### Natural Deduction for Four-Valued both Regular and Monotonic Logics

DOI: http://dx.doi.org/10.12775/LLP.2017.001

#### Abstract

The development of recursion theory motivated Kleene to create regular three-valued logics. Remove it taking his inspiration from the computer science, Fitting later continued to investigate regular three-valued logics and defined them as monotonic ones. Afterwards, Komendantskaya proved that there are four regular three-valued logics and in the three-valued case the set of regular logics coincides with the set of monotonic logics. Next, Tomova showed that in the four-valued case regularity and monotonicity do not coincide. She counted that there are 6400 four-valued regular logics, but only six of them are monotonic. The purpose of this paper is to create natural deduction systems for them. We also describe some functional properties of these logics.

#### Keywords

#### Full Text:

PDF#### References

Asenjo, F.G., “A calculus of antinomies”, Notre Dame Journal of Formal Logic, 7 (1966): 103-105. DOI: 10.1305/ndjfl/1093958482

Belnap, N.D., “A useful four-valued logic”, pages 7–37 in J.M. Dunn and G. Epstein, Modern Uses of Multiple-Valued Logic, Boston: Reidel Publishing Company, 1977. DOI: 10.1007/978-94-010-1161-7_2

Belnap, N.D., “How a computer should think”, pages 30–56 in G. Rule (ed.), Contemporary Aspects of Philosophy, Stocksfield: Oriel Press, 1977.

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

Bolotov, A., and V. Shangin, “Natural deduction system in paraconsistent setting: Proof search for PCont”, Journal of Intelligent Systems, 21 (2012): 1–24. DOI: 10.1515/jisys-2011-0021

Copi, I.M., C. Cohen, and K. McMahon, Introduction to Logic, Fourteenth Edition, Routledge, New York, 2011.

Dunn, J.M., “Intuitive semantics for first-degree entailment and coupled trees”, Philosophical Studies, 29 (1976): 149–168. DOI: 10.1007/BF00373152

Finn, V.K., “Axiomatization of some three-valued propositional calculi and their algebras” (in Russian), pages 398–438 in P. Tavanets and V. Smirnov (eds.), Philosophy in the Contemporary World. Philosophy and Logic, Moscow: Nauka Publ., 1974.

Fitting, M., “Kleene’s logic, generalized”, Journal of Logic and Computation, 1 (1991): 797-810. DOI: 10.1093/logcom/1.6.797

Fitting, M., “Kleene’s three valued logics and their children”, Fundamenta Informaticae, 20 (1994): 113–131. DOI: 10.3233/FI-1994-201234

Fitting, M., “Negation as refutation”, pages 63–70 in R. Parikh (ed.), Proceedings of the Fourth Annual Symposium on Logic in Computer Science (1989), IEEE, 1989. DOI: 10.1109/LICS.1989.39159

Font, J.M., “Belnap’s four-valued logic and De Morgan lattices”, Logic Journal of the IGPL, 5 (1997): 1–29. DOI: 10.1093/jigpal/5.3.1-e

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

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

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

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 L. Borkowski (ed.), Jan Łukasiewicz: Selected Works, Amsterdam, North-Holland Publishing Company, 1997. English translation of Łukasiewicz’s paper of 1920.

Petrukhin, Y., “Natural deduction for three-valued regular logics”, Logic and Logical Philosophy 26, 2 (2017): 197–206. DOI: 10.12775/LLP.2016.025

Pietz, A., and U. Rivieccio, “Nothing but the truth”, Journal of Philosophical Logic, 42 (2013): 125–135. DOI: 10.1007/s10992-011-9215-1

Priest, G., “Logic of paradox revisited”, Journal of Philosophical Logic, 13 (1984): 153–179. DOI: 10.1007/BF00453020

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

Priest, G., “The logic of paradox”, Journal of Philosophical Logic, 8 (1979): 219–241. DOI: 10.1007/BF00258428

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.

Zaitsev, D.V., and Y.V. Shramko, “Logical entailment and designated values” (in Russian), Logical Investigations, 11 (2004): 126–137.

ISSN: 2300-9802 (electronic version)