Algebraic Foundations of Many-Valued Reasoning: 7 (Trends in Logic, 7) - Hardcover

Roberto L. O. Cignoli; Itala M. L. D'Ottaviano; Daniele Mundici

Hardcover

ISBN 10: 0792360095 ISBN 13: 9780792360094

Publisher: Springer, 1999

Synopsis

The aim of this book is to give self-contained proofs of all basic results concerning the infinite-valued proposition al calculus of Lukasiewicz and its algebras, Chang's MV -algebras. This book is for self-study: with the possible exception of Chapter 9 on advanced topics, the only prere­ quisite for the reader is some acquaintance with classical propositional logic, and elementary algebra and topology. In this book it is not our aim to give an account of Lukasiewicz's motivations for adding new truth values: readers interested in this topic will find appropriate references in Chapter 10. Also, we shall not explain why Lukasiewicz infinite-valued propositionallogic is a ba­ sic ingredient of any logical treatment of imprecise notions: Hajek's book in this series on Trends in Logic contains the most authorita­ tive explanations. However, in order to show that MV-algebras stand to infinite-valued logic as boolean algebras stand to two-valued logic, we shall devote Chapter 5 to Ulam's game of Twenty Questions with lies/errors, as a natural context where infinite-valued propositions, con­ nectives and inferences are used. While several other semantics for infinite-valued logic are known in the literature-notably Giles' game­ theoretic semantics based on subjective probabilities-still the transi­ tion from two-valued to many-valued propositonallogic can hardly be modelled by anything simpler than the transformation of the familiar game of Twenty Questions into Ulam game with lies/errors.

About this title

Review

`Meanwhile, a rich structure theory for MV-algebras has been developed, relating them e.g. to l-groups and to nonstandard reals. The present book develops these matters in detail, and gives a coherent presentation of the core results of the last 15 years or so, also adding unpublished material of the authors. For future work on MV-algebras, this monograph will be an indispensable source.'
Mathematical Reviews, 2001

Synopsis

This unique textbook states and proves all the major theorems of many-valued propositional logic and provides the reader with the most recent developments and trends, including applications to adaptive error-correcting binary search. The book is suitable for self-study, making the basic tools of many-valued logic accessible to students and scientists with a basic mathematical knowledge who are interested in the mathematical treatment of uncertain information. Stressing the interplay between algebra and logic, the book contains material never before published, such as a simple proof of the completeness theorem and of the equivalence between Chang's MV algebras and Abelian lattice-ordered groups with unit - a necessary prerequisite for the incorporation of a genuine addition operation into fuzzy logic.Readers interested in fuzzy control are provided with a rich deductive system in which one can define fuzzy partitions, just as Boolean partitions can be defined and computed in classical logic. Detailed bibliographic remarks at the end of each chapter and an extensive bibliography lead the reader on to further specialised topics.

Edition details

• Publisher: Springer
• Publication date: 1999
• Language: English
• ISBN 10: 0792360095
• ISBN 13: 9780792360094
• Binding: Hardcover
• Number of pages: 242