Existential generalization

In predicate logic, existential generalization (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 ($\exists$) 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 $Q(x)$.

Quine

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

See also

• Inference rules