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What are the differences between propositional logic and temporal logic?

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I am currently studying temporal logic and find it much similar to propositional logic. However, temporal logic seems to be slightly more specific because it encapsulates more precise meaning symbols, like until, eventually, next etc.. My question is are there other major differences with temporal logic and propositional logic? And can everything expressed in propositional logic be expressed in temporal logic too and vice versa? Perhaps, why would one use one over the other or vice versa?

Asked By : Teodorico Levoff
Answered By : Yuval Filmus

An interpretation in propositional logic consists of assigning a truth value to every variable. In contrast an interpretation of, say, LTL consists of assigning a truth value to every variable at each point in time. So the domain of discourse is different. Propositional logic is about variables having a definite truth value, whereas LTL is about variables having a truth value depending on time.

Everything that can be expressed in propositional logic can also be expressed in LTL, by just always referring to the truth value at time zero. We say that propositional logic is interpretable in LTL. The converse isn't true – you can't express temporal concepts in propositional logic. You use temporal logic when you need to express temporal concepts. In situations where you don't need to express temporal concepts, you use propositional logic.

The language of set theory is rich enough to encompass both propositional logic (and its extension, predicate logic) and temporal logic. However, the language of set theory might be too expressive for your comfort at times (for example, it may be hard to process algorithmically), which is why we are interested in propositional logic and temporal logic.

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