Heinrich Wansing

The relationships between various modal logics based on Belnap and Dunn’s paraconsistent four-valued logic FDE are investigated. It is shown that the paraconsistent modal logic \, which lacks a primitive possibility operator \, is definitionally equivalent with the logic \, which has both \ and \ as primitive modalities. Next, a tableau calculus for the paraconsistent modal logic KN4 introduced by L. Goble is defined and used to show that KN4 is definitionally equivalent with \ without the absurdity constant. Moreover, a tableau calculus is defined for the modal bilattice logic MBL introduced and investigated by A. Jung, U. Rivieccio, and R. Jansana. MBL is a generalization of BK that in its Kripke semantics makes use of a four-valued accessibility relation. It is shown that MBL can be faithfully embedded into the bimodal logic \ over the non-modal vocabulary of MBL. On the way from \ to MBL, the Fischer Servi-style modal logic \ is defined as the set of all modal formulas valid under a modified standard translation into first-order FDE, and \ is shown to be characterized by the class of all models for \. Moreover, \ is axiomatized and this axiom system is proved to be strongly sound and complete with respect to the class of models for \. Moreover, the notion of definitional equivalence is suitably weakened, so as to show that \ and \ are weakly definitionally equivalent.

"This volume comprises the proceedings of the First All-Berlin Workshop on Nonclassical Logics and Information Processing, held at the Free University of Berlin, November 9-10, 1990. The scope of the ten papers in the volume is broad, covering various different subfields of logic - particularly nonclassical logic - and its applications in artificial intelligence. The papers are grouped according to the four major topics that emerged at the meeting: modal systems, logic programming, nonmonotonic logics, and proof theory. The classification is only a rough guide since the four areas overlap considerably."--PUBLISHER'S WEBSITE.

In this paper it is suggested to generalize our understanding of general (structural) proof theory and to consider it as a general theory of two kinds of derivations, namely proofs and dual proofs. The proposal is substantiated by (i) considerations on assertion, denial, and bi-lateralism, (ii) remarks on compositionality in proof-theoretic semantics, and (iii) comments on falsification and co-implication. The main formal result of the paper is a normal form theorem for the natural deduction proof system N2Int of the bi-intuitionistic logic 2Int. The proof makes use of the faithful embedding of 2Int into intuitionistic logic with respect to validity and shows that conversions of dual proofs can be sidestepped.

The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the unwanted disappearance of the diamond may be escaped. The emphasis is not laid on a discussion of the contentious premise of the knowability paradox, namely that all truths are possibly known, but on how from this assumption the conclusion is derived that all truths are, in fact, known. Nevertheless, the solution offered is in the spirit of the constructivist attitude usually maintained by defenders of the anti-realist premise. In order to avoid the paradoxical reasoning, a paraconsistent constructive relevant modal epistemic logic with strong negation is defined semantically. The system is axiomatized and shown to be complete

This volume presents recent advances in philosophical logic with chapters focusing on non-classical logics, including paraconsistent logics, substructural logics, modal logics of agency and other modal logics. The authors cover themes such as the knowability paradox, tableaux and sequent calculi, natural deduction, definite descriptions, identity, truth, dialetheism and possible worlds semantics. The developments presented here focus on challenging problems in the specification of fundamental philosophical notions, as well as presenting new techniques and tools, thereby contributing to the development of the field. Each chapter contains a bibliography, to assist the reader in making connections in the specific areas covered. Thus this work provides both a starting point for further investigations into philosophical logic and an update on advances, techniques and applications in a dynamic field. The chapters originate from papers presented during the Trends in Logic XI conference at the Ruhr University Bochum, June 2012.

Modal Logic, originally conceived as the logic of necessity and possibility, has developed into a powerful mathematical and computational discipline. It is the main source of formal languages aimed at analyzing complex notions such as common knowledge and formal provability. Modal and modal-like languages also provide us with families of restricted description languages for relational and topological structures; they are being used in many disciplines, ranging from artificial intelligence, computer science and mathematics via natural language syntax and semantics to philosophy. This volume presents a broad and up-to-date view of the field, with contributions covering both the foundations of modal logic itself and each of the aforementioned application areas. Complemented with an editorial introduction covering the roots of modal logic, this book is indispensable for any advanced student and researcher in non-classical logic and its applications.

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 p roofs, 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.