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In predicate logic universal instantiation[1][2][3] (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

∀ x

A ( x ) ⇒ A ( a

/

x ) ,

displaystyle forall x,A(x)Rightarrow A(a/x),

for some term a and where

A ( a

/

x )

displaystyle A(a/x)

is the result of substituting a for all free occurrences of x in A.

A ( a

/

x )

displaystyle ,A(a/x)

is an instance of

∀ x

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 independently by Gerhard Gentzen
Gerhard Gentzen
and Stanisław Jaśkowski
Stanisław Jaśkowski
in 1934." [4] Quine[edit] 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.[5] See also[edit]

Existential generalization Existential quantification Inference rules

References[edit]

^ 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. 

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