Ever since Aristotle (and even before him, see the stoics), philosophers have tried to formalize time. It is only around the fifties, starting with the work of Arthur Prior, when temporal logic is developed considerably with the development of new systems used to represent different types of time (linear time, infinite time, branching time, etc.). The foundation of possible worlds semantics was vital for the semantics of such systems. These systems have found applications in a variety of fields, the most representative being those of linguistics and computer science.
Temporal logic systems can be based on propositional logic or on first order logic. On both cases operators are added to represent the past (P and H) and the future (F and G). It is also possible to include operators to represent intervals. The most common semantics is based on the notion of moment. These moments are organized through an ulteriority relation (before/after). Hence, if I claim that m0 < m1 I am claiming that moment m0 is previous to moment m1. This ulteriority relation has different properties depending on the type of time we are working with, although it is always irreflexive. Thus, for instance, if time is transitive, the ulteriority relation will have the transitive property and syntactically transitivity axioms will be introduced ( FFA®FA and PPA®PA).
There are multimodal and bidimensional systems of temporal logic, such as the system of indeterministic time HN1, which combines temporal and modal operators and in whose semantic, evaluation is made in two indices (moment and history).
Presently, hybrid temporal logic systems are being developed. These systems increase the expressive power of temporal logic, because they allow making reference in the syntax to the moments.
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M. Vázquez (14/04/2010)
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