Abstract
The semantical structures called T×W frames were introduced in (Thomason, 1984) for the Ockhamist temporal-modal language, ℒO, 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, ℒSO, containing a further possibility operator ⋄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 ℒSO, which is shown to be strongly complete forT ×W-validity. Von Kutschera (1997) contains a finite axiomatization of T×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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Di Maio, M.C., Zanardo, A. A Gabbay-Rule Free Axiomatization of T×W Validity. Journal of Philosophical Logic 27, 435–487 (1998). https://doi.org/10.1023/A:1004284420809
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DOI: https://doi.org/10.1023/A:1004284420809