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:

:$Q(a) \to\ \exists\, Q(x) ,$

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 $\text=\text$, we could as well say that the denial $\text \ne \text$ 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. Here: p.366.

See also

*Inference rules

References

Rules of inference
Predicate logic

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