Existential instantiation

Existential instantiationType| Rule of inference

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Field| Predicate logic
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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* e

In predicate logic, existential instantiation (also called existential elimination)[1][2][3] is a rule of inference which says that, given a formula of the form [math]\displaystyle{ (\exists x) \phi(x) }[/math], one may infer [math]\displaystyle{ \phi(c) }[/math] for a new constant symbol c. The rule has the restrictions that the constant c introduced by the rule must be a new term that has not occurred earlier in the proof, and it also must not occur in the conclusion of the proof. It is also necessary that every instance of [math]\displaystyle{ x }[/math] which is bound to [math]\displaystyle{ \exists x }[/math] must be uniformly replaced by c. This is implied by the notation [math]\displaystyle{ P\left({a}\right) }[/math], but its explicit statement is often left out of explanations.

In one formal notation, the rule may be denoted by

[math]\displaystyle{ \exists x P \left({x}\right) \implies P \left({a}\right) }[/math]

where a is a new constant symbol that has not appeared in the proof.

## See also

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

## References

1. ↑ Hurley, Patrick. A Concise Introduction to Logic. Wadsworth Pub Co, 2008.
2. ↑ Copi and Cohen
3. ↑ Moore and Parker

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