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Handbook of Philosophical Logic pp 249–295Cite as

Basic Many-Valued Logic

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Part of the book series: Handbook of Philosophical Logic ((HALO,volume 2))

Abstract

Many-valued logic is a vast field with hundreds of published papers and numerous monographs devoted to it. I have attempted to keep this survey to manageable length by focusing on many-valued logic as an independent discipline. This means that such topics as the use of many-valued logics for proving the independence of axioms in propositional logic have been omitted.

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Bibliography

  1. J.F. Baldwin, editor. Fuzzy Logic. Wiley, 1996.

    Google Scholar 

  2. R. E. Bellman and L. A. Zadeh. Local and fuzzy logics. In Modern Uses of Multiple-valued Logic. D. Reidel, Dordrecht and Boston, 1977.

    Google Scholar 

  3. N. D. Belnap. A useful four-valued logic. In Modern Uses of Multiple-valued Logic. D. Reidel, Dordrecht and Boston, 1977.

    Google Scholar 

  4. D. A. Bochvar. Ob odnom tréhznacnom isčislenii i égo priménénii k analiza paradoksov klassičékogo râssirénnogo funkcional’nogo isčislénid. (On a 3-valued logical calculus and its application to the analysis of contradictions.). Matématičeskij sbornik, 4: 287–308, 1939.

    Google Scholar 

  5. C. C. Chang. Algebraic analysis of many valued logics. Transactions of the American Mathematical Society, 88: 467–490, 1958.

    Article  Google Scholar 

  6. C. C. Chang. Proof of an axiom of Lukasiewicz. Transactions of the American Mathematical Society, 87: 55–56, 1958.

    Google Scholar 

  7. C. C. Chang. A new proof of the completeness of the Lukasiewicz axioms. Transactions of the American Mathematical Society, 93: 74–80, 1959.

    Google Scholar 

  8. C. C. Chang. The axiom of comprehension in infinite valued logic. Mathematica Scandinavica, 13: 9–30. 1963.

    Google Scholar 

  9. A. Church. Review of Bochvar [Bochvar, 1939]. Journal of Symbolic Logic, 4: 98–99, 1939.

    Article  Google Scholar 

  10. R. Cignoli and D. Mundici. An elementary proof of Chang’s completeness theorem for the infinite-valued calculus of Lukasiewicz. Studia Logica, 58: 79–97, 1997.

    Article  Google Scholar 

  11. R. Cignoli. Free lattice-ordered Abelian groups and varieties of mvalgebras. In Ninth Latin American symposium on mathematical logic,pages 113–118. Universidad Nacional del Sur, 1993. Notas de Matemâtica Vol. 38.

    Google Scholar 

  12. G. Epstein. The lattice theory of Post algebras. Transactions of the American Mathematical Society, 95: 300–317, 1960.

    Article  Google Scholar 

  13. J. E. Fenstad. On the consistency of the axiom of comprehension in the Lukasiewicz infinite-valued logic. Mathematica Scandinavica, 14:65–74, 1964. [Goddard and Routley, 1973 ] L. Goddard and R. Routley. The Logic of Significance and Context. Scottish Academic Press, Edinburgh and London, 1973.

    Google Scholar 

  14. J. A. Goguen. The logic of inexact concepts. Synthese, 19: 325–373, 1969.

    Article  Google Scholar 

  15. F. Gonseth, editor. Les entretiens de Zurich sur les fondements et la müthode des sciences mathümatiques 6–9 Dücembre 1938 Leemann, Zurich, 1941. [Gottwald, 1989] S. Gottwald. Mehrwertige Logik Akademie-Verlag, Berlin, 1989. Expanded English edition: A Treatise on Many-valued Logics,RSP, 2000.

    Google Scholar 

  16. R.S. Grigolia. An algebraic analysis of n-valued systems of LukasiewiczTarski (russian). Proceedings of the University of Tbilisi A, 6–7: 121–132, 1973.

    Google Scholar 

  17. R.S. Grigolia. Algebraic analysis of Lukasiewicz-Tarski’s n-valued logical systems. In R. Wôjcicki and G. Malinowski, editors, Selected papers on Lukasiewicz sentential Calculi, pages 81–92. Ossolineum, 1977.

    Google Scholar 

  18. S. Halldén. The logic of nonsense. Uppsala Universitets drsskrift, 9: 132, 1949.

    Google Scholar 

  19. L. Hay. Axiomatization of the infinite-valued predicate calculus. Journal of Symbolic Logic, 28: 77–86, 1963.

    Article  Google Scholar 

  20. U. Höhle and E. P. Klement, editors. Non-classical logics and their applications to fuzzy subsets. Kluwer, 1995.

    Google Scholar 

  21. Z. A. Jordan. The Development of Mathematical Logic and of Logical Positivism in Poland between the Two Wars. Oxford University Press, 1945. Partially reprinted in [McCall, 1967 ].

