Temporal expressions and tense require a semantic framework treating time as an additional dimension. Tense can be analyzed as quantification over times or as context-dependent reference to utterance time, and interacts with aspect and modality in complex ways.
Natural language is saturated with temporal information: verbs are marked for tense, sentences carry aspect, and time adverbials locate events precisely. Temporal semantics asks how this information is encoded in meaning. From your study of first-order semantics, you know that truth conditions for simple sentences relate predicates to objects at a world. Adding temporal dimensions means extending those truth conditions to include times as an additional evaluation parameter — sentences are no longer just true or false at a world, but true or false at a world *and a time*.
The simplest approach treats tense as quantification over times. "It rained" means roughly: there is a time t such that t is before the utterance time, and it rained at t. The utterance time — when the sentence is spoken or written — becomes a context parameter. Past tense existentially quantifies over times before the utterance; future tense quantifies over times after it. This connects naturally to your background in temporal logic, where the operators P (it was the case that) and F (it will be the case that) function as temporal quantifiers, and the compositional semantics unpacks them accordingly.
A richer framework introduces Reichenbach's three-way distinction: Speech Time (S), Reference Time (R), and Event Time (E). Simple past ("she left") places the event before speech: E before S. But the past perfect ("she had left") adds a layer: an event before a reference time, which is itself before speech: E before R, R before S. "She had left when he arrived" makes this concrete — her leaving is E, his arriving is R, both before S. This three-way structure explains the systematic behavior of perfect and pluperfect constructions. Aspect adds yet another dimension: simple past presents an event as a completed whole ("she left"), while progressive aspect ("she was leaving") presents it as ongoing — an event viewed from the inside rather than from completion. This aspectual distinction affects how temporal adverbials attach and how tenses interact.
Temporal semantics connects most interestingly to possible worlds semantics when we consider the future tense. "It will rain tomorrow" — is this simply true or false based on present facts? If all times are equally real (eternalism/B-theory), future-tensed sentences are true or false based on what happens at future times, just as past-tensed sentences are true based on past events. If only the present is real (presentism/A-theory), future tense becomes semantically fraught — what makes a future-tensed claim true if future events don't yet exist? Some analyze future tense as quantifying over a default "most normal" continuation; others analyze it as expressing a kind of epistemic or metaphysical possibility. The relationship between formal temporal semantics and metaphysical theories of time is thus not merely analogical: the formal framework forces metaphysical commitments into the open, making them precise and subject to linguistic evidence.
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