Existential instantiation

Existential instantiation

Type	Rule of inference
Field	Predicate logic
Symbolic statement	$\exists xP\left({x}\right)\implies P\left({a}\right)$

In predicate logic, existential instantiation (also called existential elimination) is a rule of inference which says that, given a formula of the form $(\exists x)\phi (x)$, one may infer $\phi (c)$ 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 $x$ which is bound to $\exists x$ must be uniformly replaced by c. This is implied by the notation $P\left({a}\right)$, but its explicit statement is often left out of explanations.

In one formal notation, the rule may be denoted by

$\exists xP\left({x}\right)\implies P\left({a}\right)$

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