Definition:Square of Opposition/Categorical Statements

Definition
The square of opposition is a diagram whose purpose is to illustrate the relations between the various types of categorical statement.


 * $\begin{xy}

<-10em,0em>*+{\forall x: S \left({x}\right) \implies P \left({x}\right)} = "A", <10em,0em>*+{\forall x: S \left({x}\right) \implies \neg P \left({x}\right)} = "E", <-10em,-20em>*+{\exists x: S \left({x}\right) \land P \left({x}\right)} = "I", <10em,-20em>*+{\exists x: S \left({x}\right) \land \neg P \left({x}\right)} = "O",

"A";"E" **@{-} ?<*@{<} ?>*@{>} ?*!/^.8em/{\text{Contraries}}, "A";"I" **@{-} ?>*@{>} ?*!/^3.2em/{\text{Subimplicant}}, "A";"O" **@{-} ?<*@{<} ?>*@{>} ?*!/^4em/{\text{Contradictories}}, "I";"E" **@{-} ?<*@{<} ?>*@{>} ?*!/_4em/{\text{Contradictories}}, "I";"O" **@{-} ?<*@{<} ?>*@{>} ?*!/_.8em/{\text{Subcontraries}}, "E";"O" **@{-} ?>*@{>} ?*!/^-3.2em/{\text{Subimplicant}}, \end{xy}$

This therefore illustrates the relations:


 * All $S$ are $P$ is contrary to No $S$ are $P$


 * All $S$ are $P$ is contradictory to Some $S$ are not $P$


 * Some $S$ are $P$ is contradictory to No $S$ are $P$


 * Some $S$ are $P$ is subimplicant to All $S$ are $P$


 * Some $S$ are not $P$ is subimplicant to No $S$ are $P$


 * Some $S$ are $P$ is subcontrary to Some $S$ are not $P$

where $S$ and $P$ are predicates.

Vacuous Universe

 * Beware: Note that if $S$ is empty, then the square of opposition no longer holds.

Although All $S$ are $P$ is vacuously true for such an empty universe, Some $S$ are $P$ is not.

Thus Some $S$ are $P$ is no longer subimplicant to All $S$ are $P$.

Similarly, as Some $S$ are not $P$ is also false, it follows that All $S$ are $P$ and Some $S$ are not $P$ are no longer subcontrary.