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A Gabbay-Rule Free Axiomatization of T x W Validity

Maria Concetta Di Maio and Alberto Zanardo
Journal of Philosophical Logic
Vol. 27, No. 5 (Oct., 1998), pp. 435-487
Published by: Springer
Stable URL: http://www.jstor.org/stable/30226653
Page Count: 53
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Abstract

The semantical structures called T x W frames were introduced in (Thomason, 1984) for the Ockhamist temporal-modal language, $[Unrepresented Character]_{o}$, which consists of the usual propositional language augmented with the Priorean operators P and F and with a possibility operator ◇. However, these structures are also suitable for interpreting an extended language, $[Unrepresented Character]_{so}$, containing a further possibility operator $\lozenge^{s}$ which expresses synchronism among possibly incompatible histories and which can thus be thought of as a cross-history 'simultaneity' operator. In the present paper we provide an infinite set of axioms in $[Unrepresented Character]_{so}$, which is shown to be strongly complete for T x W-validity. Von Kutschera (1997) contains a finite axiomatization of T x W-validity which however makes use of the Gabbay Irreflexivity Rule (Gabbay, 1981). In order to avoid using this rule, the proof presented here develops a new technique to deal with reflexive maximal consistent sets in Henkin-style constructions.

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