predicate logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

, universal instantiation (UI; also called universal specification or universal elimination, and sometimes confused with '' dictum de omni'') is a valid

rule of inference In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of in ...

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 In mathematical logic, a universal quantification is a type of quantifier, a logical constant which is interpreted as "given any" or "for all". It expresses that a predicate can be satisfied by every member of a domain of discourse. In other w ...

but it can also be encoded in an

axiom schema In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom. Formal definition An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables ap ...

. 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 :

\forall x \, A \Rightarrow A\,

for every formula ''A'' and every term ''a'', where

A\

is the result of substituting ''a'' for each ''free'' occurrence of ''x'' in ''A''.

\, A\

is an instance of

\forall x \, A.

And as a rule of inference it is :from

\vdash \forall x A

infer

\vdash A \ .

Irving Copi Irving Marmer Copi (; né Copilovich or Copilowish; July 28, 1917 – August 19, 2002) was an American philosopher, logician, and university textbook author. Biography Copi studied under Bertrand Russell while at the University of Chicago. I ...

noted that universal instantiation "... follows from variants of rules for '

natural deduction In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axiom ...

', which were devised independently by

Gerhard Gentzen Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died ...

and Stanisław Jaśkowski in 1934."

Quine

According to

Willard Van Orman Quine Willard Van Orman Quine (; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century". ...

, universal instantiation and

existential generalization In predicate logic, existential generalization (also known as existential introduction, ∃I) is a valid rule of inference that allows one to move from a specific statement, or one instance, to a quantified generalized statement, or existential ...

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. Here: p. 366.

References

{{Reflist Rules of inference Predicate logic

Quine

See also

References