A Gabbay-Rule Free Axiomatization of T x W Validity

Journal of Philosophical Logic 27 (5):435 - 487 (1998)
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
Keywords temporal logic  branching-time  synchronism  axiomatization
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DOI 10.1023/A:1004284420809
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Stefan Wölfl (2002). Propositional Q-Logic. Journal of Philosophical Logic 31 (5):387-414.
Emiliano Lorini (2013). Temporal Logic and its Application to Normative Reasoning. Journal of Applied Non-Classical Logics 23 (4):372-399.
Niko Strobach (2007). Fooling Around with Tenses. Studies in History and Philosophy of Science Part B 38 (3):653-672.
Niko Strobach (2007). Fooling Around with Tenses. Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 38 (3):653-672.

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