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© 2009

Proof Theory for Fuzzy Logics

Book

Part of the Applied Logic Series book series (APLS, volume 36)

Table of contents

  1. Front Matter
    Pages I-VIII
  2. George Metcalfe, Nicola Olivetti, Dov Gabbay
    Pages 1-6
  3. George Metcalfe, Nicola Olivetti, Dov Gabbay
    Pages 7-35
  4. George Metcalfe, Nicola Olivetti, Dov Gabbay
    Pages 37-66
  5. George Metcalfe, Nicola Olivetti, Dov Gabbay
    Pages 67-100
  6. George Metcalfe, Nicola Olivetti, Dov Gabbay
    Pages 101-135
  7. George Metcalfe, Nicola Olivetti, Dov Gabbay
    Pages 137-175
  8. George Metcalfe, Nicola Olivetti, Dov Gabbay
    Pages 177-199
  9. George Metcalfe, Nicola Olivetti, Dov Gabbay
    Pages 201-228
  10. George Metcalfe, Nicola Olivetti, Dov Gabbay
    Pages 229-258
  11. Back Matter
    Pages 259-276

About this book

Introduction

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.

Keywords

Addition Fuzzy mathematics Gentzen Hilbert Many valued logics Ordered algebraic structures Proof theory Substructural logics Truth values Vagueness fuzzy logic mathematics proof

Authors and affiliations

  1. 1.Department of MathematicsVanderbilt UniversityNasvilleUSA
  2. 2.Université Aix-Marseille III LSIS-UMR CNRS 6168Avenue Escadrille Normandie-NiemenFrance
  3. 3.Department of Computer ScienceKing’s CollegeLondonUK

Bibliographic information

Reviews

From the reviews:

"This is a pioneering book on proofs for fuzzy logics, well-suited both for logicians who are interested in fuzzy logic and for specialists in expert systems and fuzzy logic applications who want to know more about the applications of proof theory." (V. Ya. Kreinovich, Mathematical Reviews, Issue 2009 h)

“The class of mathematical fuzzy logics is a natural extension of the class of t-norm-based [0, 1]-valued logics. … the present monograph offers a study of proof-theoretically more interesting Gentzen-type calculi for such logics. … This monograph is a well readable and up-to-date presentation of its topic, which clearly indicates which interesting results have been proved … . It is excellently written by some of the leading experts in the field.” (Siegfried J. Gottwald, Zentralblatt MATH, Vol. 1168, 2009)