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, 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 (∃) 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]

Quine[edit]

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

References[edit]

  1. ^ Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall. 
  2. ^ Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing. 
  3. ^ pg. 347. Jon Barwise and John Etchemendy, Language proof and logic Second Ed., CSLI Publications, 2008.
  4. ^ Willard van Orman Quine; Roger F. Gibson (2008). "V.24. Reference and Modality". Quintessence. Cambridge, Mass: Belknap Press of Harvard University Press.  Here: p.366.