# Existential generalization

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Type Rule of inference Predicate logic There exists a member $\displaystyle{ x }$ in a universal set with a property of $\displaystyle{ Q }$

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."

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]

## 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 $\displaystyle{ \text{Socrates}=\text{Socrates} }$, we could as well say that the denial $\displaystyle{ \text{Socrates} \ne \text{Socrates} }$ implies $\displaystyle{ \exists x \, x \ne x }$. 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.[4]