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SYNTAX: The old information in a sentence that forms the background for the new information, that is usually called the focus. If we see a sentence as the answer to a question, then the presupposition is the information that was already part of the question. EXAMPLE:

Q: Who is taking care of Mary?
A: John is taking care of Mary

The presupposition of sentence A is boldfaced. The focus is John. The topic, Mary in this case, is part of the presupposition.

SEMANTICS: Condition that has to be fulfilled for a sentence to be either true or false. The sentence has no truth value if the presupposition fails to hold. Presuppositions can be triggered by specific words or constructions, like definite descriptions, factive predicates and clefts. The presupposition of a sentence A is entailed by both A and the negation of A. EXAMPLE:

 A:  It is John who is taking care of Mary

-A: It isnít John who is taking care of Mary
P: Someone is taking care of Mary

P is the presupposition of both A and its negation -A, triggered by the cleft construction. Both A and -A are infelicitous if the presupposition is not satisfied.
LIT. Levinson, Stephen C. (1983)