Algebraic Foundations of Many-valued Reasoning

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

ISBN 10: 0792360095 / ISBN 13: 9780792360094

Published by Kluwer Academic Publishers, 1999

About this Item

States and proves various theorems of many-valued propositional logic. This text provides developments and trends, including applications to adaptive error-correcting binary search. It contains material, such as a simple proof of completeness theorem and of the equivalence between Chang's MV algebras and Abelian lattice-ordered groups with unit. Series: Trends in Logic. Num Pages: 233 pages, 1 black & white illustrations, biography. BIC Classification: PBCD. Category: (P) Professional & Vocational; (UP) Postgraduate, Research & Scholarly; (UU) Undergraduate. Dimension: 265 x 189 x 22. Weight in Grams: 660. . 1999. Hardback. . . . . Books ship from the US and Ireland.

Seller Inventory # V9780792360094

Bibliographic Details

• Title: Algebraic Foundations of Many-valued Reasoning
• Publisher: Kluwer Academic Publishers
• Publication Date: 1999
• Language: English
• ISBN 10: 0792360095
• ISBN 13: 9780792360094
• Binding: Hardcover
• Condition: New

About this title

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.

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