Existential generalization

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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 to a quantified generalized statement. In first-order logic, it is often used as a rule for the existential quantifier (∃) in formal proofs.

Example: "Rover loves to wag his tail. Therefore, something loves to wag its tail."

In the Fitch-style calculus:

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

Where a replaces all free instances of x within Q(x).[3]


Universal instantiation and Existential Generalization are two aspects of a single principle, for instead of saying that '(x)(x=x)' implies 'Socrates is Socrates', we could as well say that the denial 'Socrates≠Socrates' implies '(∃x)(x≠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]

See also[edit]


  1. ^ Hurley
  2. ^ Copi and Cohen
  3. ^ pg. 347. Jon Barwise and John Etchemendy, Language proof and logic Second Ed., CSLI Publications, 2008.
  4. ^ Quine,W.V.O., Quintessence, Extensionalism, Reference and Modality, P366