1st Edition

Proof Theory Sequent Calculi and Related Formalisms

By Katalin Bimbo Copyright 2014
    386 Pages 13 B/W Illustrations
    by Chapman & Hall

    Although sequent calculi constitute an important category of proof systems, they are not as well known as axiomatic and natural deduction systems. Addressing this deficiency, Proof Theory: Sequent Calculi and Related Formalisms presents a comprehensive treatment of sequent calculi, including a wide range of variations. It focuses on sequent calculi for various non-classical logics, from intuitionistic logic to relevance logic, linear logic, and modal logic.

    In the first chapters, the author emphasizes classical logic and a variety of different sequent calculi for classical and intuitionistic logics. She then presents other non-classical logics and meta-logical results, including decidability results obtained specifically using sequent calculus formalizations of logics.

    The book is suitable for a wide audience and can be used in advanced undergraduate or graduate courses. Computer scientists will discover intriguing connections between sequent calculi and resolution as well as between sequent calculi and typed systems. Those interested in the constructive approach will find formalizations of intuitionistic logic and two calculi for linear logic. Mathematicians and philosophers will welcome the treatment of a range of variations on calculi for classical logic. Philosophical logicians will be interested in the calculi for relevance logics while linguists will appreciate the detailed presentation of Lambek calculi and their extensions.

    Proofs and proof theory
    Proofs of all kinds
    Early history of proof theory in a nutshell
    Proofs as calculations

    Classical first-order logic
    The sequent calculus LK
    An axiom system for FOL
    Equivalence of LK and K
    Interpretations, soundness and completeness

    Variants of the first sequent calculi
    Intuitionistic logic and other modifications
    Sequent calculi with multisets and sets
    Sequent calculi with no structural rules
    One-sided sequent calculi
    Uniform sequent calculi
    Disjunction property
    Translations between classical and intuitionistic logics

    Sequent calculi for non-classical logics
    Associative Lambek calculus
    Extensions of the associative Lambek calculus
    Relevant implication and pure entailment
    Non-distributive logic of relevant implication
    Linear logic
    Positive logic of relevant implication
    Sequent calculi for modal logics
    Merge calculi

    Consecution calculi for non-classical logics
    Non-associative Lambek calculus
    Structurally free logics
    More implicational relevance logics
    Positive entailment logics
    Calculi with multiple right-hand side

    Display calculi and hypersequents
    Display logics with star
    Display logic for linear logic
    Display logic for symmetric gaggles
    Hypersequent calculi

    Cut rules and cut theorems
    Uniform cut theorem
    Mix, multiple and single cuts
    Constants and the cut
    Display cut
    Cut theorem via normal proofs
    Cut theorem via interpretations
    Analytic cut
    Consequences of the cut theorem and uses of the cut rules

    Some other proof systems
    Natural deduction systems
    Tableau systems
    Resolution systems

    Applications and applied calculi
    Sequent calculi for mathematical theories
    Typed and labeled calculi

    Appendix: Some supplementary concepts




    Katalin Bimbo

    "Katalin Bimbo is one of the leading relevance logicians in the world today and indeed one of the leading non-classical logicians in general. Her book on proof theory takes readers through standard (classical) proof theory and beyond, including proof theory for some of the most important non-classical logics. The discussion is brilliantly executed. All graduate students interested in logic should study this book and all faculty too. I plan to use the book often."
    —Jc Beall, Professor of Philosophy, University of Connecticut, and Professorial Fellow, Northern Institute of Philosophy, University of Aberdeen