Logic for Computer Scientists/Modal Logic/Temporal Logics

Temporal Logics
The two modalities $$\square$$ and $$\diamond $$ cannot be used to distinguish between past and future. For this we need a multi-modal logic with the following $$\square$$-operators and the corresponding $$\diamond$$-operators:
 * $$ [F] A $$: $$ A$$ holds always in the future
 * $$ [P]A$$: $$ A$$ holds always in the past
 * $$ [A]A$$: $$ A$$ holds always
 * $$ \langle F\rangle A $$: $$ A$$ holds somewhere in the future
 * $$ \langle P\rangle A $$: $$ A$$ holds somewhere in the past
 * $$ \langle A\rangle A $$: $$ A$$ holds somewhere

The semantics is then given as before, by giving constraints for the three reachability relations or by giving appropriate axioms, e.g. In addition there are many other aspects of temporal logics. E.g. one can distinguish between left- and rightlinear structures or between dense and discrete time structures.
 * $$[F] A \to [F][F]A$$: Transitivity; an analog axiom holds for the two other $$\square$$-operators.
 * $$A \to [F] \langle P\rangle A$$: if we go from a time point $$t$$ in the future $$t'$$, we can go back in the past to the time point where $$A$$ was true.
 * $$[A] A \leftrightarrow [F]A \land[P] A $$: connection of past with future.