Ossolineum, 1984. 292 p.
Wojcicki R. Theory of Logical Calculi: Basic Theory of Consequence Operations. Dordrecht: Kluwer, 1988. 474 p.