The two modalities and cannot be used to distinguish between past and future. For this we need a multi-modal logic with the following -operators
- : holds always in the future
- : holds always in the past
- : holds always
and the corresponding -operators:
- : holds somewhere in the future
- : holds somewhere in the past
- : holds somewhere
The semantics is then given as before, by giving constraints for the three reachability relations or by giving appropriate axioms, e.g.
- : Transitivity; an analog axiom holds for the two other -operators.
- : if we go from a time point in the future , we can go back in the past to the time point where was true.
- : connection of past with future.
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.