COMPLETENESS VIA COMPLETENESS: SINCE AND UNTIL

In this paper we give finite axiomatizations of the set of all valid formulas in the formalism with S and U , for the class of the well-ordered flows of time and for the frame consisting of the natural numbers. These axiom systems are orthodox in the sense that they only use the standard derivation rules of Modus Ponens, Temporal Generalization and Substitution. An essential use is made of the fact that the language with S and U is expressively complete over the frames involved.