Universal instantiation

In logic universal instantiation (UI, also called "Dictum de omni") is an 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. 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."

In symbols the rule as an axiom schema is

$(\forall x\,A(x))\Rightarrow A(a/x),$

for some term a and where $A(a/x)$ is the result of substituting a for all free occurrences of x in A.

And as a rule of inference it is

from ⊢ ∀x A infer ⊢ A(a/x),

with A(a/x) the same as above.

Irving Copi noted that universal instantiation "...follows from variants of rules for 'natural deduction', which were devised independently by Gerhard Gentzen and Stanislaw Jaskowski in 1934." -pg. 71. Symbolic Logic; 5th ed.