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: | \(\ds \forall x:\) | \(\ds \map S x \) | \(\ds \implies \) | \(\ds \map P x \) | For every thing: if it has $S$, then it also has $P$ | ||
| \((E)\) | $:$ | Universal Negative: | \(\ds \forall x:\) | \(\ds \map S x \) | \(\ds \implies \) | \(\ds \neg \map P x \) | For every thing: if it has $S$, then it does not also have $P$ | ||
| \((I)\) | $:$ | Particular Affirmative: | \(\ds \exists x:\) | \(\ds \map S x \) | \(\ds \land \) | \(\ds \map P x \) | There is a thing which has $S$ and also has $P$ | ||
| \((O)\) | $:$ | Particular Negative: | \(\ds \exists x:\) | \(\ds \map S x \) | \(\ds \land \) | \(\ds \neg \map P x \) | 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 presented in abbreviated form as:
- $\map {\mathbf \Phi} {S, P}$
where $\mathbf \Phi$ 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:
| \(\ds \map {\mathbf A} {S, P} \) | denotes | All $S$ are $P$ | |||||||
| \(\ds \map {\mathbf E} {S, P} \) | denotes | No $S$ are $P$ | |||||||
| \(\ds \map {\mathbf I} {S, P} \) | denotes | Some $S$ are $P$ | |||||||
| \(\ds \map {\mathbf O} {S, P} \) | 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 a categorical statement 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).
Sources
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $4$: Propositional Functions and Quantifiers: $4.1$: 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): categorical proposition
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): categorical proposition