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."

Example: "Alice made herself a cup of tea. Therefore, Alice made someone a cup of tea."

Example: "Alice made herself a cup of tea. Therefore, someone made someone a cup of tea."

In the Fitch-style calculus:


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

where $$Q(a)$$ is obtained from $$Q(x)$$ by replacing all its free occurrences of $$x$$ (or some of them) by $$a$$.

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 $$\text{Socrates} \ne \text{Socrates}$$ implies $$\exists x \, x \ne x$$. 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.