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One of the oldest systems of paraconsistent logic is the set of so-called C-systems of Newton da Costa, and this has been generalized into a family of systems now known as logics of formal inconsistencies by Walter Carnielli, Marcelo Coniglio and João Marcos. The characteristic notion in these systems is the so-called consistency operator which, roughly speaking, indicates how gluts are behaving. One natural question then is to ask if we can let not only gluts but also gaps be around and generalize the notion of consistency into classicality. This is already considered by Andréa Loparić and da Costa in the style of C-systems. The aim of this paper is to develop a family of systems that generalizes the system of Loparić and da Costa which may be called logics of formal classicality.

In two recent articles, Norihiro Kamide introduces unusual variants of Nelson’s paraconsistent logic and its classical extension. Kamide’s systems, IP and CP, are unusual insofar as double negations in these logics behave as intuitionistic and classical negations, respectively. In this article we present Hilbert-style axiomatizations of both IP and CP. The axiom system for IP is shown to be sound and complete with respect to a four-valued Kripke semantics, and the axiom system for CP is characterized by four-valued truth tables. Moreover, we note some properties of IP and CP, and emphasize that these logics are unusual also because they are contra-classical and inconsistent but nontrivial. We point out that Kamide’s approach exemplifies a general method for obtaining contra-classical logics, and we briefly speculate about a linguistic application of Kamide’s logics.

The present note revisits the joint work of Leonard Goddard and Richard Routley on significance logics with the aim of shedding new light on their understanding by studying them under the lens of recent semantic developments, such as the plurivalent semantics developed by Graham Priest. These semantics allow sentences to receive one, more than one, or no truth-value at all from a given carrier set. Since nonsignificant sentences are taken to be neither true nor false, i.e. truth-value gaps, in this essay we show that with the aid of plurivalent semantics it is possible to straightforwardly instantiate Goddard and Routley's understanding of how the connectives should work within significance logics. Fil: Szmuc, Damián Enrique. Instituto de Investigaciones Filosóficas - Sadaf; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Filosofía y Letras; Argentina Fil: Omori, Hitoshi. Kyoto University; Japón

We outline the rather complicated history of attempts at axiomatizing Jaśkowski’s discussive logic $$\mathbf {D_2}$$ and show that some clarity can be had by paying close attention to the language we work with. We then examine the problem of axiomatizing $$\mathbf {D_2}$$ in languages involving discussive conjunctions. Specifically, we show that recent attempts by Ciuciura are mistaken. Finally, we present an axiomatization of $$\mathbf {D_2}$$ in the language Jaśkowski suggested in his second paper on discussive logic, by following a remark of da Costa and Dubikajtis. We also deal with an interesting variant of $$\mathbf {D_2}$$ , introduced by Ciuciura, in which negation is also taken to be discussive.

In this introduction to the special issue “40 years of FDE”, we offer an overview of the field and put the papers included in the special issue into perspective. More specifically, we first present various semantics and proof systems for FDE, and then survey some expansions of FDE by adding various operators starting with constants. We then turn to unary and binary connectives, which are classified in a systematic manner (affirmative/negative, extensional/intensional). First-order FDE is also briefly revisited, and we conclude by listing some open problems for future research.

For one of the most well-known many-valued logics FDE, there are several semantics, including the star semantics by Richard Routley and Valerie Routley, the two-valued relational semantics by Michael Dunn and the four-valued semantics by Nuel Belnap. The last semantics inspired Yaroslav Shramko and Heinrich Wansing to introduce the trilattice SIXTEEN\(_3\). In this article, we offer two alternative semantical presentations for SIXTEEN\(_3\), by applying the Routleys’ semantics and the Dunn semantics. Based on our new semantics, we discuss related systems with less truth values, as well as the relation to FDE-based modal logics.

In this introduction, we offer an overview of main systems developed in the growing literature on connexive logic, and also point to a few topics that seem to be collecting attention of many of those interested in connexive logic. We will also make clear the context to which the papers in this special issue belong and contribute.

In this paper, we investigate an extended first-order Belnap-Dunn logic with classical negation. We introduce a Gentzen-type sequent calculus FBD+ for this logic and prove theorems for syntactically and semantically embedding FBD+ into a Gentzen-type sequent calculus for first-order classical logic. Moreover, we show the cut-elimination theorem for FBD+ and prove the completeness theorems with respect to both valuation and many-valued semantics for FBD+.

The paper explores the connection between Soren Hallden's logic of nonsense (and its expansions) and Logics of Formal Inconsistency (LFIs), one of the main traditions in paraconsistency. Although not much attention has been payed by paraconsistentists, Hallden's logic can be nicely connected to LFIs. The main result of the paper is a reformulation of an expansion of logic of nonsense studied by Lennart Aqvist and Krister Segerberg in the light of LFIs. More specifically, we present a three-valued semantics and a Hilbert-style system, and prove soundness and completeness results. We also observe some definability results related to the consistency and just-true operators, and briefly discuss an 'interpretation' of truth values of the three-valued semantics in view of the recent work of Graham Priest on plurivalent semantics.

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.