Bochvar's Three-Valued Logic and Literal Paralogics: Their Lattice and Functional Equivalence

Alexander Karpenko, Natalya Tomova

DOI: http://dx.doi.org/10.12775/LLP.2016.029

Abstract


In the present paper, various features of the class of propositional literal paralogics are considered. Literal paralogics are logics in which the paraproperties such as paraconsistence, paracompleteness and paranormality, occur only at the level of literals; that is, formulas that are propositional letters or their iterated negations. We begin by analyzing Bochvar’s three-valued nonsense logic B3 , which includes two isomorphs of the propositional classical logic CPC. The combination of these two ‘strong’ isomorphs leads to the construction of two famous paralogics P1 and I1, which are functionally equivalent. Moreover, each of these logics is functionally equivalent to the fragment of logic B3 consisting of external formulas only. In conclusion, we structure a four-element lattice of three-valued paralogics with respect to the possession of paraproperties.


Keywords


Bochvar’s logic B3; isomorphs, extended formulas; paraconsistent logics P1 and P12 ; paracomplete logics I1 and I12; paranormal logic TK1; strong and weak modus ponens; lattice of paralogics

Full Text:

PDF

References


Anshakov, O., and S. Rychkov, “On finite-valued propositional logical calculi”, Notre Dame Journal of Formal Logic, 36, 4 (1995): 606–629.DOI: 10.1305/ndjfl/1040136920

Arruda, A.I., “On the imagionary logic of N.A. Vasil’ev”, pages 3–24 in A.I. Arruda, N.C.A. da Costa, and R. Chuaqui (eds.), Non-Classical logics, Model Theory and Computability, Nort-Holland, Amsterdam, 1977.

Avron, A., “Natural 3-valued logic – x characterization and proof theory”, The Journal of Symbolic Logic, 56, 1 (1991): 276–294. DOI: 10.2307/2274919

Batens, D., “Paraconsistent extensional propositional logics”, Logique et Analyse, 90–91 (1980): 127–139.

Bochvar, A.D., 1938, “On a three-valued calculus and its application to analysis of paradoxes of classical extended functional calculus”, History and Philosophy of Logic, 2 (1981): 87–112. DOI: 10.1080/01445348108837023

Boscaino, E.G., Os Cálculos Paraconsistentes P 1 e β 2 , thesis, PUC-S. Paulo, 1992.

Brunner, A.B.M., and W.A. Carnielli, “Anti-intuitionism and paraconsistency”, Journal of Applied Logics, 3, 1 (2005): 161–184. DOI: 10.1016/j.jal.2004.07.016

Carnielli, W.A., “Possible-translations semantics for paraconsistent logics”, pages 149–163 in D. Batens, C. Mortensen, G. Priest, and J.-P. van Bendegem (eds.), Frontiers of Paraconsistent Logic, Baldock Research Studies Press, 2000.

Carnielli, W.A., J.Marcos, “A Taxonomy of C-systems”, pages 1–94 in W.A.Carnielli, M.E.Coniglio, and I.M.L. D’Ottaviano (eds.), Paraconsistency: The Logical Way to the Inconsistent, vol. 228 of Lecture Notes in Pure and Applied Mathematics, New York, 2002. DOI: 10.1201/9780203910139.pt1

Ciucci, D., and D. Dubois, “A map of dependencies among three-valued logics”, Information Sciences, 250 (2013): 162–177. DOI: 10.1016/j.ins.2013.06.040

Ciuciura, J. “A weakly-intuitionistic logic I1”, Logical Investigations, 21, 2 (2015): 53–60.

Ciuciura, J., “Paraconsistency and Sette’s calculus P1”, Logic and Logical Philosophy, 24, 2 (2015): 265–273. DOI: 10.12775/LLP.2015.003

da Costa, N.C.A., and E.H. Alves, “Relations between paraconsistent logic and many-valued logic”, Bulletin of the Section of Logic, 10, 4 (1981): 185–191.

