In logic

Logic

In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

 universal instantiation (UI, also called universal specification, and sometimes confused with Dictum de omni

Dictum de omni et nullo

In Aristotelean logic, dictum de omni et nullo is the principle that whatever is affirmed or denied of a whole kind K may be affirmed or denied any subkind of K...

) is an inference

Inference

Inference is the act or process of deriving logical conclusions from premises known or assumed to be true. The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic.Human inference Inference is the act or process of deriving logical conclusions...

 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

Axiom schema

In mathematical logic, an axiom schema generalizes the notion of axiom.An axiom schema is a formula in the language of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which...

 is

for some term a and where is the result of substituting a for all free occurrences of x in A.

And as a rule of inference

Rule of inference

In logic, a rule of inference, inference rule, or transformation rule is the act of drawing a conclusion based on the form of premises interpreted as a function which takes premises, analyses their syntax, and returns a conclusion...

 it is

from ⊢ ∀x A infer ⊢ A(a/x),

with A(a/x) the same as above.

Irving Copi

Irving Copi

Irving Marmer Copi was an American philosopher, logician, and university textbook author....

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

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...

', which were devised independently by Gerhard Gentzen

Gerhard Gentzen

Gerhard Karl Erich Gentzen was a German mathematician and logician. He had his major contributions in the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus...

 and Stanislaw Jaskowski

Stanislaw Jaskowski

Stanisław Jaśkowski 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–Warsaw School of Logic. Upon his death his name was added to the Genius Wall of fame...

in 1934."