Definition:Categorical Statement

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


In the above:

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


Predicate of Categorical Statement

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


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

  • 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).


Sources