On the da Costa, Dubikajtis and Kotas’ system of the discursive logic, D*2

Janusz Ciuciura

doi:10.12775/LLP.2005.015

Authors

Janusz Ciuciura University of Łódź

DOI:

https://doi.org/10.12775/LLP.2005.015

Keywords

discursive (discussive) logic, D2, paraconsistent logic, labelled tableaux

Abstract

In the late forties, Stanisław Jaśkowski published two papers on the discursive (or discussive) sentential calculus, D₂. He provided a definition of it by an interpretation in the language of S5 of Lewis. The known axiomatization of D₂ with discursive connectives as primitives was introduced by da Costa, Dubikajtis and Kotas in 1977. It turns out, however, that one of the axioms they used is not a thesis of the real Jaśkowski’s calculus. In fact, they built a new system, D*₂ for short, that differs from D₂ in many respects. The aim of this paper is to introduce a direct Kripke-type semantics for the system, axiomatize it in a new way and prove soundness and completeness theorems. Additionally, we present labelled tableaux for D*₂.

Author Biography

Janusz Ciuciura, University of Łódź

Department of Logic

References

Achtelik G., L. Dubikajtis, E. Dudek, J. Kanior, “On Independence of Axioms of Jaśkowski Discussive Propositional Calculus”, Reports on Mathematical Logic 11: 3–11, 1981.

Ciuciura, J., “History and Development of the Discursive Logic”, Logica Trianguli 3 (1999): 3–31.

Ciuciura, J., “Logika dyskusyjna”, Principia 35–36: 279–291, 2003.

Ciuciura, J., “A New Real Axiomatization of D2”, 1 st Congress on Universal Logic, Montreux, 31. 03–03. 04. 2005, an abstract available at http://www.uni-log.org/one2.html

Ciuciura, J., “Labelled Tableaux for D2”, Bulletin of the Section of Logic 33 (4): 223-236, 2004.

N.C.A. da Costa, Lech Dubikajtis, “A New Axiomatization for the Discursive Propositional Calculus”. In: A.I. Arruda, N.C.A. da Costa, R. Chuaqui (eds.), Non Classical Logics, Model Theory and Computability, North-Holland Publishing, Amsterdam 1977, pp. 45–55.

Fitting, M.C., First-Order Logic and Automated Theorem Proving, Springer, 1996 (first edition, 1990).

Goré, R., “Tableau Methods for Modal and Temporal Logics”. In: M. D’Agostino, D. Gabbay, R. Haenle and J. Possegga (eds.), Handbook of Tableau Methods, Kluwer Academic Publishers, Dordrecht-Boston-London, 1999, pp. 297–396.

Jaśkowski, S., “A Propositional Calculus for Inconsistent Deductive Systems”, Logic and Logical Philosophy, 7 (1): 35-56, 2001.

Jaśkowski, S., “On the Discussive Conjunction in the Propositional Calculus for Inconsistent Deductive Systems”, Logic and Logical Philosophy, 7 (1): 57-59, 2001.

Kotas, J., “Discussive Sentential Calculus of Jaśkowski”, Studia Logica 34 (2): 149-168, 1975.

Kotas, J., N.C.A. da Costa, “On Some Modal Logical Systems Sefined 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, North-Holland Publishing, Amsterdam 1977, pp. 57–73.