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Possible worlds semantics interprets modal expressions by quantifying over alternative worlds: a statement is necessary if true in all possible worlds, contingent if true in some but not all. This framework provides truth conditions for modals, counterfactuals, and intensional contexts.
Practice translating English modal sentences into possible worlds notation; study accessibility relations between worlds to formalize different modal systems (K, S4, S5).
Possible worlds are abstract models, not metaphysical entities; the framework is a mathematical tool for reasoning about modality, not a claim about hidden universes.
From your work in Montague semantics, you know that sentence meaning can be modeled as functions from contexts to truth values, and from modal semantics you know that words like "must," "might," "necessarily," and "possibly" resist simple truth-functional analysis — their truth depends not just on what is actually the case but on what could or must be the case. Possible worlds semantics provides the formal machinery that makes modal semantics precise. The core idea is elegant: treat "possible world" as a primitive — an abstract index representing a complete way the world could be — and evaluate modal sentences by quantifying over these indices.
The key definitions fall out immediately. A proposition is necessarily true if it is true in every possible world accessible from the world of evaluation. It is possibly true if it is true in at least one accessible world. "Accessible from" is doing critical technical work here: not every world is relevant for every modal claim. The accessibility relation R between worlds is what distinguishes modal systems. In system K, R is unrestricted. In S4, R is reflexive and transitive (any world accessible from an accessible world is itself accessible). In S5, R is an equivalence relation, meaning all worlds access all worlds — the most permissive and philosophically controversial system. Which system is correct depends on what kind of modality you are modeling: epistemic necessity (what an agent must believe given their knowledge) uses different accessibility than metaphysical necessity (what could not fail to be true).
Counterfactuals — "If it had rained, the match would have been cancelled" — are one of the framework's most powerful applications. The problem is that the antecedent is false in the actual world, which makes simple material implication vacuously true for all counterfactuals. David Lewis's solution is to evaluate counterfactuals by looking at the closest possible worlds where the antecedent holds — the worlds that are most similar to the actual world except for the relevant change — and checking whether the consequent holds there. This gives non-trivial truth conditions that match our intuitions about when counterfactuals are true.
A persistent misconception — important enough that it appears in the Core Idea — is that possible worlds semantics makes claims about the physical existence of alternative universes. It does not. Possible worlds are abstract indices in a formal model, no different in status from the numbers used in arithmetic. You can use the framework without any commitment about whether non-actual worlds "really exist" in any deep sense. The metaphysics of possible worlds (realism vs. ersatzism vs. fictionalism) is a separate philosophical debate; the semantics works the same way regardless of which metaphysical view you hold.
When you work with this framework, the practical skill is translating between English modal sentences and their formal representations. "It must be raining" becomes □P (box P: P is true at all accessible worlds). "It might be raining" becomes ◇P (diamond P: P is true at some accessible world). Intensional contexts — belief reports, attitude verbs, temporal operators — all become amenable to possible worlds treatment, which is why Montague's program depends on this foundation. The unifying power of the framework is what makes it central to formal semantics.