In predicate logic universal instantiation (UI, also called universal specification, 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. 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 is 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.
- Copi and Cohen
- Moore and Parker
- pg. 71. Symbolic Logic; 5th ed.
- Quine,W.V.O., Quintessence, Extensionalism, Reference and Modality, P366