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Implicit assumptions about the world. They are required to make an utterance meaningful.
Do you want to do it again? (Presupposition: You have done it already, at least once.)
My wife is pregnant. (Presupposition: I have a wife.)
Negation of a sentence does not change its presuppositions: I want to do it again and I don't want to do it again both mean that I have done it already one or more times; My wife is pregnant and My wife is not pregnant both mean that I have a wife.
If presuppositions of a sentence do not comply to the actual state of affairs, then it makes both the sentence and its negation false.
Much of the information that is exchanged in discourse happens in the form of presuppositions.
This differs from entailment and implication.
Pragmatics