Definition:Categorical Statement

Definition
Let $S$ and $P$ be predicates.

A categorical statement is a statement that can be expressed in one of the following ways in natural language:


 * $(A): \quad$ Every $S$ is $P$


 * $(E): \quad$ No $S$ is $P$


 * $(I): \quad$ Some $S$ is $P$


 * $(O): \quad$ Some $S$ is not $P$

In modern predicate logic, they are interpreted as:


 * $(A): \quad \forall x: S \left({x}\right) \implies P \left({x}\right)$


 * $(E): \quad \forall x: S \left({x}\right) \implies \neg P \left({x}\right)$


 * $(I): \quad \exists x: S \left({x}\right) \land P \left({x}\right)$


 * $(O): \quad \exists x: S \left({x}\right) \land \neg P \left({x}\right)$

where $S \left({x}\right)$ and $P \left({x}\right)$ are propositional functions.

It is assumed that all $x$ are elements of a universal set.

Linguistic Note
The letters $A$, $E$, $I$ and $O$ are assigned to the various categorical statements from the first and second vowels to appear in the Latin words affirmo (I affirm) and nego (I deny).

Also known as
Some sources refer to this as a categorical sentence. However, the word statement is generally preferred as the latter term has a more precise definition.

Also see

 * Existential Import
 * Categorical Syllogism
 * Square of Opposition