Existential generalization

Rule of inference in predicate logic

Existential generalizationType| Rule of inference

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Field| Predicate logic
Statement| There exists a member  x {\displaystyle x} in a universal set with a property of  Q {\displaystyle Q}
Symbolic statement|  Q ( a ) -> ∃ x Q ( x ) , {\displaystyle Q(a)\to \ \exists {x}\,Q(x),}
Transformation rules

Propositional calculus
Rules of inference (List)

* 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
* Modus non excipiens
* Negation introduction

Rules of replacement

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

Predicate logic
Rules of inference

* 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 ( ∃ {\displaystyle \exists } ) 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 ) -> ∃ x Q ( x ) , {\displaystyle Q(a)\to \ \exists {x}\,Q(x),}

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

## Quine

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

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* List of rules of inference
* Universal generalization

## References

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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.
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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