- Quine
- See also
- References
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
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.
Irving Copi noted that universal instantiation "...follows from variants of rules for 'natural deduction', which were devised independently by Gerhard Gentzen and Stanisław Jaśkowski in 1934." [4]Quine
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.[5]
See also
- Existential generalization
- Existential quantification
- Inference rules
References
2. ^Hurley{{full citation needed|date=November 2014}}
3. ^Moore and Parker{{full citation needed|date=November 2014}}
4. ^Copi, Irving M. (1979). Symbolic Logic, 5th edition, Prentice Hall, Upper Saddle River, NJ
5. ^{{cite book |author1=Willard Van Orman Quine |author1-link=Willard Van Orman Quine|author2=Roger F. Gibson |title=Quintessence |contribution= V.24. Reference and Modality |location=Cambridge, Mass |publisher=Belknap Press of Harvard University Press |year=2008 |url=http://www.worldcat.org/title/quintessence-basic-readings-from-the-philosophy-of-wv-quine/oclc/728954096 }} Here: p. 366.
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