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 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
A ( x ) ⇒ A ( a
x ) ,
displaystyle forall x,A(x)Rightarrow A(a/x),
for some term a and where
A ( a
is the result of substituting a for all free occurrences of x in A.
A ( a
is an instance of
A ( x ) .
displaystyle forall x,A(x).
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
Existential generalization Existential quantification Inference rules
^ Irving M. Copi; Carl Cohen; Kenneth McMahon (Nov 2010). Introduction to Logic. Pearson Education. ISBN 978-0205820375. [page needed] ^ Hurley[full citation needed] ^ Moore and Parker[full citation needed] ^ Copi, Irving M. (1979). Symbolic Logic, 5th edition, Prentice Hall, Upper Saddle River, NJ ^ Willard van Orman Quine; Roger F. Gibson (2008). "V.24. Reference and Modality". Quintessence. Cambridge, Mass: Belknap Press of Harvard University Press.