Proof theory : Sequent calculi and related formalisms
Katalin Bimbo, University of Alberta, Edmonton, Canada
- x, 376 p. : ill. ; 25 cm
- Discrete mathematics and its applications .
Bibliografie p. 364 Index p. 375
1. Proofs and proof theory -- 2. Classical first-order logic -- 3. Variants of the first sequent calculi -- 4. Sequent calculi for non-classical logics -- 5. Consecution calculi for non-classical logics -- 6. Display calculi and hypersequents -- 7. Cut rules and cut theorems -- 8. Some other proof systems -- 9. Applications and applied calculi.
"Sequent calculi constitute an interesting and important category of proof systems. They are much less known than axiomatic systems or natural deduction systems are, and they are much less known than they should be. Sequent calculi were designed as a theoretical framework for investigations of logical consequence, and they live up to the expectations completely as an abundant source of meta-logical results. The goal of this book is to provide a fairly comprehensive view of sequent calculi -- including a wide range of variations. The focus is on sequent calculi for various non-classical logics, from intuitionistic logic to relevance logic, through linear and modal logics. A particular version of sequent calculi, the so-called consecution calculi, have seen important new developments in the last decade or so. The invention of new consecution calculi for various relevance logics allowed the last major open problem in the area of relevance logic to be solved positively: pure ticket entailment is decidable. An exposition of this result is included in chapter 9 together with further new decidability results (for less famous systems). A series of other results that were obtained by J. M. Dunn and me, or by me in the last decade or so, are also presented in various places in the book. Some of these results are slightly improved in their current presentation. Obviously, many calculi and several important theorems are not new. They are included here to ensure the completeness of the picture; their original formulations may be found in the referenced publications. This book contains very little about semantics, in general, and about the semantics of non-classical logic in particular"--