Английская Википедия:Existential generalization

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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 (<math>\exists</math>) 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:

<math> Q(a) \to\ \exists{x}\, Q(x) ,</math>

where <math>Q(a)</math> is obtained from <math>Q(x)</math> by replacing all its free occurrences of <math>x</math> (or some of them) by <math>a</math>.

Quine

According to Willard Van Orman Quine, universal instantiation and existential generalization are two aspects of a single principle, for instead of saying that <math>\forall x \, x=x</math> implies <math>\text{Socrates}=\text{Socrates}</math>, we could as well say that the denial <math>\text{Socrates} \ne \text{Socrates}</math> implies <math>\exists x \, x \ne x</math>. 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

References

Шаблон:Logic-stub