Presuppositions are semantic contents that must be true for an utterance to have a truth value, distinct from what is asserted. "The king of France is bald" presupposes there is a king of France; if false, the whole utterance fails to be true or false (or context must repair it). Understanding presupposition projection and how it interacts with assertion explains semantic phenomena and communication success.
When you make a statement, you are not just asserting one thing — you are simultaneously taking a background of other things for granted. Presuppositions are the background assumptions that a sentence requires to be in place for the main claim to even get off the ground as a truth-apt assertion. The classic example is Bertrand Russell's: "The present king of France is bald." This sentence presupposes that France currently has a king. If it doesn't, the sentence doesn't just come out false — it fails to make a determinate claim at all. Philosophers call this presupposition failure: the utterance misfires rather than merely being wrong.
This is distinct from what the sentence asserts, which is the main claim the speaker is committing themselves to as true. Assertion and presupposition carry different semantic weight. If I assert "John has stopped smoking," I presuppose that John was smoking (if he never smoked, the assertion is neither true nor false in a well-defined way), and I assert that he no longer is. You can reject a presupposition with a presupposition denial: "John hasn't stopped smoking — he never started." This is different from just saying "that's false." You're not denying the assertion; you're pulling out the presuppositional prop from under it.
One of the most diagnostically useful tests for presuppositions is that they survive negation. Both "The king of France is bald" and "The king of France is not bald" presuppose that there is a king of France. If negation killed the presupposition, we'd say the negative form lacks it — but it doesn't. Assertions, by contrast, flip under negation: asserting the positive commits you to something the negative denies. This survival-under-negation test distinguishes presupposed content from what is merely conversationally implied (which you may have studied as implicature).
Presupposition projection is the phenomenon where presuppositions of embedded clauses bubble up to the whole sentence. "Sam knows that there's life on Mars" presupposes that there is life on Mars — the presupposition of the embedded clause "there's life on Mars" projects through the factive verb "knows." Factive verbs (know, realize, discover) are major presupposition triggers. So are definite descriptions, iteratives ("again," "still"), and change-of-state verbs ("stop," "start"). Knowing which constructions trigger presuppositions helps you see how much context-setting is happening implicitly in ordinary conversation, and why removing a false presupposition can feel more disruptive than simply contradicting an assertion.
You've already studied the pragmatics-semantics boundary from your prerequisites, and presupposition sits right on that boundary. Some presuppositions are semantic — built into the meaning of the words themselves, regardless of context. Others are pragmatic presuppositions: things the speaker takes for granted in the context of utterance, which listeners are expected to accommodate. When context can supply what's missing (a friend has clearly been telling you about a new relationship and then says "he told me he loves me"), listeners automatically accommodate the presupposition without even noticing. Presupposition accommodation is ubiquitous, invisible when it succeeds, and disruptive when it fails — which is why tracking presuppositional structure is essential to understanding how communication works and breaks down.
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