# 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, or one instance, to a quantified generalized statement, or existential proposition. In first-order logic, it is often used as a rule for the existential quantifier (${\displaystyle \exists }$) in formal proofs.

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

In the Fitch-style calculus:

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

Where ${\displaystyle a}$ replaces all free instances of ${\displaystyle x}$ within ${\displaystyle Q(x)}$.[3]

## Quine

According to Willard Van Orman Quine, universal instantiation and existential generalization are two aspects of a single principle, for instead of saying that ${\displaystyle \forall x\,x=x}$ implies ${\displaystyle {\text}={\text}}$, we could as well say that the denial ${\displaystyle {\text}\neq {\text}}$ implies ${\displaystyle \exists x\,x\neq x}$. 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]