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Algebraic Foundations of Many-Valued Reasoning : Trends in Logic - Roberto Cignoli

Algebraic Foundations of Many-Valued Reasoning

Trends in Logic

Hardcover Published: 30th November 1999
ISBN: 9780792360094
Number Of Pages: 233

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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.

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`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these matters in detail, and gives a coherent presentation of the core results of the last 15 years or so, also adding unpublished material of the authors.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

Introductionp. 1
Basic notionsp. 7
Chang completeness theoremp. 31
Free MV-algebrasp. 51
Lukasiewicz [infinity]-valued calculusp. 77
Ulam's gamep. 103
Lattice-theoretical propertiesp. 111
MV-algebras and [lambda]-groupsp. 139
Varieties of MV-algebrasp. 157
Advanced topicsp. 179
Further Readingsp. 197
Bibliographyp. 203
Indexp. 225
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792360094
ISBN-10: 0792360095
Series: Trends in Logic
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 233
Published: 30th November 1999
Publisher: Springer
Country of Publication: NL
Dimensions (cm): 23.5 x 15.5  x 1.91
Weight (kg): 0.53

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