In fact, some of the most interesting Temporal Logics are not extensions of FOL.
It would be more proper to say that Temporal Logic is an extension of Propositional Logic with modal operator acting over paths. Those modal operators are analogous to the universal and existential operators found in FOL, but not the same. The most expressive Temporal Logics will have both pairs of dual operators.
Anyway, this is a great write-up. Happy to help you think through nuances.
In fact, some of the most interesting Temporal Logics are not extensions of FOL.
It would be more proper to say that Temporal Logic is an extension of Propositional Logic with modal operator acting over paths. Those modal operators are analogous to the universal and existential operators found in FOL, but not the same. The most expressive Temporal Logics will have both pairs of dual operators.
Anyway, this is a great write-up. Happy to help you think through nuances.