Natural Deduction for Three-Valued Regular Logics

Yaroslav Petrukhin

DOI: http://dx.doi.org/10.12775/LLP.2016.025

Abstract


IIn this paper, I consider a family of three-valued regular logics: the well-known strong and weak S.C. Kleene’s logics and two intermediate logics, where one was discovered by M. Fitting and the other one by E. Komendantskaya. All these systems were originally presented in the semantical way and based on the theory of recursion. However, the proof theory of them still is not fully developed. Thus, natural deduction systems are built only for strong Kleene’s logic both with one (A. Urquhart, G. Priest, A. Tamminga) and two designated values (G. Priest, B. Kooi, A. Tamminga). The purpose of this paper is to provide natural deduction systems for weak and intermediate regular logics both with one and two designated values.

Keywords


natural deduction; regular logic; Kleene’s logic; three-valued logic

Full Text:

PDF

References


Bochvar, D.A., “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, 2 (1981): 87–112. English translation of Bochvar’s paper of 1938. DOI: 10.1080/01445348108837023

Fitting, M., “Kleene’s three valued logics and their children”, Fundamenta Informaticae, 20 (1992): 113–131.

Karpenko, A.S., The Development of Many-Valued Logic (in Russian), LKI, 2010.

Kleene, S.C., “On a notation for ordinal numbers”, The Journal of Symbolic Logic, 3 (1938): 150–155. DOI: 10.2307/2267778

Kleene, S.C., Introduction to Metamathematics, Sixth Reprint, Wolters-Noordhoff Publishing and North-Holland Publishing Company, 1971.

Komendantskaya, E.Y., “Functional expressibility of regular Kleene’s logics” (in Russian), Logical Investigations, 15 (2009): 116–128.

Kooi, B., and A. Tamminga, “Completeness via correspondence for extensions of the logic of paradox”, The Review of Symbolic Logic, 5 (2012): 720–730. DOI: 10.1017/S1755020312000196

Łukasiewicz, J., “On three-valued logic”, pages 87–88 in Selected Works, L. Borkowski (ed.), Amsterdam, North-Holland Publishing Company, 1997 (English translation of Łukasiewicz’s paper of 1920).

Mendelson, E., Introduction to Mathematical Logic, Fourth Edition, Chapman & Hall, 1997.

Petrukhin, Y.I., “Correspondence analysis for first degree entailment”, Logical Investigations, 22, 1 (2016): 108–124.

Priest, G., “Paraconsistent logic”, in Handbook of Philosophical Logic, Second Edition, Vol. 6, M. Gabbay and F. Guenthner (eds.), Dordrecht, Kluwer, 2002. DOI: 10.1007/978-94-017-0460-1_4

Tamminga, A., “Correspondence analysis for strong three-valued logic”, Logical Investigations, 20 (2014): 255–268.

Tomova, N.E., “About four-valued regular logics” (in Russian), Logical Investigations, 15 (2009): 223–228.

Urquhart, A., “Basic many-valued logic”, in Handbook of Philosophical Logic, Second Edition, Vol. 2, M. Gabbay and F. Guenthner (eds.), Dordrecht, Kluwer, 2001. DOI: 10.1007/978-94-017-0452-6_4








Print ISSN: 1425-3305
Online ISSN: 2300-9802

Partnerzy platformy czasopism