    Google Scholar 

  22. S. C. Kleene. On a notation for ordinal numbers. Journal of Symbolic Logic, 3: 150–155, 1938.

    Article  Google Scholar 

  23. S. C. Kleene. Introduction to Metamathematics. Van Nostrand, Amsterdam and Princeton, 1952.

    Google Scholar 

  24. J. Los and R. Suszko. Remarks on sentential logics. Indagationes Math, 20: 177–183, 1958.

    Google Scholar 

  25. J. Lukasiewicz and A. Tarski. Untersuchungen über den Aussagenkalkül. Comptes rendus de la Société des Sciences et des Lettres de Varsovie, 23:1–21, 1930. English translation in [Lukasiewicz, 1970 ].

    Google Scholar 

  26. J. Lukasiewicz. Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls. Comptes rendus de la Société des Sciences et des Lettres de Varsovie, 23:51–77, 1930. English translation in[Lukasiewicz, 1970 ].

    Google Scholar 

  27. J. Lukasiewicz. Selected Works. North-Holland, Amsterdam, 1970. Edited by L. Borkowski.

    Google Scholar 

  28. G. Malinowksi. Many-valued Logics. Oxford, 1993.

    Google Scholar 

  29. S. McCall, editor. Polish Logic 1920–1939. Oxford University Press, 1967.

    Google Scholar 

  30. R. McNaughton. A theorem about infinite-valued sentential logic. Journal of Symbolic Logic, 16: 1–13, 1951.

    Article  Google Scholar 

  31. C. A. Meredith. The dependence of an axiom of Lukasiewicz. Transactions of the American Mathematical Society, 87: 54, 1958.

    Google Scholar 

  32. A. Mostowski. Models of set theory. In Foundational Studies, Selected Works, volume 1. North-Holland, Amsterdam, 1979. Lectures delivered in Varenna, September 1968.

    Google Scholar 

  33. D. Mundici. Interpretation of AF C’-algebras in Lukasiewicz sentential calculus. Journal of Functional Analysis, 65: 15–63, 1986.

    Article  Google Scholar 

  34. D. Mundici. The logic of Ulam’s game with lies. In Christina Bicchieri and Maria Luisa Dalla Chiara, editors, Knowledge, belief and strategic interaction. Cambridge University Press, 1992.

    Google Scholar 

  35. D. Mundici. Ulam’s game, Lukasiewicz logic and AF C`-algebras. Fundamenta Informaticae, 18: 151–161, 1993.

    Google Scholar 

  36. D. Mundici. A constructive proof of McNaughton’s theorem in infinite-valued logic. Journal of Symbolic Logic, 59: 596–602, 1994.

    Article  Google Scholar 

  37. G. Panti. A geometric proof of the completeness of the Lukasiewicz calculus. Journal of Symbolic Logic, 60: 563–578, 1995.

    Article  Google Scholar 

  38. J. Pavelka. On fuzzy logic, I,II,III. Zeitschrift fiir mathematische Logik und Grundlagen der Mathematik, 25:45–52, 119–134, 447–464, 1979.

    Google Scholar 

  39. E. Post. Introduction to a general theory of elementary propositions. American Journal of Mathematics, 43: 163–185, 1921.

    Article  Google Scholar 

  40. H. Rasiowa. Many-valued algorithmic logic as a tool to investigate programs. In Modern Uses of Multiple-valued Logic. D. Reidel, Dordrecht and Boston, 1977.

    Google Scholar 

  41. W. Rautenberg. 2-element matrices. Studia Logica, 40: 315–353, 1981.

    Article  Google Scholar 

  42. N. Rescher. Many-valued Logic. McGraw-Hill, New York, 1969.

    Google Scholar 

  43. D.C. Rine, editor. Computer Science and Multiple-valued Logic. North-Holland, 1984.

    Google Scholar 

  44. A. Rose and J. B. Rosser. Fragments of many-valued statement calculi. Transactions of the American Mathematical Society, 87: 1–53, 1958.

    Article  Google Scholar 

  45. I. Rosenberg. La structure des fonctions de plusieurs variables sur un ensemble fini. C. R. Acad. Sci. Paris, Ser. A.B, 260: 3817–3819, 1965.

    Google Scholar 

  46. I. Rosenberg. fiber die funktionale Vollständigkeit in den mehrwertigen Logiken (Struktur der Funktionen von mehreren Veränderlichen auf endlichen Mengen). Rozpravy Cs. Akademie Ved. Ser. Math. Nat. Sci, 80: 3–93, 1970.

    Google Scholar 

  47. I. Rosenberg. Completeness properties of multiple-valued logic algebras. In David C. Rine, editor, Computer Science and Multiple-valued logic, pages 150–192. North-Holland, 1984.

    Google Scholar 

  48. J. B. Rosser and A. R. Turquette. Many-valued Logics. North-Holland, Amsterdam, 1952.

    Google Scholar 

  49. J. B. Rosser. Axiomatisation of infinite-valued logics. Logique et Analyse, 3: 137–153, 1960.

    Google Scholar 

  50. A. Salwicki. Formalised algorithmic languages. Bull. Acad. Pol. Sci., Ser. Math. Astron. Phys., 18: 227–232, 1970.

    Google Scholar 

  51. B. Scarpellini. Die Nicht-Axiomatisierbarkeit des unendlichwertigen Prädikaten-kalküls von Lukasiewicz. Journal of Symbolic Logic, 27: 159–170, 1962.

