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 quanti ...

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

valid Validity or Valid may refer to: Science/mathematics/statistics: * Validity (logic), a property of a logical argument * Scientific: ** Internal validity, the validity of causal inferences within scientific studies, usually based on experiments ** ...

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 ...

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 Quantification may refer to: * Quantification (science), the act of counting and measuring * Quantification (machine learning) In machine learning and data mining, quantification (variously called ''learning to quantify'', or ''supervised preva ...

for the universal quantifier 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 ...

. It is one of the basic principles used in

quantification theory 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 ...

. 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 noted that universal instantiation "...

follows from Follows is a surname. Notable people with the surname include: * Dave Follows (1941–2003), British cartoonist * Denis Follows (1908–1983), British sports administrator * Geoffrey Follows (1896–1983), British colonial administrator * Megan Fo ...

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 a ...

', which were devised independently by Gerhard Gentzen and

Stanisław Jaśkowski Stanisław Jaśkowski (22 April 1906, in Warsaw – 16 November 1965, in Warsaw) was a Polish logician who made important contributions to proof theory and formal semantics. He was a student of Jan Łukasiewicz and a member of the Lwów–War ...

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