Petr Hajek on Mathematical Fuzzy Logic.pdf
This volume celebrates the work of Petr Hajek on mathematical fuzzy logic and presents how his efforts have influenced prominent logicians who are continuing his work. The book opens with a discussion on Hajek's contribution to mathematical fuzzy logic and with a scientific biography of him, progresses to include two articles with a foundation flavour, that demonstrate some important aspects of Hajek's production, namely, a paper on the development of fuzzy sets and another paper on some fuzzy versions of set theory and arithmetic. Articles in the volume also focus on the treatment of vagueness, building connections between Hajek's favorite fuzzy logic and linguistic models of vagueness. Other articles introduce alternative notions of consequence relation, namely, the preservation of truth degrees, which is discussed in a general context, and the differential semantics. For the latter, a surprisingly strong standard completeness theorem is proved. Another contribution also looks at two principles valid in classical logic and characterize the three main t-norm logics in terms of these principles. Other articles, with an algebraic flavour, offer a summary of the applications of lattice ordered-groups to many-valued logic and to quantum logic, as well as an investigation of prelinearity in varieties of pointed lattice ordered algebras that satisfy a weak form of distributivity and have a very weak implication. The last part of the volume contains an article on possibilistic modal logics defined over MTL chains, a topic that Hajek discussed in his celebrated work, Metamathematics of Fuzzy Logic, and another one where the authors, besides offering unexpected premises such as proposing to call Hajek's basic fuzzy logic HL, instead of BL, propose a very weak system, called SL as a candidate for the role of the really basic fuzzy logic. The paper also provides a generalization of the prelinearity axiom, which was investigated by Hajek in the context of fuzzy logic.
Franco Montagna is Professor of Mathematical Logic at the University of Siena. He is author of 110 publications which appeared in prestigious scientific journals of Logic, Algebra and Computer Science. Franco Montagna was the leader of several research projects and participates very often at scientific meetings as an invited speaker. He is an Editor of Soft Computing. His main interest is many-valued logic. In this field, he obtained relevant results with a high number of citations.
Chapter 1. Introduction; Francesc Esteva, Lluis Godo, Siegfried Gottwald and Franco Montagna.- Chapter 2. Petr Hajek: a scientific biography; Zuzana Hanikova.- Part I. Foundational aspects of Mathematical Fuzzy Logic.- Chapter 3. The logic of fuzzy set theory: a historical approach; Siegfried Gottwald.- Chapter 4. Set theory and arithmetic in fuzzy logic; Libor Behounek and Zuzana Hanikova.- Chapter 5. Bridges Between Contextual Linguistic Models of Vagueness and T-Norm Based Fuzzy Logic; Christian G. Fermuller and Christoph Roschger.- Part II. Semantics and consequence relation in Many-Valued Logic.- Chapter 6. Consequence and degrees of truth in many-valued logic; Josep Maria Font.- Chapter 7. The differential semantics of Lukasiewicz syntactic consequence; Daniele Mundici.- Chapter 8. Two principles in many-valued logic; Stefano Aguzzoli and Vincenzo Marra.- Part III. Algebra for Many-Valued Logic.- Chapter 9. How do '-groups and po-groups appear in algebraic and quantum structures?; Anatolij Dvurecenskij.- Chapter 10. Semi-linear Varieties of Lattice-Ordered Algebras; Antonio Ledda, Francesco Paoli and Constantine Tsinakis.- Part IV. More recent trends.- Chapter 11. On possibilistic modal logics defined over MTL-chains; Felix Bou, Francesc Esteva and Lluis Godo.- Chapter 12. The quest for the basic fuzzy logic; Petr Cintula, Rostislav Horcik and Carles Noguera.- A Bibliography of Petr Hajek.