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 ($\mathrm{\exists }$) 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:

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

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 $\mathrm{\forall }xx=x$ implies $\text{Socrates}=\text{Socrates}$, we could as well say that the denial $\text{Socrates}\ne \text{Socrates}$ implies $\mathrm{\exists }xx\ne x$. 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.

See also

• List of rules of inference
• Universal generalization

References

Template:Logic-stub