# Existential generalization

In predicate logic, existential generalization[1][2] (also known as existential introduction, ∃I) is a valid rule of inference that allows one to move from a specific statement to a quantified generalized statement. In first-order logic, it is often used as a rule for the existential quantifier (∃) in formal proofs.

Example: "Rover loves to wag his tail. Therefore, something loves to wag its tail."

In the Fitch-style calculus:

$Q(a) \to\ \exists{x}\, Q(x)$

Where a replaces all free instances of x within Q(x).[3]

## Quine

Universal instantiation and Existential Generalization are two aspects of a single principle, for instead of saying that '(x)(x=x)' implies 'Socrates is Socrates', we could as well say that the denial 'Socrates≠Socrates' implies '(∃x)(x≠x)'. The principle embodied in these two operations is the link between quantifications and the singular statements that are related to them as instances. Yet it is a principle only by courtesy. It holds only in the case where a term names and, furthermore, occurs referentially.[4]