© 2015

Petr Hájek on Mathematical Fuzzy Logic

  • Franco Montagna


  • Offers new developments in fuzzy logic

  • Presents the work of prominent logicians and how their own work has been influenced by that of Petr Ha’jek

  • Provides a range of articles from an overview of the development of fuzzy set theory to surprising proposition to call Hajek's basic fuzzy logic H, instead of BL


Part of the Outstanding Contributions to Logic book series (OCTR, volume 6)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Introduction

    1. Front Matter
      Pages 1-1
    2. Francesc Esteva, Lluís Godo, Siegfried Gottwald, Franco Montagna
      Pages 3-20
    3. Zuzana Haniková
      Pages 21-38
  3. Foundational Aspects of Mathematical Fuzzy Logic

    1. Front Matter
      Pages 39-40
    2. Libor Běhounek, Zuzana Haniková
      Pages 63-89
    3. Christian G. Fermüller, Christoph Roschger
      Pages 91-114
  4. Semantics and Consequence Relation in Many-Valued Logic

    1. Front Matter
      Pages 115-116
    2. Stefano Aguzzoli, Vincenzo Marra
      Pages 159-174
  5. Algebra for Many-Valued Logic

    1. Front Matter
      Pages 175-176
    2. Antonio Ledda, Francesco Paoli, Constantine Tsinakis
      Pages 207-221
  6. More Recent Trends

    1. Front Matter
      Pages 223-224
    2. Félix Bou, Francesc Esteva, Lluís Godo
      Pages 225-244
    3. Petr Cintula, Rostislav Horčík, Carles Noguera
      Pages 245-290
  7. Back Matter
    Pages 291-318

About this book


This volume celebrates the work of Petr Hájek 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 Hájek'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 Hájek'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 Hájek'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 surprising 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 flavor, 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 Hájek discussed in his celebrated work, Metamathematics of Fuzzy Logic, and another one where the authors, besides of offering unexpected premises such as proposing to call Hájek'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 Hájek in the context of fuzzy logic.


Algebraic Logic Fuzzy Logic and Vagueness Fuzzy sets Ha'jek and Modal Fuzzy Logic MV-algebras Partially Ordered Groups Set Theory and Arithmetic over Fuzzy Logic Standard Completeness Substructural Fuzzy Logics Truth Degrees

Editors and affiliations

  • Franco Montagna
    • 1
  1. 1.Department of Information Engineering anUniversity of SienaSienaItaly

About the editors

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.

Bibliographic information

  • Book Title Petr Hájek on Mathematical Fuzzy Logic
  • Editors Franco Montagna
  • Series Title Outstanding Contributions to Logic
  • Series Abbreviated Title Outstanding Contributions
  • DOI
  • Copyright Information Springer International Publishing Switzerland 2015
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-319-06232-7
  • Softcover ISBN 978-3-319-38295-1
  • eBook ISBN 978-3-319-06233-4
  • Series ISSN 2211-2758
  • Series E-ISSN 2211-2766
  • Edition Number 1
  • Number of Pages XII, 318
  • Number of Illustrations 9 b/w illustrations, 0 illustrations in colour
  • Topics Mathematical Logic and Foundations
  • Buy this book on publisher's site
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