Existential Quantification

Tags

logic

Is a quantification \[ \exists x \; P(x) \] which [see page 18, reads] as "there exists an \( x \) for which the predicate P(x) holds". In this expression the free-variable \( x \) is bound by the quantifier \( \exists x \).

If the predicate is true for some value the truth-set must not be empty. If the predicate is not true for some value the truth set is a strict subset of the universe of discourse (and may be the empty set).

This quantification can be [see page 20, bounded] by another set or predicate. For example \( \exists x \in A \; P(x) \) is equivalent ot \( \exists x (x \in A \land P(x)) \).