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Completeness Theory for Propositional Logics

Witold A. Pogorzelski, Piotr Wojtylak (Autoren)

Buch | Softcover

VIII, 178 Seiten

2008 | 2008
Springer Basel (Verlag)
978-3-7643-8517-0 (ISBN)

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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. Completeness means the possibility of getting all correct and reliable sc- mata of inference by use of logical methods. The word 'all', seemingly neutral, is here a crucial point of distinction. Assuming the de?nition as given by E. Post we get, say, a global notion of completeness in which the reliability refers only to syntactic means of logic and outside the correct schemata of inference there are only inconsistent ones. It is impossible, however, to leave aside local aspects of the notion when we want to make it relative to some given or invented notion of truth. Completeness understood in this sense is the adequacy of logic in relation to some semantics, and the change of the logic is accompanied by the change of its semantics. Such completeness was e?ectively used by J. ?ukasiewicz and investigated in general terms by A. Tarski and A. Lindenbaum, which gave strong foundations for research in logic and, in particular, for the notion of consequence operation determined by a logical system. The choice of logical means, by use of which we intend to represent logical inferences, is also important. Most of the de?nitions and results in completeness theory were originally developed in terms of propositional logic. Propositional formal systems ?nd many applications in logic and theoretical computer science.

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.

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.

From the reviews:

"The book provides a uniform treatment of the variety of results centered around the completeness property. ... book is a good introduction to the problems of completeness. A wealth of examples, comments and theorems well elucidate various difficult aspects of the theory. ... From the methodological viewpoint, the book applies the tools that were elaborated in metalogic ... . AAL also offers subtle tools for tackling some of the problems raised in the book." (Janusz M. Czelakowski, Mathematical Reviews, Issue 2010 c)

"The book is written with exceptional clarity and precision. This combination makes it accessible to a wide spectrum of potential readers, and hence it can be recommended to anyone interested in formal logic. ... the book may stimulate to further research by opening new fields of investigation and introducing new concepts and ideas. Finally, one cannot miss the extensive and up-to-date bibliography which is included in the book. Summing up, the book ... offers a deep and intelligible exposition of completeness theory in propositional logics." (Tomasz Polacik, Studia Logica, Vol. 95, 2010)

Erscheint lt. Verlag	17.4.2008
Reihe/Serie	Studies in Universal Logic
Zusatzinfo	VIII, 178 p.
Verlagsort	Basel
Sprache	englisch
Maße	170 x 240 mm
Gewicht	375 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika
Themenwelt	Mathematik / Informatik ► Mathematik ► Logik / Mengenlehre
Schlagworte	Completeness • Consequence operation • Hardcover, Softcover / Mathematik/Grundlagen • HC/Mathematik/Grundlagen • Logic • Logical matrix • Mathematische Logik • Post-completeness • Proof • Structural completeness • Universal logic
ISBN-10	3-7643-8517-0 / 3764385170
ISBN-13	978-3-7643-8517-0 / 9783764385170
Zustand	Neuware