In mathematical logic, a witness is a specific value t to be substituted for variable x of an existential statement of the form ∃x φ(x) such that φ(t) is true.
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Examples
For example, a theory T of arithmetic is said to be inconsistent if there exists a proof in T of the formula "0=1". The formula I(T), which says that T is inconsistent, is thus an existential formula. A witness for the inconsistency of T is a particular proof of "0 = 1" in T.
Boolos, Burgess, and Jeffrey (2002:81) define the notion of a witness with the example, in which S is an n-place relation on natural numbers, R is an n-place recursive relation, and ↔ indicates logical equivalence (if and only if):
" A y such that R holds of the xi may be called a 'witness' to the relation S holding of the xi (provided we understand that when the witness is a number rather than a person, a witness only testifies to what is true)." In this particular example, B-B-J have defined s to be (positively) recursively semidecidable, or simply semirecursive.Henkin witnesses
In predicate calculus, a Henkin witness for a sentence
Relation to game semantics
The notion of witness leads to the more general idea of game semantics. In the case of sentence