Existential generalization

Short description: Rule of inference in predicate logic

Existential generalizationType| Rule of inference

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Field| Predicate logic
Statement| There exists a member x in a universal set with a property of Q
Transformation rules

Propositional calculus
Rules of inference

* Implication introduction / elimination (modus ponens)
* Biconditional introduction / elimination
* Conjunction introduction / elimination
* Disjunction introduction / elimination
* Disjunctive / hypothetical syllogism
* Constructive / destructive dilemma
* Absorption / modus tollens / modus ponendo tollens

Rules of replacement

* Associativity
* Commutativity
* Distributivity
* Double negation
* De Morgan's laws
* Transposition
* Material implication
* Exportation
* Tautology
* Negation introduction

Predicate logic

* Universal generalization / instantiation
* Existential generalization / instantiation

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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."

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:

Q(a)-> ∃xQ(x),

where Q(a) is obtained from Q(x) by replacing all its free occurrences of x (or some of them) by a.[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 ∀xx=x implies Socrates=Socrates, we could as well say that the denial Socrates≠Socrates implies ∃xx≠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

* Existential instantiation
* List of rules of inference
* Universal generalization

## 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. ISBN 9780534145156. 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.

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