Concept
Universal instantiation
Overview
In predicate logic, universal instantiation (UI; also called universal specification or universal elimination, and sometimes confused with dictum de omni) 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
∀
x
A
⇒
A
{
x
↦
t
}
,
{\displaystyle \forall x\,A\Rightarrow A\{x\mapsto t\},}
for every formula A and every term t, where
A
{
x
↦
t
}
{\displaystyle A\{x\mapsto t\}}
is the result of substituting t for each free occurrence of x in A.
A
{
x
↦
t
}
{\displaystyle \,A\{x\mapsto t\}}
is an instance of
∀
x
A
.
{\displaystyle \forall x\,A.}
And as a rule of inference it is
from
⊢
∀
x
A
{\displaystyle \vdash \forall xA}
infer
⊢
A
{
x
↦
t
}
.
{\displaystyle \vdash A\{x\mapsto t\}.}
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."
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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Between the phenomena, then, which exist at any instant, and the phenomena which exist at the succeeding instant, there is an invariable order of succession; and, as we said in speaking of the general uniformity of the course of nature, this web is composed of separate fibres; this collective order is made up of particular sequences, obtaining invariably among the separate parts. To certain facts, certain facts…
John Stuart Mill, A System of Logic
States what Universal instantiation is.
In the proposition, “Wine is good,” it is understood with equal readiness, though for somewhat different reasons, that the assertion is not intended to be universal, but particular.(28) As is observed by Professor Bain,(29) the chief examples of Indefinite propositions occur “with names of material, which are the subjects sometimes of universal, and at other times of particular predication. ‘Food is chemically…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Universal instantiation is.
Examples of the latter are: 'The mind is no elephant:' 'A lion is no table;' propositions which are correct but absurd, exactly like the identical propositions: 'A lion is a lion;' 'Mind is mind.' Propositions like these are undoubtedly the truth of the immediate, or, as it is called, Qualitative judgment. But they are not judgments at all, and can only occur in a subjective thought where even an untrue abstraction…
Georg Wilhelm Friedrich Hegel, The Logic of Hegel
States what Universal instantiation is.
Related ideas
- First-order logic
- Reference
- Validity (logic)
- Rule of inference
- Willard Van Orman Quine
- Logical consequence
- Quantifier (logic)
- Natural deduction
- Universal quantification
- Stanisław Jaśkowski
Place in the tradition
Part of Outline of logic.