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LINEAR TIME IN HYPERSEQUENT FRAMEWORK

Published online by Cambridge University Press:  29 March 2016

ANDRZEJ INDRZEJCZAK
ANDRZEJ INDRZEJCZAK*
Affiliation:
DEPARTMENT OF LOGIC INSTITUTE OF PHILOSOPHY UNIVERSITY OF LODZ KOPCIŃSKIEGO 16/18 90–232 ŁÓDŹ, POLANDE-mail: indrzej@filozof.uni.lodz.pl
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Abstract

Hypersequent calculus (HC), developed by A. Avron, is one of the most interesting proof systems suitable for nonclassical logics. Although HC has rather simple form, it increases significantly the expressive power of standard sequent calculi (SC). In particular, HC proved to be very useful in the field of proof theory of various nonclassical logics. It may seem surprising that it was not applied to temporal logics so far. In what follows, we discuss different approaches to formalization of logics of linear frames and provide a cut-free HC formalization of Kt4.3, the minimal temporal logic of linear frames, and some of its extensions. The novelty of our approach is that hypersequents are defined not as finite (multi)sets but as finite lists of ordinary sequents. Such a solution allows both linearity of time flow, and symmetry of past and future, to be incorporated by means of six temporal rules (three for future-necessity and three dual rules for past-necessity). Extensions of the basic calculus with simple structural rules cover logics of serial and dense frames. Completeness is proved by Schütte/Hintikka-style argument using models built from saturated hypersequents.

Keywords

03F0303B44proof theorytemporal logichypersequent calculi
Type
Communication
Information
Bulletin of Symbolic Logic , Volume 22 , Issue 1 , March 2016 , pp. 121 - 144
Copyright
Copyright © The Association for Symbolic Logic 2016 

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