On many-valued logics

Khatami, Seyed Mohammad Amin; Eslami, Esfandiar

doi:10.30465/lsj.2021.36264.1359

Document Type : Research

Authors

¹ Department of Computer Science, Birjand University of Technology, Birjand, Iran

² Shahid Bahonar University of Kerman

https://doi.org/10.30465/lsj.2021.36264.1359

Abstract

In the early 19th century, the ''principle of bivalence'' of the Aristotelian logic was challenged. Of course, Aristotle himself was questioned the applicability of this principle to propositions concerning future contingents, and he answered it via something like as modalities of possibility. However, Aristotle did not abandon the principle and it has not received much attention till the Renaissance. From Renaissance to the early 19th century, some philosophical considerations to this issue were developed. Rejecting the principle of bivalence implies alternative accounts of various kinds of logics such as many-valued logics in the context of logic. In this article, we first survey the development of many-valued logics by reviewing motivational ideas behind many-valued logics together with examining the aims and scopes of some of these logics. Then, we devote the rest of the paper to study various aspects of "truth value sets" and "interpretation of logical connectives" in many-valued logics to obtain a more comprehensive view on these logics.

Keywords

References

اسلامی، ا. (1391)، منطق فازی و کاربردهای آن، انتشارات دانشگاه شهید باهنر کرمان.

خاتمی، س. م. ا. و پورمهدیان، م. (1398)، ”منطق پیوسته“، منطق پژوهی، شماره 19، 89-120.

McKeon, R. ed. (1941), The Basic Works of Aristotle, New York, Random House.

Boehner, P. ed. (1945), The Tractatus de Praedestinatione et de Praescientia Dei et de Futuris Contingentibus of William Ockham, O.F.M. New York, The Franciscan Institute.

Baudry, L. ed. (1989), The Quarrel over Future Contingents, Dordrecht, Kluwer.

MacColl, H. (1906), Symbolic Logic and Its Applications. London, Longmans, Green.

Hartshorne, C. and Weiss, P. ed. (1933), Collected Papers of Charles Sanders Peirce, vol. 4 ,Cambridge, MA, Harvard.

Venanzio, R. and Vergauwen, R. (1997), “Possible worlds with impossible objects: The imaginary logic of N. A. Vasil'ev”, Logique Et Analyse, Vol. 40, No. 159, 225–248.

Łukasiewicz, J. (1920), “On three-valued logic”, In Borkowski, 1970, 87–88.

Borkowski , L. ed. (1970). Jan Łukasiewicz Selected Works, Amsterdam, North-Holland.

Łukasiewicz, J. (1925), “Demonstration of compatibility of the axioms of the theory of deduction”, Annales de la Société Polonaise de Mathématique, Vol. 3, 149.

Wajsberg, M. (1931), “Axiomatizationof the three-valued propositional calculus”, In McCall, 1967, 264–284.

McCall ,S. ed. (1967), Polish Logic 1920–1939, London, Oxford Univ. Press.

Rose, A. and J. B. Rosser, (1958), “Fragmentsof manyvaluedstatement calculi”, Trans AMS., Vol. 87, 1–53.

Meredith, C. A. (1958), “The dependence of an axiom of Łukasiewicz”, Trans AMS., Vol. 87, 54.

Chang, C. C. (1958), “Proof of an axiom of Lukasiewicz”, Trans AMS., Vol. 87, 55-56.

Stone, M. H. (1936). “The theory of representations for Boolean algebras", Trans AMS., Vol. 40, 37–111.

Chang, C.C. and Keisler, H.J. (1966), Continuous Model Theory, Princeton University Press.

Post, E. L. (1921), “Introduction to a general theory of elementary propositions”, American J. Math. Vol. 43, No. 3, 163–185.

Ben Yaacov, I. and Berenstein, A. and Henson C.W. and Usvyatsov, A. (2008), “Model theory for metric structures”, Model theory with applications to algebra and analysis, Vol. 2, London Mathematical Society Lecture Note Series, Vol. 350, Cambridge University Press, 315-427.

Gottwald, S. (2001), A Treatise on Many-valued Logics, Baldock, UK:Research Studies Press.

Mancosu, P. ed., (1998), From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s, NewYork: Oxford Univ. Press.

Feferman, S. and Dawson, J. and Kleene, S. and Moore, G. and Solovay, R. and Heijenoort, J. ed., (2001), Kurt Godel Collected Works, Vol. I, Publications 1929-1936, NewYork: Oxford Univ. Press.

Dummett, M. (1959), “A propositional calculus with denumerable matrix”, J. Symb. Logic., Vol. 24, No. 2, 97–106.

Bochvar, D. A. (1938), Trans. Bergmann, M. (1981), “On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus”, History and Philosophy of Logic, Vol. 2, 87–112.

Moh , S. K. (1954), “Logical paradoxes for many-valued systems”, .J Symb. Logic, Vol. 19, 37–40.

Kleene, S. C. (1938), “On a notation forordinal numbers”, J. Symb. Logic, Vol. 3, 150–155.

Kleene, S. C. (1952), Introduction to Metamathematics, Amsterdam: North-Holland.

Zadeh, L. A. (1965), “Fuzzy sets”, Inf. Control, Vol. 8, No. 3, 338–353.

Goguen, J. A. (1968-69), “The logic of inexact concepts”, Synthese, Vol. 19, No. 3-4, 325–73.

Hájek, P. (1998), Metamathematics of Fuzzy Logic, Springer Science.

Tarski, A. (1936) , “The concept of truth in formalized languages”, In Woodger, 1956: 152–278.

Woodger, J. H., ed. (1956), Logic, Semantics, Metamathematics: Papers from 1923 to 1938 by Alfred Tarski. Oxford,

Chang, C. C. (1959), “A New Proof of the Completeness of the Lukasiewicz Axioms”, Transactions of the American Mathematical Society, Vol. 93, No. 1, pp. 74-80.

Menger, K. (1942), “Statistical metrics”, Proc. Natl. Acad. Sci. USA., Vol. 28, No. 12, 535–37.

Esteva, F. and Godo, L. (2001), “Monoidal t-norm based logic: Towards a logic of left-continuous t-norms”, Fuzzy Sets and Systems, Vol. 124, 271–288.

Klement, E. P. and Mesiar, R. and Pap, E. (2013), Triangular norms, Springer Science & Business Media.

Logical Studies

On many-valued logics

References

References

Volume 12, Issue 1
April 2021
Pages 59-99

On many-valued logics

References

References

Volume 12, Issue 1April 2021Pages 59-99

Volume 12, Issue 1
April 2021
Pages 59-99