On Saturated Calculi for a Linear Temporal Logic

A new type of finitary and infinitary sequential calculi (named the saturated ones) for a linear temporal logic are introduced. Non-logical axioms in saturated calculi are some sequents, indicating the saturation of the derivation process in these calculi. The finitary saturation suggests that ”nothing new” can be obtained continuing the derivation process. An infinitary saturated calculus instead of an ω-type rule of inference has an infinite set of ”saturated” sequents, but the form of these sequents is uniform. The saturation presents the unique deductive principle both for finitary and infinitary cases. The derivability in a finitary saturated calculus serves as a finitary completeness criterion for the first order linear temporal logic.