Universal instantiation

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.

Universal instantiation

Type	Rule of inference
Field	Predicate logic
Symbolic statement	${\displaystyle \forall x\,A\Rightarrow A\{x\mapsto a\}}$

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 ${\displaystyle \vdash \forall xA}$ infer ${\displaystyle \vdash A\{x\mapsto a\}.}$

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]