Hitoshi Omori

Classical Negation and Expansions of Belnap---Dunn Logic

We investigate the notion of classical negation from a non-classical perspective. In particular, one aim is to determine what classical negation amounts to in a paracomplete and paraconsistent four-valued setting. We first give a general semantic characterization of classical negation and then consider an axiomatic expansion BD+ of four-valued Belnap–Dunn logic by classical negation. We show the expansion complete and maximal. Finally, we compare BD+ to some related systems found in the literature, specifically a four-valued modal logic of Béziau and the logic of classical implication and a paraconsistent de Morgan negation of Zaitsev.

Generalizing Functional Completeness in Belnap-Dunn Logic

One of the problems we face in many-valued logic is the difficulty of capturing the intuitive meaning of the connectives introduced through truth tables. At the same time, however, some logics have nice ways to capture the intended meaning of connectives easily, such as four-valued logic studied by Belnap and Dunn. Inspired by Dunn’s discovery, we first describe a mechanical procedure, in expansions of Belnap-Dunn logic, to obtain truth conditions in terms of the behavior of the Truth and the False, which gives us intuitive readings of connectives, out of truth tables. Then, we revisit the notion of functional completeness, which is one of the key notions in many-valued logic, in view of Dunn’s idea. More concretely, we introduce a generalized notion of functional completeness which naturally arises in the spirit of Dunn’s idea, and prove some fundamental results corresponding to the classical results proved by Post and Słupecki.

da Costa Meets Belnap and Nelson

There are various approaches to develop a system of paraconsistent logic, and those we focus on in this paper are approaches of da Costa, Belnap, and Nelson. Our main focus is da Costa, and we deal with a system that reflects the idea of da Costa. We understand that the main idea of da Costa is to make explicit, within the system, the area in which you can infer classically. The aim of the paper is threefold. First, we introduce and present some results on a classicality operator which generalizes the consistency operator of Logics of Formal Inconsistency. Second, we show that we can introduce the classicality operator to the systems of Belnap. Third, we demonstrate that we can generalize the classicality operator above to the system of Nelson. The paper presents both the proof theory and semantics for the systems to be introduced, and also establishes some completeness theorems.

From Paraconsistent Logic to Dialetheic Logic

The only condition for a logic to be paraconsistent is to invalidate the so-called explosion. However, the understanding of the only connective involved in the explosion, namely negation, is not shared among paraconsistentists. By returning to the modern origin of paraconsistent logic, this paper proposes an account of negation, and explores some of its implications. These will be followed by a consideration on underlying logics for dialetheic theories, especially those following the suggestion of Laura Goodship. More specifically, I will introduce a special kind of paraconsistent logic, called dialetheic logic, and present a new system of paraconsistent logic, which is dialetheic, by expanding the Logic of Paradox of Graham Priest. The new logic is obtained by combining connectives from different traditions of paraconsistency, and has some distinctive features such as its propositional fragment being Post complete. The logic is presented in a Hilbert-style calculus, and the soundness and completeness results are established.

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.