\begin{definition}[Definition:Categorical Statement]
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:
{{begin-axiom}}
{{axiom | n  = A
        | lc = Universal Affirmative:
        | t  = Every $S$ is $P$
}}
{{axiom | n  = E
        | lc = Universal Negative:
        | t  = No $S$ is $P$
}}
{{axiom | n  = I
        | lc = Particular Affirmative:
        | t  = Some $S$ is $P$
}}
{{axiom | n  = O
        | lc = Particular Negative:
        | t  = Some $S$ is not $P$
}}
{{end-axiom}}
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:
{{begin-axiom}}
{{axiom | n  = A
        | lc = Universal Affirmative:
        | q  = \forall x
        | ml = \map S x
        | mo = \implies
        | mr = \map P x
        | rc = For every thing: if it has $S$, then it also has $P$
}}
{{axiom | n  = E
        | lc = Universal Negative:
        | q  = \forall x
        | ml = \map S x
        | mo = \implies
        | mr = \neg \map P x
        | rc = For every thing: if it has $S$, then it does not also have $P$
}}
{{axiom | n  = I
        | lc = Particular Affirmative:
        | q  = \exists x
        | ml = \map S x
        | mo = \land
        | mr = \map P x
        | rc = There is a thing which has $S$ and also has $P$
}}
{{axiom | n  = O
        | lc = Particular Negative:
        | q  = \exists x
        | ml = \map S x
        | mo = \land
        | mr = \neg \map P x
        | rc = There is a thing which has $S$ that does not also have $P$
}}
{{end-axiom}}
In the above:
:$\map S x$ and $\map P x$ are propositional functions
:all $x$ belong to a specified universal of discourse.
\end{definition}