Existential generalization

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Existential generalization
TypeRule of inference
FieldPredicate logic
StatementThere exists a member [math]\displaystyle{ x }[/math] in a universal set with a property of [math]\displaystyle{ Q }[/math]

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 ([math]\displaystyle{ \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]\displaystyle{ Q(a) \to\ \exists{x}\, Q(x) , }[/math]

where [math]\displaystyle{ Q(a) }[/math] is obtained from [math]\displaystyle{ Q(x) }[/math] by replacing all its free occurrences of [math]\displaystyle{ x }[/math] (or some of them) by [math]\displaystyle{ a }[/math].[3]


According to Willard Van Orman Quine, universal instantiation and existential generalization are two aspects of a single principle, for instead of saying that [math]\displaystyle{ \forall x \, x=x }[/math] implies [math]\displaystyle{ \text{Socrates}=\text{Socrates} }[/math], we could as well say that the denial [math]\displaystyle{ \text{Socrates} \ne \text{Socrates} }[/math] implies [math]\displaystyle{ \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.[4]

See also


  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. https://archive.org/details/studyguidetoacco00burc. 
  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, Massachusetts: Belknap Press of Harvard University Press. OCLC 728954096.  Here: p.366.