Universal instantiation

In logic universal instantiation (UI, sometimes confused with 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.