Existential generalization
| Transformation rules |
|---|
| Propositional calculus |
| Rules of inference |
| Rules of replacement |
| Predicate logic |
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:
Where a replaces all free instances of x within Q(x).[3]
Quine
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 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
References
- ↑ Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic. Prentice Hall.
- ↑ Hurley, Patrick (1991). A Concise Introduction to Logic 4th edition. Wadsworth Publishing.
- ↑ pg. 347. Jon Barwise and John Etchemendy, Language proof and logic Second Ed., CSLI Publications, 2008.
- ↑ 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.