Toshiharu Waragai

On Béziau’s logic Z

In [1] Beziau 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.

SOME NEW RESULTS ON PCL1 AND ITS RELATED SYSTEMS

In [Waragai & Shidori, 2007], a system of paraconsistent logic called PCL1, which takes a similar approach to that of da Costa, is proposed. The present paper gives further results on this system and its related systems. Those results include the concrete condition to enrich the system PCL1 with the classical negation, a comparison of the concrete notion of “behaving classically” given by da Costa and by Waragai and Shidori, and a characterisation of the notion of “behaving classically” given by Waragai and Shidori.

On the Propagation of Consistency in Some Systems of Paraconsistent Logic

The present paper offers some results on the propagation of consistency in two systems of Logics of Formal Inconsistency(LFIs). One is the system Bk of Avron, which is an extension of the base system mbC of Carnielli, Coniglio and Marcos, and the other is an extension of Bk to the predicate calculus which will be referred to as Bk ∗. We first present a new characterization of consistency operator in Bk. This reflects the intuition of the consistency operator quite well. Second, we prove that two kinds of propagation of consistency known in the literature are actually equivalent to certain forms of de Morgan’s laws without any occurrence of the consistency operator. Finally, we extend the result in Bk to Bk ∗.

A Note on Majkić's Systems

The present note offers a proof that systems developed by Majkić are actually extensions of intuitionistic logic, and therefore not paraconsistent.

Negative Modalities in the Light of Paraconsistency

Modality and non-classical negation have some interesting connections. One of the most famous connections is the relation between S4-modality and intuitionistic negation. In this chapter, we focus on the negative modalities in the perspective of paraconsistency. The basic idea here is to consider the negative modality defined as ‘not necessarily’ or equivalently ‘possibly not’ where ‘not’ is classical negation, and ‘necessarily’ and ‘possibly’ are modalities in modal logics. This chapter offers a solution to the problem of axiomatizing systems of modal logic such as D and S4 in terms of negative modalities. One of the upshots of this solution is that we may consider the semantics of paraconsistency with the help of various considerations known in the literature of modal logics related to D and S4.