Existential generalization

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 (${\displaystyle \exists }$) in formal proofs.

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

In the Fitch-style calculus:

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

Where ${\displaystyle a}$ replaces all free instances of ${\displaystyle x}$ within ${\displaystyle Q(x)}$.[3]

Quine

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

• Inference rules

References

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, Massachusetts: Belknap Press of Harvard University Press. OCLC 728954096. Here: p.366.