Devyatkin L. Yu., A.S. Karpenko, and V.M. Popov, “Three-valued characteristic matrices of classical propositional logic” (in Russian), Proceedings of the research logical seminar of Institute of Philosophy Russian Academy of Science, XVIII (2007): 50–62.

Devyatkin, L.Yu., Three-valued semantics for the classical propositional logic (in Russian), Moscow, Institute of Philosophy of RAS, 2011.

D’Ottaviano, I.M.L., and H. de A. Feitosa, “Paraconsistent logics and translations”, Synthese, 125 (2000): 77–95.

Finn, V.K., “An axiomatization of some three-valued logics and their algebras” (in Russian), pages 398–438 in P.V. Tavanetz and V.A. Smirnov (eds.), Philosophy and Logic, Moscow, NAUKA Publishers, 1974.

Finn, V.K., “Criterion of functional completeness for B3 ”, Studia Logica, 33, 2 (1974): 121–125.

Finn, V.K., O.M. Anshakov, R. Grigolia, and M.I. Zabezhailo, “Many-valued logics as fragments of formalized semantics”, Acta Philisophica Fennica, 35 (1982): 239–272.

Finn V.K., and R. Grigolia, “Nonsense logics and their algebraic properties”, Theoria, LIX, 1–3 (1993): 207–273.

Jaśkowski, S., 1948, “A propositional calculus for inconsistent deductive systems”, Studia Logica, 24 (1969): 143–157; and Logic and Logical Philosophy, 7 (1999): 35–56. DOI: 10.12775/LLP.1999.003

Karpenko, A.S., “Paraconsistent structure inside of many-valued logic”, Synthese, 66 (1986): 63–69. DOI: 10.1007/BF00413579

Karpenko, A.S., “Jaśkowski’s criterion and three-valued paraconsistent logics”, Logic and Logical Philosophy, 7 (1999): 81–86. DOI: 10.12775/LLP.1999.006

Karpenko, A.S., “The classification of propositional calculi”, Studia Logica, 66, 2 (2000): 253–271.

Karpenko, A.S., “A maximal paraconsistent logic: The combination of two three-valued isomorphs of classical propositional logic”, pages 181–187 in D. Batens, C. Mortensen, G. Priest, and J.-P. van Bendegem (eds.), Frontiers of Paraconsistent Logic, Baldock Research Studies Press, 2000.

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

Kleene, S.C., “On a notation for ordinal numbers”, The Journal of Symbolic Logic, 3, 4 (1938): 150–155. DOI: 10.2307/2267778

Kleene, S.C., Introduction to Metamathematics, D. Van Nostrand Company, N.Y., 1952.

Lewin, R.A., I.F. Mikenberg, and M.G. Schwarze, “Algebraization of paraconsistent logic P 1 ”, The Journal of Non-Classical Logic, 7, 2 (1990): 79–88.

Lewin, R.A., and I.F. Mikenberg, “Literal-paraconsistent and literalparacomplete matrices”, Math. Log. Quart., 52, 5 (2006): 478–493. DOI: 10.1002/malq.200510044

Loparic, A., and N.C.A. da Costa, “Paraconsistency, paracompleteness and induction”, Logique et Analyse, 113 (1986): 73–80.

Łukasiewicz J., 1930, “Philosophical remarks on many-valued systems of propositional logic”, pages 153–178 in J. Łukasiewicz, Selected Works, North-Holland and PWN, Amsterdam and Warszawa, 1970.

Łukasiewicz, J., and A. Tarski, 1930, “Investigations into the sentential calculus”, pages 131–152 in J. Łukasiewicz, Selected Works, North-Holland and PWN, Amsterdam and Warszawa, 1970.

Marcos, J., “Possible translations semantics”, pages 119–128 in W.A. Carnielli, F.M. Dionísio, and P. Mateus (eds.), Proceedings of CombLog’04, Workshop on Combination of Logics: Theory and Applications, Instituto Superior Técnico, Lisboa, 2004.

