Universal instantiation

In predicate logic, universal instantiation<ref>Template:Cite bookTemplate:Page needed</ref><ref>Hurley, Patrick. A Concise Introduction to Logic. Wadsworth Pub Co, 2008.</ref><ref>Moore and ParkerTemplate:Full citation needed</ref> (UI; also called universal specification or universal elimination,Template:Cn and sometimes confused with dictum de omni)Template:Cn 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 schema. 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."

Formally, the rule as an axiom schema is given as

<math>\forall x \, A \Rightarrow A\{x \mapsto t\},</math>

for every formula A and every term t, where <math>A\{x \mapsto t\}</math> is the result of substituting t for each free occurrence of x in A. <math>\, A\{x \mapsto t\}</math> is an instance of <math>\forall x \, A.</math>

And as a rule of inference it is

from <math>\vdash \forall x A</math> infer <math>\vdash A \{ x \mapsto t \} .</math>

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."<ref>Copi, Irving M. (1979). Symbolic Logic, 5th edition, Prentice Hall, Upper Saddle River, NJ</ref>

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.<ref>Template:Cite book Here: p. 366.</ref>

See also

• Existential instantiation
• Existential quantification