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.
|Rules of inference|
|Rules of replacement|
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 free occurrences of x in A. is an instance of
And as a rule of inference it is
- from ⊢ ∀x A infer ⊢ A(a/x),
with A(a/x) the same as above.
According to Willard Van Orman Quine, 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.
- Irving M. Copi; Carl Cohen; Kenneth McMahon (Nov 2010). Introduction to Logic. Pearson Education. ISBN 978-0205820375.
- Moore and Parker
- 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. Here: p. 366.