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.

Existential generalization

Type	Rule of inference
Field	Predicate logic
Statement	There exists a member $x$ in a universal set with a property of $Q$
Symbolic statement	$Q(a)\to \ \exists {x}\,Q(x),$

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 ${\text{Socrates}}\neq {\text{Socrates}}$ 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