Completeness Theory for Propositional Logics [electronic resource]/
by Witold A. Pogorzelski, Piotr Wojtylak.
- Basel: Birkhäuser Basel, 2008.
- digital.
- Studies in Universal Logic .
- Studies in Universal Logic .
Introduction -- 1. Basic notions: Propositional languages -- Abstract algebras -- Preliminary lattice-theoretical notions -- Propositional logics -- Brief exposition of the most important propositional logics -- 2. Semantic methods in propositional logic: Preordered sets -- Preordered algebras -- Logical matrices -- Adequacy -- Propositional logic and lattice theory -- 3. Completeness of propositional logic: Generalized completeness -- Post-completeness -- The problem of uniqueness of Lindenbaum extensions -- Some related concepts -- 4. Characterization of propositional connectives: Cn-definitions -- The system (D) -- Variants -- The system (I) -- Classical logic -- Appendix: The fundamental metatheorem for the classical propositional logic -- A proof system for the classical logic .
The book develops the theory of one of the most important notions in the methodology of formal systems. Particularly, completeness plays an important role in propositional logic where many variants of the notion have been defined. Global variants of the notion mean the possibility of getting all correct and reliable schemata of inference. Its local variants refer to the notion of truth given by some semantics. A uniform theory of completeness in its general and local meaning is carried out and it generalizes and systematizes some variety of the notion of completeness such as Post-completeness, structural completeness and many others. This approach allows also for a more profound view upon some essential properties (e.g. two-valuedness) of propositional systems. For these purposes, the theory of logical matrices, and the theory of consequence operations is exploited .