Algebraic Foundations of Many-Valued Reasoning
Springer Science & Business Media, Mar 9, 2013 - Mathematics - 233 pages
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.
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6-group a e G a e y A-valuation abelian group assume atom boolean algebra C*-algebras Chapter coincides complete MV-algebra completeness theorem Cont(X Corollary defined Definition denote desired conclusion direct product element following conditions formula free MV-algebras function f functor given Hausdorff space Hence homomorphism hyperarchimedean ideal of L(A immediate consequence implicative filter induction infinite-valued calculus integer intersection isomorphism lattice-ordered Lemma Lindenbaum algebra linear logic many-valued Many-valued logics maſcimal maximal ideals McNaughton functions monoid MUNDICI MV-chain MV-equations MV-term NOLA nonempty notation obtain one-one operations ordered abelian group prime ideal proper ideal Proposition 3.1.4 propositional calculus prove quotient Rad(A real numbers satisfying the following Schauder semisimple sequence simple MV-algebras simplexes stonean ideal strong unit subalgebra subdirect product subset Suppose surjective tautology theory toric varieties trivial Ulam game unimodular variables vectors whence