existsand

Existsand is a term sometimes used in logical discussions to denote the existential conjunction of two predicates under a single existential quantifier. In formal notation, existsand is equivalent to the standard expression ∃x (P(x) ∧ Q(x)): there exists an element x such that P(x) and Q(x) hold simultaneously.

Existsand is distinct from the expression (∃x P(x)) ∧ (∃x Q(x), which only asserts that there is some element satisfying P and possibly a separate element satisfying Q. The existsand form requires a single witness that satisfies both predicates.

In practice, existsand is not a standard operator in mainstream predicate logic. It appears mainly as a pedagogical device or in informal discussions to highlight the difference between having a single witness versus separate witnesses. When used, its meaning is usually clarified by defining the scope and providing the equivalent standard form ∃x (P(x) ∧ Q(x)).

Example: ∃x (Musical(x) ∧ Author(x)) asserts there is some x who is both a musician and an author. By contrast, ∃x Musical(x) ∧ ∃x Author(x) says there exists a musician and there exists an author, not necessarily the same person.

See also: existential quantifier, conjunction, predicate logic.