# 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:

(I affirm)*A*ff*I*rmo**n**(I deny).*E*g*O*

## Sources

- 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $4.1$: Singular Propositions and General Propositions - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{III}$: The Logic of Predicates $(1): \ 4$: Propositions with more than one predicate expression - 1995: Merrilee H. Salmon:
*Introduction to Logic and Critical Thinking*: $\S 10.2$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**syllogism** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**syllogism**