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Universal instantiation

Summary

In predicate logic, universal instantiation[1][2][3] (UI; also called universal specification or universal elimination, and sometimes confused with dictum de omni) is a valid rule of inference from a truth about each member of a class of individuals to the truth about a particular individual of that class. It is generally given as a quantification rule for the universal quantifier but it can also be encoded in an axiom schema. It is one of the basic principles used in quantification theory.

Example: "All dogs are mammals. Fido is a dog. Therefore Fido is a mammal."

Formally, the rule as an axiom schema is given as

${\displaystyle \forall x\,A\Rightarrow A\{x\mapsto a\},}$

for every formula A and every term a, where ${\displaystyle A\{x\mapsto a\}}$ is the result of substituting a for each free occurrence of x in A. ${\displaystyle \,A\{x\mapsto a\}}$ is an instance of ${\displaystyle \forall x\,A.}$

And as a rule of inference it is

from ⊢ ∀x A infer ⊢ A{xa}.

Irving Copi noted that universal instantiation "...follows from variants of rules for 'natural deduction', which were devised independently by Gerhard Gentzen and Stanisław Jaśkowski in 1934."[4]

Quine

According to Willard Van Orman Quine, universal instantiation and existential generalization are two aspects of a single principle, for instead of saying that "∀x x = x" implies "Socrates = Socrates", we could as well say that the denial "Socrates ≠ Socrates" implies "∃x x ≠ x". The principle embodied in these two operations is the link between quantifications and the singular statements that are related to them as instances. Yet it is a principle only by courtesy. It holds only in the case where a term names and, furthermore, occurs referentially.[5]