The primary purpose of this book is to provide the reader with a comprehensive coverage of theoretical foundations of fuzzy set theory and fuzzy logic, as well as a broad overview of the increasingly important applications of these novel areas of mathematics. Although it is written as a text for a course at the graduate or upper division undergraduate level, the book is also suitable for self-study and for industry-oriented courses of continuing education. No previous knowledge of fuzzy set theory (...) and fuzzy logic is required for understanding the material covered in the book. Although knowledge of basic ideas of classical set theory and classical logic is useful, fundamentals of these subject areas are briefly overviewed in the book. In addition, basic ideas of neural networks, genetic algorithms, and rough sets are also explained. This makes the book virtually self-contained. Throughout the book, many examples are used to illustrate concepts, methods, and generic applications as they are introduced. Each chapter is followed by a set of exercises, which are intended to enhance readers' understanding of the material presented in the chapter. Extensive and carefully selected bibliography, together with bibliographical notes at the end of each chapter and a bibliographical subject index, is an invaluable resource for further study of fuzzy theory and applications. (shrink)

Two families of many-valued modal logics are investigated. Semantically, one family is characterized using Kripke models that allow formulas to take values in a finite many-valued logic, at each possible world. The second family generalizes this to allow the accessibility relation between worlds also to be many-valued. Gentzen sequent calculi are given for both versions, and soundness and completeness are established.

We present a Kripke model for Girard's Linear Logic (without exponentials) in a conservative fashion where the logical functors beyond the basic lattice operations may be added one by one without recourse to such things as negation. You can either have some logical functors or not as you choose. Commutatively and associatively are isolated in such a way that the base Kripke model is a model for noncommutative, nonassociative Linear Logic. We also extend the logic by adding a coimplication operator, (...) similar to Curry's subtraction operator, which is resituated with Linear Logic's contensor product. And we can add contraction to get nondistributive Relevance Logic. The model rests heavily on Urquhart's representation of nondistributive lattices and also on Dunn's Gaggle Theory. Indeed, the paper may be viewed as an investigation into nondistributive Gaggle Theory restricted to binary operations. The valuations on the Kripke model are three valued: true, false, and indifferent. The lattice representation theorem of Urquhart has the nice feature of yielding Priestley's representation theorem for distributive lattices if the original lattice happens to be distributive. Hence the representation is consistent with Stone's representation of distributive and Boolean lattices, and our semantics is consistent with the Lemmon-Scott representation of modal algebras and the Routley-Meyer semantics for Relevance Logic. (shrink)

Many results in fuzzy logic depend on the mathematical structure the truth value set obeys. In this textbook the algebraic foundations of many-valued and fuzzy reasoning are introduced. The book is self-contained, thus no previous knowledge in algebra or in logic is required. It contains 134 exercises with complete answers, and can therefore be used as teaching material at universities for both undergraduated and post-graduated courses. Chapter 1 starts from such basic concepts as order, lattice, equivalence and residuated lattice. It (...) contains a full section on BL-algebras. Chapter 2 concerns MV-algebra and its basic properties. Chapter 3 applies these mathematical results on Lukasiewicz-Pavelka style fuzzy logic, which is studied in details; besides semantics, syntax and completeness of this logic, a lot of examples are given. Chapter 4 shows the connection between fuzzy relations, approximate reasoning and fuzzy IF-THEN rules to residuated lattices. (shrink)

In this paper we give relational semantics and an accompanying relational proof system for a variety of intuitionistic substructural logics, including linear logic with exponentials. Starting with the semantics for FL as discussed in [13], we developed, in [11], a relational semantics and a relational proof system for full Lambek calculus. Here, we take this as a base and extend the results to deal with the various structural rules of exchange, contraction, weakening and expansion, and also to deal with an (...) involution operator and with the operators ! and ? of linear logic. To accomplish this, for each extension X of FL we develop a Kripke-style semantics, RelKripkeX semantics, as a bridge to relational semantics. The RelKripke semantics consists of a set with distinguished elements, ternary relations and a list of conditions on the relations. For each extension X, RelKripkeX semantics is accompanied by a Kripke-style valuation system analogous to that in [13]. Soundness and completeness theorems with respect to FLX hold for RelKripkeX-models. Then, in the spirit of the work of Orlowska [16], [17], and Buszkowski & Orlowska [4], we develop relational logic RFLX for each extension X. The adjective relational is used to emphasize the fact that RFLX has a semantics wherein formulas are interpreted as relations. We prove that a sequent Γ→α in FLX is provable if, a translation, t<εvu, has a cut-complete proof tree which is fundamental. This result is constructive: that is, if a cut-complete proof tree for tεvu is not fundamental, we can use the failed proof search to build a relational countermodel for t and from this, build a RelKripkeX countermodel for γ1[sdot ]…[sdot ]γn⊃α. (shrink)