Completeness Theory for Propositional Logics

1. Front Matter

   Pages i-viii

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2. Basic notions

   Pages 1-40

3. Semantic methods in propositional logic

   Pages 41-89

4. Completeness of propositional logics

   Pages 91-130

5. Characterizations of propositional connectives

   Pages 131-152

6. Back Matter

   Pages 153-178

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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.

• Completeness
• Consequence operation
• Logical matrix
• Post-completeness
• Structural completeness
• Universal Logic
• logic
• proof

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 Połacik, Studia Logica, Vol. 95, 2010)

• Institute of Mathematics, University of Bialystok, Bialystok, Poland

  Witold A. Pogorzelski

• Institute of Mathematics, Silesian University, Katowice, Poland

  Piotr Wojtylak

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