Definition:Universal Statement
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.
Examples
All Men are Mortal
The classical example of a universal statement is the statement:
- All men are mortal.
It has the logical form:
- $\forall x: \map {\operatorname {Man} } x \implies \map {\operatorname {Mortal} } x$
where $\operatorname {Man}$ and $\operatorname {Mortal}$ are the predicates is a man and is mortal respectively.
Also known as
A universal statement can also be referred to as:
- a universal sentence, or more wordily, a sentence of a universal character
- a general sentence
- a general statement.
Also see
- Results about universal statements can be found here.
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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): quantifier
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): quantifier