Existential instantiation

Short description: Rule of inference in predicate logic

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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In predicate logic, existential instantiation (also called existential elimination)[1][2] is a rule of inference which says that, given a formula of the form (∃x)ϕ(x), one may infer ϕ(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 ∃x must be uniformly replaced by c. This is implied by the notation P(a), but its explicit statement is often left out of explanations.

In one formal notation, the rule may be denoted by

∃xP(x)⟹P(a)

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

## See also

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

## References

1. ↑ Hurley, Patrick. A Concise Introduction to Logic (11th ed.). Wadsworth Pub Co, 2008. Pg. 454. ISBN 978-0-8400-3417-5
2. ↑ Copi, Irving M.; Cohen, Carl (2002). Introduction to logic (11th ed.). Upper Saddle River, N.J.: Prentice Hall. ISBN 978-0-13-033737-5. https://archive.org/details/studyguideintrod0000mill/mode/2up?q=instantiation.

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