Algebraic Foundations of Many-Valued Reasoning
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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Chang completeness theorem
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abelian group arbitrary atom boolean algebra C*-algebras Chang's Chapter coincides compact Hausdorff space complete MV-algebra completeness theorem Computer Cont(X Corollary defined Definition denote desired conclusion direct product equation holds Farey finite number following conditions formula free MV-algebras FreeK functor given group homomorphism Hausdorff space Hence hyperarchimedean i-group immediate consequence implicative filter induction infinite-valued calculus integer intersection of maximal isomorphism lattice-ordered Lemma Lindenbaum algebra linear many-valued logics Mathematics maximal ideals McNaughton functions monoid Mundici MV-chain MV-equations MV-term natural order Nola nonempty nontrivial MV-algebra notation obtain one-one operations ordered abelian group prime ideal proper ideal Proposition 3.1.4 propositional calculus prove quotient Rad(A rational numbers real numbers satisfying the following Schauder semisimple sequence simple MV-algebras stonean ideal strong unit subalgebra subdirect product subset Suppose surjective tautology theory tion topology toric varieties trivial Ulam game unimodular unique variables Xi vectors whence
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Philosophy of Logic
Limited preview - 2006