    Article  Google Scholar 

  52. D. Scott. Completeness and axiomatizability in many-valued logic. In Proceedings of the Tarski Symposium. Proceedings of Symposia in Pure Mathematics, Vol. XXV, American Mathematical Society, Rhode Island, 1974.

    Google Scholar 

  53. D. Scott. Does many-valued logic have any use? In S. Körner, editor, Philosophy of Logic. Blackwell, Oxford, 1976.

    Google Scholar 

  54. D.J. Shoesmith and T.J. Smiley. Deducibility and manyvaluedness. J. of Symbolic Logic, 36: 610–622, 1971.

    Article  Google Scholar 

  55. T. Skolem. Bemerkungen zum Komprehensionsaxiom. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 3: 1–17, 1957.

    Google Scholar 

  56. J. Slupecki. Der volle dreiwertige Aussagenkalkül. Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, 29:9–11, 1936. English translation in [McCall, 1967 ].

    Google Scholar 

  57. J. Slupecki. Completeness criterion for systems of many-valued propositional calculus (in Polish). Comptes rendus des Séances de la Société des Sciences et des Lettres de Varsovie, 32:102–109, 1939. English translation in Studia Logica, 30, 153–157, 1972.

    Article  Google Scholar 

  58. T. Smiley. Comment on Scott [1976]. In S. Körner, editor, Philosophy of Logic. Blackwell, Oxford, 1976.

    Google Scholar 

  59. T. Traczyk. Axioms and some properties of Post algebras. Colloq. Math, 10: 193–209, 1963.

    Google Scholar 

  60. T. Traczyk. An equational definition of a class of Post algebras. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys., 12: 147–150, 1964.

    Google Scholar 

  61. S. Ulam. Adventures of a Mathematician. Scribner, New York, 1976. [Urquhart, 1973 ] A. Urquhart. An interpretation of many-valued logic. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 19: 111–114, 1973.

    Google Scholar 

  62. A. Urquhart. A finite matrix whose conseqeunce relation is not finitely axiomatizable. Rep. Math. Logic, 9: 71–71, 1977.

    Google Scholar 

  63. B. van Fraassen. Singular terms, truth value gaps and free logic. Journal of Philosophy, 63: 481–495, 1966.

    Article  Google Scholar 

  64. C. I. Wade. Post algebras and rings. Duke Math. J., 12: 389–395, 1945.

    Google Scholar 

  65. M. Wajsberg. Aksjomatyzacja trójwartsciowego rachunku zdarh (Axiomatisation of the 3-valued propositioal calculus). Comptes rendus des séances de la Sociéeé des Sciences et des Lettres de Varsovie, 24:125–148, 1931. English translation in [McCall, 1967 ].

    Google Scholar 

  66. R. B. White. The consistency of the axiom of comprehension in the infinite-valued predicate logic of Lukasiewicz. Journal of Philosophical Logic, 8: 509534, 1979.

    Google Scholar 

  67. R. Wójcicki and G. Malinowski, editors. Selected Papers on Lukasiewicz Sentential Calculi. Ossolineum, Wroclaw, 1977.

    Google Scholar 

  68. R. Wójcicki. Some remarks on the consequence operation in sentential logics. Fundamenta Mathematica, 68: 269–279, 1970.

    Google Scholar 

  69. R. Wójcicki. Theory of Logical Calculi: Basic Theory of Consequence Operations. Kluwer, Dordrecht and Boston, 1988.

    Google Scholar 

  70. R. G. Wolf. A survey of many-valued logic (1966–1974). In Modern Uses of Multiple-valued Logic. D. Reidel, Dordrecht and Boston, 1977.

    Google Scholar 

  71. A. Wroňski. On finitely based consequence operations. Studia Logica, 35: 453–458, 1976.

    Article  Google Scholar 

  72. A. Wroňski. A three element matrix whose consequence operation is not finitely based. Bulletin of the Section of Logic, Polish Academy of Sciences, 8: 68–71, 1979.

    Google Scholar 

  73. L. A. Zadeh. Fuzzy sets. Information and Control, 8: 338–353, 1965.

    Article  Google Scholar 

  74. L. A. Zadeh. Fuzzy sets, Fuzzy logics and Fuzzy systems. World Scientific, Singapore, 1996.

    Google Scholar 

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Authors and Affiliations

  1. University of Toronto, Canada

    Alasdair Urquhart

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  1. Alasdair Urquhart
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Editors and Affiliations

  1. King’s College, London, USA

    D. M. Gabbay

  2. Centrum für Informations- und Sprachverarbeitung, Ludwig-Maximilians-Universität München, Germany

    F. Guenthner

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Urquhart, A. (2001). Basic Many-Valued Logic. In: Gabbay, D.M., Guenthner, F. (eds) Handbook of Philosophical Logic. Handbook of Philosophical Logic, vol 2. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0452-6_4

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  • DOI: https://doi.org/10.1007/978-94-017-0452-6_4

  • Publisher Name: Springer, Dordrecht

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