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Presuppositions are background assumptions that must be true for a sentence to have a truth value. 'The king of France is bald' presupposes a unique king of France exists; if false, the statement is neither true nor false. Presuppositions survive under negation, conditionals, and questions—a key test distinguishing them from assertions, revealing a fundamental division in linguistic content.
From your study of Montague semantics, you know that sentences are evaluated for truth conditions relative to possible worlds — a sentence is true if the world is as the sentence describes it. Montague semantics handles two-valued logic cleanly: every proposition is either true or false. Presupposition introduces a complication: some sentences carry background assumptions that, if false, make the sentence neither true nor false — they fail to express a complete proposition at all. This is called presupposition failure, and it forces us to distinguish between what a sentence *asserts* and what it *takes for granted*.
Bertrand Russell's famous example remains the clearest entry point. *The king of France is bald* — said today — presupposes that there is a unique king of France. The *assertion* is that this king is bald. If France has no king, the sentence is not false; it is semantically defective. There is nothing to be true or false about. This is in sharp contrast to an ordinary assertion like *France is a monarchy* — that sentence is simply false if France is not a monarchy. The presupposition is different in kind: it is a precondition for the sentence to enter the true/false game at all.
The diagnostic test for presuppositions — and the core of what makes them theoretically interesting — is projection under embedding. An ordinary assertion that P is true gets negated when you negate the sentence: *France is a monarchy* is true; *France is not a monarchy* says the opposite. But presuppositions survive negation, conditionals, and questions. Consider: *John stopped smoking* presupposes that John used to smoke. Now embed it: *John didn't stop smoking* still carries the presupposition that he used to smoke. *Did John stop smoking?* still presupposes prior smoking. *If John stopped smoking, he'll feel better* still presupposes prior smoking. The presupposition "projects" out of these presupposition holes even when the sentence is negated, questioned, or conditionalized. This projection behavior is what distinguishes presuppositions from ordinary entailments (which do not survive negation) and from implicatures (which are defeasible and pragmatic rather than semantic).
Different presupposition triggers — the lexical items that introduce presuppositions — produce slightly different projection behaviors. Definite descriptions (the king) trigger existence and uniqueness presuppositions. Factive verbs like *know*, *realize*, *regret* presuppose the truth of their complement (*John knows it's raining* presupposes it is raining). Change-of-state verbs like *stop*, *begin*, *continue* presuppose a prior state. Cleft constructions (*It was Mary who won*) presuppose that someone won. The challenge for formal semantics — going beyond Montague's two-valued framework — is to model how these presuppositions interact with context, how they can be accommodated (silently accepted by the listener even when not previously established), and when they cancel rather than project. This is the bridge from Montague semantics into formal pragmatics and dynamic semantics, where the meaning of an utterance is understood as an update to a context state rather than a static truth condition.