predicate logic First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables ove ...

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

rule of inference Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the Logical form, logical structure of Validity (logic), valid arguments. If an argument with true premises follows a ...

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", "for all", "for every", or "given an arbitrary element". It expresses that a predicate can be satisfied by e ...

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 variabl ...

. 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 ''t'', where

A\

is the result of substituting ''t'' 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 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 ...

', which were devised independently by Gerhard Gentzen 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 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