Existential instantiation explained

Existential instantiation
Type:	Rule of inference
Field:	Predicate logic
Symbolic Statement:	\exists x P \left({x}\right) \implies P \left({a}\right)

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

(\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 x P \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

Notes and References

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