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Proof analysis in non-classical logics

Published online by Cambridge University Press:  18 December 2009

Costas Dimitracopoulos
Affiliation:
University of Athens, Greece
Ludomir Newelski
Affiliation:
Uniwersytet Wroclawski, Poland
Dag Normann
Affiliation:
Universitetet i Oslo
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Summary

Introduction. The development of sequent systems for non-classical, in particular, modal, logics, started in the 1950s, with the work of Curry [1952] who provided a system with cut elimination and a decision procedure for S4, and Kanger [1957], who gave sequent calculi and decision procedures for T, S4, S5 with the use of “spotted formulas”, i.e., formulas indexed by natural numbers.

Difficulties in the Gentzen-style formalization of modal logic were, however, encountered at a very elementary level, for instance in the search of an adequate cut-free sequent calculus for the modal logic S5. These difficulties are well witnessed by the ongoing present interest in the problem, with two more proposals presented in this Colloquium (Restall [2005], Stouppa [2005]).

The lack of a general solution has justified an overall pessimistic attitude towards the possibility of applying Gentzen's systems to non-classical logics, as is shown in the following passages:

Gentzen's methods do not provide anything like a universal approach to logic … There are certain standard logics to which these methods do not apply in as direct a fashion … For example, consider the logics B and S5. The Kripke models for these are symmetric … Such things effectively destroy all possibility of a good, simple cut-free Gentzen system.

Fitting (1983, p. 4)

The other tradition that should be mentioned is that of proof theory. Gentzen methods have never really flourished in modal logic.

Bull and Segerberg (1984, p. 7)
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Chapter
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Logic Colloquium 2005 , pp. 107 - 128
Publisher: Cambridge University Press
Print publication year: 2007

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