Sequent-systems for modal logic

Journal of Symbolic Logic 50 (1):149-168 (1985)
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Abstract

The purpose of this work is to present Gentzen-style formulations of S5 and S4 based on sequents of higher levels. Sequents of level 1 are like ordinary sequents, sequents of level 1 have collections of sequents of level 1 on the left and right of the turnstile, etc. Rules for modal constants involve sequents of level 2, whereas rules for customary logical constants of first-order logic with identity involve only sequents of level 1. A restriction on Thinning on the right of level 2, which when applied to Thinning on the right of level 1 produces intuitionistic out of classical logic (without changing anything else), produces S4 out of S5 (without changing anything else). This characterization of modal constants with sequents of level 2 is unique in the following sense. If constants which differ only graphically are given a formally identical characterization, they can be shown inter-replaceable (not only uniformly) with the original constants salva provability. Customary characterizations of modal constants with sequents of level 1, as well as characterizations in Hilbert-style axiomatizations, are not unique in this sense. This parallels the case with implication, which is not uniquely characterized in Hilbert-style axiomatizations, but can be uniquely characterized with sequents of level 1. These results bear upon theories of philosophical logic which attempt to characterize logical constants syntactically. They also provide an illustration of how alternative logics differ only in their structural rules, whereas their rules for logical constants are identical

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Citations of this work

Proofnets for S5: sequents and circuits for modal logic.Greg Restall - 2007 - In C. Dimitracopoulos, L. Newelski & D. Normann (eds.), Logic Colloquium 2005. Cambridge: Cambridge University Press. pp. 151-172.
Sequent-systems and groupoid models. I.Kosta Došen - 1988 - Studia Logica 47 (4):353 - 385.

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References found in this work

On engendering an illusion of understanding.Dana Scott - 1971 - Journal of Philosophy 68 (21):787-807.
Foundations of Mathematical Logic.William Craig - 1963 - Journal of Symbolic Logic 45 (2):377-378.
Introduction to Metamathematics.Ann Singleterry Ferebee - 1968 - Journal of Symbolic Logic 33 (2):290-291.
Modal logic, the Lewis-modal systems.J. Jay Zeman - 1973 - Revue Philosophique de la France Et de l'Etranger 163:479-479.

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