Marcos, J., “Possible translations semantics”, preprint, 2005. http://sqig.math.ist.utl.pt/pub/MarcosJ/04-M-pts.pdf

Marcos, J., “On a problem of da Costa”, pages 53–69 in G. Sica (ed.), Essays on the Foundations of Mathematics and Logic, vol. 2, Polimetrica, 2005.

Mortensen, C., “Paraconsistency and C1 ”, pages 289–305 in G. Priest, R. Routley, and J. Norman (eds.), Paraconsistent Logic. Essays on the Inconsistent, Philosophia Verlag, Munich, 1989.

Piróg-Rzepecka, K., “A predicate calculus with formulas which lose sense and the corresponding propositional calculus”, Bulletin of the Section of Logic, 2 (1973): 22–29.

Popov, V.M., “On a three-valued paracomplete logic” (in Russian), Logical Investigations, 9 (2002): 175–178.

Popov, V.M., “On the logics related to A. Arruda’s system V1”, Logic and Logical Philosophy, 7 (1999): 87–90. DOI: 10.12775/LLP.1999.007

Post, E.L., “Introduction to a general theory of elementary propositions”, American Journal of Mathematics, 43, 3 (1921): 163–185. DOI: 10.2307/2370324

Priest, G., K. Tanaka, and Z. Weber, “Paraconsistent logic”, Stanford Encyclopedia of Philosophy, 2013. http://plato.stanford.edu/entries/logic-paraconsistent

Puga, L.Z., and N.C.A. da Costa, “On the imaginary logic of N.A. Vasiliev”, Z. Math. Logik Grundl. Math, 34 (1988): 205–211. DOI: 10.1002/malq.19880340304

Pynko, A.P., “Algebraic study of Sette’s maximal paraconsistent logic”, Studia Logica, 54, 1 (1995): 89–128. DOI: 10.1007/BF01058534

Rescher, N., Many-Valued Logic, N.Y., McGraw Hill, 1969. Reprinted: Gregg Revivals, Aldershot, 1993.

Sette, A.M., “On propositional calculus P1 ”, Mathematica Japonica, 18 (1973): 173–180.

Sette, A.M., and E.H. Alves, “On the equivalence between some systems of non-classical logic”, Bulletin of the Section of Logic, 25 (1996): 68–72.

Sette, A.M., and W.A. Carnielli, “Maximal weakly-intuitionistic logics”, Studia Logica, 55, 1 (1995): 181–203. DOI: 10.1007/BF01053037

Shestakov, V.I., “Modeling operations of propositional calculus through the relay contact circuit” (in Russian), Logical investigations, 2 (1959): 315–351.

Shestakov, V.I., “On the relationship between certain three-valued logical calculi” (in Russian), Uspekhi Mat. Nauk, 19, 2, 116 (1964): 177–181.

Shestakov, V.I., “On one fragment of D.A. Bochvar’s calculus” (in Russian), Information issues of semiotics, linguistics and automatic translation. VINITI, 1 (1971): 102–115.

Tomova, N.E., “Implicative extensions of regular Kleene logics” (in Russian), Logical Ivestigations, 16 (2010): 233–258.

Tomova, N.E., “Natural p-logics” (in Russian), Logical Investigations, 17 (2011): 256–268.

Tomova, N.E., “A lattice of implicative extensions of regular Kleene’s logics”, Report on Mathematical Logic, 47 (2012): 173–182. DOI: 10.4467/20842589RM.12.008.0689

Tomova, N.E., “Natural implication and modus ponens principle”, Logical Investigations, 21, 1 (2015): 138–143.

Tomova, N.E., “Erratum to: Natural implication and modus ponens principle”, Logical Investigations, 21, 2 (2015): 186–187.








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

Partnerzy platformy czasopism