Times in tense logic (Q1038658)

The author presents a way to get quantification over times in the context of elementary tense logics which have as tense operators only \(P\) and \(F\) within their primitive vocabulary. To reach this goal he shows that the usual semantics of temporal logic frames can, without loss of completeness, be reduced to the subclass of such frames which have maximally consistent sets of tense formulas as their elements. In this new context he introduces time-series as suitable subsets of his frames, represents the structure of time by them, and is able to define the earlier-than and later-than relations within these time-series. Also, time-series with continuum many time points become possible, however, depending in their structure also from the countable first-order language the tense logic is based upon. Even time operators like ``until'' and ``since'' become definable. This result does not contradict the older non-definability result of \textit{H. Kamp} [Tense logic and the theory of linear orders. Ph.D. Thesis, UCLA (1968)] because the models Kamp's proof had been based upon do not exist in the author's context.