Completeness Theory for Propositional Logics
Completeness is one of the most important notions in logic and the foundations of mathematics. Many variants of the notion have been de?ned in literature. We shallconcentrateonthesevariants,andaspects,of completenesswhicharede?ned in propositional logic.
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eries is devoted to the universal approach to logic and the development of a general theory of logics. It covers topics such as global set-ups for fundamental theorems of logic and frameworks for the study of logics, in particular logical matrices, Kripke structures, combination of logics, categorical logic, abstract proof theory, consequence operators, and algebraic logic. It includes also books with historical and philosophical discussions about the nature and scope of logic. Three types of books will appear in the series: graduate textbooks, research monographs, and volumes with contributed papers.
Witold A. Pogorzelski Piotr Wojtylak
Completeness Theory for Propositional Logics
Birkhäuser Basel · Boston · Berlin
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Contents Introduction
vii
1 Basic notions 1.1 Propositional languages . . . . . . . . 1.2 Abstract algebras . . . . . . . . . . . . 1.3 Preliminary lattice-theoretical notions 1.4 Propositional logics . . . . . . . . . . . 1.5 Brief exposition of the most important
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2 Semantic methods in propositional logic 2.1 Preordered sets . . . . . . . . . . . . 2.2 Preordered algebras . . . . . . . . . 2.3 Logical matrices . . . . . . . . . . . 2.4 Adequacy . . . . . . . . . . . . . . . 2.5 Propositional logic and lattice theory
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3 Completeness of propositional logics 3.1 Generalized completeness . . . . . . . . . 3.2
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