Definition:Universal Statement
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Definition
A universal statement is one which expresses the fact that all objects (in a particular universe of discourse) have a particular property.
That is, a statement of the form:
- $\forall x: \map P x$
where:
- $\forall$ is the universal quantifier
- $P$ is a predicate symbol.
It means:
- All $x$ (in some given universe of discourse) have the property $P$.
Note that if there exist no $x$ in this particular universe, $\forall x: \map P x$ is always true: see vacuous truth.
Bound Variable
In the universal statement:
- $\forall x: \map P x$
the symbol $x$ is a bound variable.
Thus, the meaning of $\forall x: \map P x$ does not change if $x$ is replaced by another symbol.
That is, $\forall x: \map P x$ means the same thing as $\forall y: \map P y$ or $\forall \alpha: \map P \alpha$.
And so on.
Also known as
A universal statement can also be referred to as a universal sentence, or more wordily, a sentence of a universal character.
Also see
Sources
- 1946: Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences (2nd ed.) ... (previous) ... (next): $\S 1.3$: Universal and Existential Sentences
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 3$: Statements and conditions; quantifiers