Existential generalization

Existential generalization

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

In predicate logic, existential generalization¹² (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."

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:

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

where ${\displaystyle Q(a)}$ is obtained from ${\displaystyle Q(x)}$ by replacing all its free occurrences of ${\displaystyle x}$ (or some of them) by ${\displaystyle a}$.³

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 ${\displaystyle {\text{Socrates}}={\text{Socrates}}}$, we could as well say that the denial ${\displaystyle {\text{Socrates}}\neq {\text{Socrates}}}$ 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