On Béziau’s logic Z

Hitoshi Omori, Toshiharu Waragai

DOI: http://dx.doi.org/10.12775/LLP.2008.017

Abstract


In [1] Béziau developed the paraconsistent logic Z, which is definitionally equivalent to the modal logic S5 (cf. Remark 2.3), and gave an axiomatization of the logic Z: the system HZ. In the present paper, we prove that some axioms of HZ are not independent and then propose another axiomatization of Z. We also discuss a new perspective on the relation between S5 and classical propositional logic (CPL) with the help of the new axiomatization of Z. Then we conclude the paper by making a remark on the paraconsistency of HZ.

Keywords


paraconsistent logic Z; modal logic S5; classical propositional logic

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References


Béziau, J.-Y., “The paraconsistent logic Z. A possible solution to Jaśkowski’s problem”, Logic and Logical Philosophy 15 (2006), 2, 99–111.

Jaśkowski, S., “Rachunek zdań dla systemów dedukcyjnych sprzecznych”, Studia Societatis Scientiarum Torunensis, Sectio A, Vol. I, No. 5 (1948), 57–77.

Jaśkowski, S., “A Propositional calculus for inconsistent deductive systems”, Logic and Logical Philosophy 7 (1999), 35–56; translation of [2] by O. Wojtasiewicz with corrections and notes by J. Perzanowski.

Rasiowa, H., and R. Sikorski, The Mathematics of Metamathematics, Monografie Matematyczne, tom 41, Warszawa, Polish Scientific Publishers, 1963.

Waragai, T., and T. Shidori, “A system of paraconsistent logic that has the notion of ‘behaving classically’ in terms of the law of double negation and its relation to S5”, pp. 177–187 in: Handbook of Paraconsistency, J.-Y. Béziau, W.A. Carnielli and D. Gabbay (eds.), College Publications, 2007.




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