Universal instantiation

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

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

for some term a and where A(a/x) 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.

Irving Copi noted that universal instantiation "...follows from variants of rules for 'natural deduction', which were devised independently by Gerhard Gentzen and Stanislaw Jaskowski 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 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.[5]

See also[edit]

References[edit]

  1. ^ Copi and Cohen
  2. ^ Hurley
  3. ^ Moore and Parker
  4. ^ pg. 71. Symbolic Logic; 5th ed.
  5. ^ Quine,W.V.O., Quintessence, Extensionalism, Reference and Modality, P366