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Various connexive FDE-based modal logics are studied. Some of these logics contain a conditional that is both connexive and strict, thereby highlighting that strictness and connexivity of a conditional do not exclude each other. In particular, the connexive modal logics cBK\(^{-}\), cKN4, scBK\(^{-}\), scKN4, cMBL, and scMBL are introduced semantically by means of classes of Kripke models. The logics cBK\(^{-}\) and cKN4 are connexive variants of the FDE-based modal logics BK\(^{-}\) and KN4 with a weak and a strong implication, respectively. The system cMBL is a connexive variant of the modal bilattice logic MBL. The latter is a modal extension of Arieli and Avron’s logic of logical bilattices and is characterized by a class of Kripke models with a four-valued accessibility relation. In the systems scBK\(^{-}\), scKN4, and scMBL, the conditional is both connexive and strict. Sound and complete tableau calculi for all these logics are presented and used to show that the entailment relations of the systems under consideration are decidable for finite premise set. Moreover, the logics \(\mathbf {cBK}^-\) and \(\mathbf {cMBL}\) are shown to be algebraizable. The algebraizability of \(\mathbf {cMBL}\) is derived from proving \(\mathbf {cMBL}\) to be definitionally equivalent to \(\mathbf {MBL}\). All connexive modal logics studied in this paper are decidable, paraconsistent, and inconsistent but non-trivial logics.

Graham Priest takes the relation of identity to be non-transitive. In this paper, we are going to discuss several consequences of identity as a non-transitive relation. We will consider the Henkin-style completeness proof for classical first-order logic with a non-transitive “identity” predicate, Leibniz-identity in Priest’s second-order minimal logic of paradox, and the question whether or not identity of individuals should be defined as Leibniz-identity.

In this introductory note, we place the new essays on Belnap-Dunn logic, FDE, of the present volume against the background of the development of FDE. This note is an invitation to study the volume. It presents a chronological perspective on Belnap-Dunn logic and a slightly idiosyncratic list of further research topics.

The present note contains a critical discussion of the methodology of paraconsistent logic in general and “the central optimisation problem of paraconsistent logics” in particular. It is argued that there exist several reasons not to consider classical logic as the reference logic for developing systems of paraconsistent logic, and it is suggested to weaken a certain maximality condition that may be seen as essential for “optimisation”, which is a methodology in the tradition of Newton da Costa. It is argued that the guiding motivation for the development of paraconsistent logics should be neither epistemological nor ontological, but informational. Moreover, it is pointed out that there are other notions of maximality and other methodologies. A methodology due to Graham Priest and Richard Routley and another methodology that focuses on a minimal shrinkage of expressiveness relative to a given reference logic are considered in some detail.

This paper deals with some criticism that has been put forward against strong, constructive negation in comparison to a certain example of Galois connected negations. The general background to this discussion is the informational interpretation of substructural logics, and the key issue is whether there exists an asymmetry or not between positive and negative information and between verification and falsification. The present paper confirms the view that a symmetrical conception is adequate for both direct and indirect variants of verification and falsification.

This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an introductory paper that surveys Prawitz's numerous contributions to proof theory and proof-theoretic semantics and puts his work into a somewhat broader perspective, both historically and systematically. Chapters include either in-depth studies of certain aspects of Dag Prawitz's work or address open research problems that are concerned with core issues in structural proof theory and range from philosophical essays to papers of a mathematical nature. Investigations into the necessity of thought and the theory of grounds and computational justifications as well as an examination of Prawitz's conception of the validity of inferences in the light of three dogmas of proof-theoretic semantics are included. More formal papers deal with the constructive behaviour of fragments of classical logic and fragments of the modal logic S4 among other topics. In addition, there are chapters about inversion principles, normalization of proofs, and the notion of proof-theoretic harmony and other areas of a more mathematical persuasion. Dag Prawitz also writes a chapter in which he explains his current views on the epistemic dimension of proofs and addresses the question why some inferences succeed in conferring evidence on their conclusions when applied to premises for which one already possesses evidence.

Chapter 1. Semantic Defectiveness: A Dissolution of Semantic Pathology Bradley Armour-Garb and James A. Woodbridge.- Chapter 2. Emptiness and discharge in sequent calculus and natural deduction Michael Arndt and Luca Tranchini.- Chapter 3. The Knowability Paradox in the light of a Logic for Pragmatics Massimiliano Carrara and Daniele Chi.- Chapter 4. A Dialetheic Interpretation of Classical Logic Massimiliano Carrara and Enrico Martino.- Chapter 5. Strongly semantic information as information about the truth Gustavo Cevolani.- Chapter 6. Priest's Motorbike and Tolerant Identity Pablo Cobreros, Paul Egre, David Ripley and Robert van Rooij.- Chapter 7. How to unify Russellian and Strawsonian definite descriptions Marie Duzi.- Chapter 8. Tableau Metatheorem for Modal Logics Tomasz Jarmuzek.- Chapter 9. On the Essential Flatness of Possible Worlds Neil Kennedy.- Chapter 10. Collective Alternatives Franz von Kutschera.- Chapter 11. da Costa meets Belnap and Nelson Hitoshi Omori and Katsuhiko Sano.- Chapter 12. Explicating the Notion of Truth within Transparent Intensional Logic Jiri Raclavsky.- Chapter 13. Leibnizian intensional semantics for syllogistic reasoning Robert van Rooij.- Chapter 14. Inter-Model Connectives and Substructural Logics Igor Sedlar.