# 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:

 $(A)$ $:$ Universal Affirmative: Every $S$ is $P$ $(E)$ $:$ Universal Negative: No $S$ is $P$ $(I)$ $:$ Particular Affirmative: Some $S$ is $P$ $(O)$ $:$ Particular Negative: Some $S$ is not $P$

In this context, the word is has the meaning of the is of predication:

is $P$ means has the property $P$, or belongs to the class of things that have the property $P$
is not $P$ means does not have the property $P$, or does not belong to the class of things that have the property $P$.

The word has could equally well be used:

has $P$ for is $P$
does not have $P$ for is not $P$.

In modern predicate logic, they are denoted as:

 $(A)$ $:$ Universal Affirmative: $\displaystyle \forall x:$ $\displaystyle \map S x$ $\displaystyle \implies$ $\displaystyle \map P x$ For every thing: if it has $S$, then it also has $P$ $(E)$ $:$ Universal Negative: $\displaystyle \forall x:$ $\displaystyle \map S x$ $\displaystyle \implies$ $\displaystyle \neg \map P x$ For every thing: if it has $S$, then it does not also have $P$ $(I)$ $:$ Particular Affirmative: $\displaystyle \exists x:$ $\displaystyle \map S x$ $\displaystyle \land$ $\displaystyle \map P x$ There is a thing which has $S$ and also has $P$ $(O)$ $:$ Particular Negative: $\displaystyle \exists x:$ $\displaystyle \map S x$ $\displaystyle \land$ $\displaystyle \neg \map P x$ There is a thing which has $S$ that does not also have $P$

In the above:

$\map S x$ and $\map P x$ are propositional functions
all $x$ belong to a specified universal of discourse.

### Abbreviation of Categorical Statement

A categorical statement connecting $S$ and $P$ can be abbreviated as:

$\mathbf{\Phi} \left({S, P}\right)$

where $\Phi$ is one of either $\mathbf{A}$, $\mathbf{E}$, $\mathbf{I}$ or $\mathbf{O}$, signifying Universal Affirmative, Universal Negative, Particular Affirmative and Particular Negative respectively.

Thus:

$\mathbf{A} \left({S, P}\right)$ denotes All $S$ are $P$
$\mathbf{E} \left({S, P}\right)$ denotes No $S$ are $P$
$\mathbf{I} \left({S, P}\right)$ denotes Some $S$ are $P$
$\mathbf{O} \left({S, P}\right)$ denotes Some $S$ are not $P$.

### Subject of Categorical Statement

The symbol $S$ can be referred to as the subject of $\map {\mathbf {\Phi} } {S, P}$.

### Predicate of Categorical Statement

The symbol $P$ can be referred to as the predicate of $\map {\mathbf {\Phi} } {S, P}$.

## 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.

Some sources use the term categorical proposition.

## Also see

• Results about categorical statements can be found here.

## 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)
nEgO (I deny).