|Rules of inference|
|Rules of replacement|
In predicate logic universal instantiation (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 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
for some term a and where is the result of substituting a for all 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.
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.