Algebraization of Jaśkowski’s Paraconsistent Logic D2

Janusz Ciuciura

doi:10.1515/slgr-2015-0036

.blurhash-client-img { display: none !important; }

Algebraization of Jaśkowski’s Paraconsistent Logic D2

Studies in Logic, Grammar and Rhetoric

Volume 42 (2015): Issue 1 (September 2015)

By: Janusz Ciuciura

Open Access

|Nov 2015

Abstract

The aim of this paper is to present an algebraic approach to Jaśkowski’s paraconsistent logic D2. We present: a D2-discursive algebra, Lindenbaum- Tarski algebra for D2 and D2-matrices. The analysis is mainly based on the results obtained by Jerzy Kotas in the 70s.

References

Arieli, O., Avron, A., & Zamansky, A. (2011). Ideal Paraconsistent Logics. Studia Logica, 99(1-3), 31-60.10.1007/s11225-011-9346-y
Search in Google Scholar
Béziau, J.-Y. (2006). The paraconsistent logic Z. A possible solution to Jaśkowski’s problem. Logic and Logical Philosophy, 15(2), 99-111.
Search in Google Scholar
Chellas, B. F. (1980). Modal Logic: An Introduction. Cambridge: Cambridge University Press.10.1017/CBO9780511621192
Search in Google Scholar
Ciuciura, J. (2008). Negations in the Adjunctive Discursive Logic. Bulletin of the Section of Logic, 3 7(3-4), 143-160.
Search in Google Scholar
Ciuciura, J. (2005). On the da Costa, Dubikajtis and Kotas’ system of the discursive logic, D∗ 2. Logic and Logical Philosophy, 14(2), 235-252.10.12775/LLP.2005.015
Search in Google Scholar
Cvetko-Vah, K. (2011). On Strongly Symmetric Skew Lattices. Algebra universalis, 66(1-2), 99-113. 10.1007/s00012-011-0143-2
Search in Google Scholar
da Costa, N.C.A., & Dubikajtis, L. (1977). New Axiomatization for the Discussive Propositional Calculus. In A.I. Arruda, N.C.A. da Costa, & R. Chuaqui, (Eds), Non Classical Logics, Model Theory and Computability (pp. 45-55). Amsterdam: North-Holland Publishing.
Search in Google Scholar
da Costa, N.C.A., & Dubikajtis, L. (1968). Sur la logique discursive de Jaśkowski. Bulletin de L’Académie Polonaise des Sciences (Série des sciences math., astr. et phys.), 16(7), 551-557.
Search in Google Scholar
Hawranek, J. (1980). A Matrix Adequate for S5 with MP and RN. Bulletin of the Section of Logic, 9(3), 122-124.
Search in Google Scholar
Hughes, G.E., & Cresswell, M. J. (1996). A New Introduction to Modal Logic. London: Routledge.10.4324/9780203290644
Search in Google Scholar
Jaśkowski, S. (1948). Rachunek zdań dla systemow dedukcyjnych sprzecznych. Societatis Scientiarum Torunensis, Sect. A, I, 5, 57-77 (in Polish).
Search in Google Scholar
Jaśkowski, S. (1969). Propositional Calculus for Contradictory Deductive Systems. Studia Logica, 24, 143-157 (the first English translation of (Jaśkowski, 1948)).10.1007/BF02134311
Search in Google Scholar
Jaśkowski, S. (1999a). A Propositional Calculus for Inconsistent Deductive Systems. Logic and Logical Philosophy, 7(1), 35-56 (the second English translation of (Jaśkowski, 1948)).10.12775/LLP.1999.003
Search in Google Scholar
Jaśkowski, S. (1949). O koniunkcji dyskusyjnej w rachunku zdań dla systemow dedukcyjnych sprzecznych. Societatis Scientiarum Torunensis, Sect. A, I, 8, 171-172 (in Polish).
Search in Google Scholar
Jaśkowski, S. (1999b). On the Discussive Conjuntion in the Propositional Calculus for Inconsistent Deductive Systems. Logic and Logical Philosophy, 7(1), 57-59 (English translation of (Jaśkowski, 1949)).10.12775/LLP.1999.004
Search in Google Scholar
Jordan, P. (1949). Uber Nichtkommutative Verb¨ande. Arch. Math., 2, 56-59.10.1007/BF02036754
Search in Google Scholar
Kotas, J. (1975a). Discussive Sentential Calculus of Jaśkowski. Studia Logica, 34(2), 149-168.10.1007/BF02123384
Search in Google Scholar
Kotas, J. (1971). On the algebra of classes of formulae of Jaśkowski’s discussive system. Studia Logica, 27(1), 81-90.10.1007/BF02282553
Search in Google Scholar
Kotas, J. (1975b). On Quantity of Logical Values in the Discussive D2 System and in Modular Logic. Studia Logica, 33(3), 273-275.10.1007/BF02123281
Search in Google Scholar
Kotas, J., & da Costa, N.C.A. (1977). On Some Modal Logical Systems Defined in Connexion with Jaśkowski’s Problem. In A.I. Arruda, N.C.A. da Costa, & R. Chuaqui (Eds). Non Classical Logics, Model Theory and Computability (pp. 57-73). Amsterdam: North-Holland Publishing.
Search in Google Scholar
Leech, J. (1989). Skew lattices in rings. Alg. Universalis, 26, 48-72.10.1007/BF01243872
Search in Google Scholar
Nasieniewski, M., & Pietruszczak, A. (2012). On the weakest modal logics defining Jaśkowski’s logic D2 and the D2-consequence. Bulletin of the Section of Logic, 41(3-4), 215-232.
Search in Google Scholar
Priest, G., Tanaka, K., & Weber, Z. (1996). Paraconsistent Logic. In Zalta, E.N. (Principal Ed.). Stanford Encyclopedia of Philosophy, available at http://plato.stanford.edu/entries/logic-paraconsistent/
Search in Google Scholar
Rasiowa, H., (1974). An algebraic approach to non-classical logics. Studies in Logic and the Foundations of Mathematics, vol. 78. North-Holland-Warszawa- Amsterdam: PWN.
Search in Google Scholar