An axiomatization for until and since over the reals without the IRR rule

Studia Logica 51 (2):165 - 193 (1992)
We give a Hilbert style axiomatization for the set of formulas in the temporal language with Until and Since which are valid over the real number flow of time. The axiomatization, which is orthodox in the sense of only having the usual temporal rules of inference, is complete with respect to single formulas.
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Marcelo Finger & Dov M. Gabbay (1992). Adding a Temporal Dimension to a Logic System. Journal of Logic, Language and Information 1 (3):203-233.
Mark Reynolds (2010). Metric Temporal Reasoning with Less Than Two Clocks. Journal of Applied Non-Classical Logics 20 (4):437-455.
Mark Reynolds (2010). The Complexity of Temporal Logic Over the Reals. Annals of Pure and Applied Logic 161 (8):1063-1096.
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