Existential instantiation

Existential instantiation

Type	Rule of inference
Field	Predicate logic

In predicate logic, existential instantiation (also called existential elimination) 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