This book presents the author’s recent investigations of the two main concepts of negation developed in the constructive logic: the negation as reduction to absurdity (L.E.J. Brouwer) and the strong negation (D. Nelson) are studied in the setting of paraconsistent logic. The paraconsistent logics are those, which admit inconsistent but non-trivial theories, i.e., the logics which allow making inferences in non-trivial fashion from an inconsistent set of hypotheses. Logics in which all inconsistent theories are trivial are called explosive. In the intuitionistic logic Li, the negation is defined as reduction to absurdity. The concept of strong negation is realized in the Nelson logic N3. Both logics are explosive and have paraconsistent analogs: Johansson’s logic Lj and paraconsistent Nelson’s logic N4. It will be shown that refusing the explosion axiom "contradiction implies everything" does not lead to decrease of the expressive power of a logic. To understand, which new expressive possibilities have the logics Lj and N4 as compared to the explosive logics Li and N3, we study the lattices of extensions of the logics Lj and N4. This is the first case when lattices of paraconsistent logics are systematically investigated. The study is based on algebraic methods, demonstrates the remarkable regularity and the similarity of structures of both lattices of logics, and gives essential information on the paraconsistent nature of logics Lj and N4. The methods developed in this book can be applied for investigation of other classes of paraconsistent logics. – 1. Introduction.- Part I. Reductio ad Absurdum.- 2. Minimal Logic. Preliminary Remarks.- 3. Logic of Classical Refutability.- 4. The Class of Extensions of Minimal Logic.- 5. Adequate Algebraic Semantics for Extensions of Minimal Logic.- 6. Negatively Equivalent Logics.- 7. Absurdity as Unary Operator.- Part II. Strong Negation.- 8. Semantical Study of Paraconsistent Nelson's Logic.- 9. N4-Lattices.- 10. The Class of N4-Extensions. M.-M. V.

La logique paraconsistante permet de raisonner en présence de contradictions et offre par là même une nouvelle conception de la notion de preuve qui s’avère particulièrement intéressante pour l’intelligence artificielle. – Introduction. – 1. Qu’est-ce que la logique paraconsistante ?; – 2. Systèmes de preuves paraconsistantes (Déduction naturelle; Calcul des séquents); – 3. Interprétations et applications des systèmes de preuves paraconsistants.

In a world plagued by disagreement and conflict, one might expect that the exact sciences of logic and mathematics would provide a safe harbor. In fact these disciplines are rife with internal divisions between different, often incompatible, systems. Do these disagreements admit of resolution? Can such resolution be achieved without disturbing assumptions that the theorems of logic and mathematics state objective truths about the real world? In this original and historically rich book John Woods explores apparently intractable disagreements in logic and the foundations of mathematics and sets out conflict resolution strategies that evade or disarm these stalemates. An important sub-theme of the book is the extent to which pluralism in logic and the philosophy of mathematics undermines realist assumptions. This book makes an important contribution to such areas of philosophy as logic, philosophy of language and argumentation theory. It will also be of interest to mathematicians and computer scientists. – Contents : – 1. Conflict in the abstract sciences; – 2. Modalities; – 3. Managing inconsistency; – 4. Semantic intuitions; – 5. Sets and truths; – 6. Fiction; – 7. Currying liars; – 8. Normativity. – Includes bibliographical references (p. 341-358) and index.