Algebraic Foundations of Many-Valued ReasoningThe 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. |
Contents
Introduction | 1 |
Chang completeness theorem | 31 |
Łukasiewicz ovalued calculus | 77 |
Copyright | |
11 other sections not shown
Other editions - View all
Algebraic Foundations of Many-Valued Reasoning R.L. Cignoli,Itala M. d'Ottaviano,Daniele Mundici No preview available - 2010 |
Algebraic Foundations of Many-Valued Reasoning R.L. Cignoli,Itala M. d'Ottaviano,Daniele Mundici No preview available - 2012 |
Common terms and phrases
a₁ abelian group arbitrary assume atom b₁ BELLUCE boolean algebra C*-algebras Chapter coincides complete MV-algebra completeness theorem Computing Cont(X Corollary defined Definition denote desired conclusion direct product element following conditions formula free MV-algebra functor given Hausdorff space Hence homomorphism hyperarchimedean ideal of L(A implicative filter induction infinite-valued calculus integer intersection isomorphism l-group l-group homomorphism l-ideal lattice L(A lattice-ordered Lemma LETTIERI Lindenbaum algebra linear Łukasiewicz Łukasiewicz logic many-valued logics maximal ideal McNaughton functions MUNDICI MV-algebra MV-chain MV-term n-dimensional simplexes n-valued NOLA nonempty notation obtain one-one operations ordered abelian groups perfect MV-algebras prime ideal Proof Proposition 3.1.4 propositional calculus prove Rad(A satisfying the following Schauder semisimple sequence simple MV-algebras stonean ideal strong unit subalgebra subdirect product subset Suppose tautology tautology problem theory topology toric varieties trivial Ulam game unimodular triangulation variables vectors vertices whence ΕΙ