Fuzzy logics are many-valued logics that are well suited to reasoning in the context of vagueness. They provide the basis for the wider field of Fuzzy Logic, encompassing diverse areas such as fuzzy control, fuzzy databases, and fuzzy mathematics. This book provides an accessible and up-to-date introduction to this fast-growing and increasingly popular area. It focuses in particular on the development and applications of "proof-theoretic" presentations of fuzzy logics; the result of more than ten years of intensive work by researchers in the area, including the authors. In addition to providing alternative elegant presentations of fuzzy logics, proof-theoretic methods are useful for addressing theoretical problems (including key standard completeness results) and developing efficient deduction and decision algorithms. Proof-theoretic presentations also place fuzzy logics in the broader landscape of non-classical logics, revealing deep relations with other logics studied in Computer Science, Mathematics, and Philosophy. The book builds methodically from the semantic origins of fuzzy logics to proof-theoretic presentations such as Hilbert and Gentzen systems, introducing both theoretical and practical applications of these presentations. – 1. Introduction.- 2. The Semantic Basis.- 3. Hilbert Systems.- 4. Gentzen Systems.- 5. Syntactic Eliminations.- 6. Fundamental Logics.- 7. Uniformity and Efficiency.- 8. First-Order Logics.- 9. Further Topics. M.-M. V.

Introduction; – Lakos's Proposal; – Some New Machinery; – The Generic Fuzzy Logic for Non-Scalar Hedges FLh; – Conclusion.

Axé sur le thème de la logique du polonais Jan Łukasiewicz (1878-1956), ce Cahier thématique de Philosophia Scientiae se divise en deux sections. La première inclut un liminaire suivi de quatre articles thématiques (Section 1 : « La syllogistique de Łukasiewicz : entre tradition et modernité ») ; la seconde, trois articles hors thème, la présentation de la traduction d'un compte rendu de Paul Ehrenfest sur L'Algèbre de la logique de Louis Couturat, et enfin cette traduction elle-même (Section 2 : « Varia »).

Dans le présent article, on analyse l’assertion, avancée par Jan Łukasiewicz, que la véracité ou la fausseté des propositions portant sur les événements futurs contingents implique le déterminisme. Pour éviter le déterminisme, il faut, selon Łukasiewicz, rejeter la logique classique (binaire) et remplacer cette logique par la logique polyvalente (trivalente). La conception défendue par Łukasiewicz est examinée en rapport avec la thèse proposée par Susan Haack, selon laquelle la véracité des propositions portant sur les événements futurs n’implique aucun déterminisme. Dans l’article, je démontre que le point de vue de Susan Haack est erroné, car elle ne prend pas en considération toutes les prémisses admises par Łukasiewicz. La prémisse la plus importante concerne la sémantique des verbes du futur. Or, j’essaie de prouver que Łukasiewicz s’appuie sur la sémantique dite peircienne et non ockhamienne. J’essaie de prouver que si on prend en considération la prémisse concernant les verbes du futur, le raisonnement qui a mené Jan Łukasiewicz à créer la logique trivalente s’avère correct. Dans la dernière partie de l’article, je signale que la logique trivalente proposée par Łukasiewicz pose certaines difficultés et je postule la nécessité de trouver, dans l’avenir, une autre solution du problème de la valeur logique des phrases sur le futur. (Auteur)

The article deals with Jan Lukasiewicz’s thesis that the truth or falsity of some propositions about the future—future contingents—entails determinism. According to Lukasiewicz, determinism could be avoided by rejecting the classical logic (two-valued logic) and replacing it with manyvalued logic (three-valued logic). Lukasiewicz’s position is taken under scrutiny from the point of view of Susan Haack’s thesis that the truth values of future contingents do not entail any determinism. I argue in the paper that Haack’s view is incorrect because it does not take into account all premises accepted by Lukasiewicz. The most important of them concerns the semantics of future verbs. Lukasiewicz, as I try to show, assumes the so called Peircean semantics and not the Ockhamist one. Next, I try to argue that it is possible to preserve the validity of Lukasiewicz’s reasoning which led him to the three-valued logic by taking into account his assumptions regarding the meaning of future verbs. In the last part of my paper, I point out that three-valued logic proposed by Lukasiewicz has some flaws and, therefore, one needs to look for another solution of the problem of future contingents.

This revised and considerably expanded 2nd edition, published in 2008, brings together a wide range of topics, including modal, tense, conditional, intuitionist, many-valued, paraconsistent, relevant, and fuzzy logics. Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as tableau proofs, and their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area. – Contents : Preface to the first edition; Preface to the second edition; Mathematical prolegomenon. – Part I. Propositional Logic: – 1. Classical logic and the material conditional; – 2. Basic modal logic; – 3. Normal modal logics; – 4. Non-normal modal logics; strict conditionals; – 5. Conditional logics; – 6. Intuitionist logic; – 7. Many-valued logics; – 8. First degree entailment; – 9. Logics with gaps, gluts, and worlds; – 10. Relevant logics; – 11. Fuzzy logics; 11a. Appendix: Many valued modal logics; Postscript: An historical perspective on conditionals. – Part II. Qualification and Identity: – 12. Classical logic; – 13. Free logic; – 14. Constant domain modal logics; – 15. Variable domain modal logics; – 16. Necessary identity in modal logic; – 17. Contingent identity in modal logic; – 18. Non-normal modal logics; – 19. Conditional logics; – 20. Intuitionist logic; – 21. Many-valued logics; – 22. First degree entailment; – 23. Logics with gaps, gluts, and worlds; – 24. Relevant logics; – 25. Fuzzy logics. – Postscript: A methodological coda. – Includes bibliographical references (p. 587-602) and index.