Definition:Square of Opposition

Definition

The square of opposition is a diagram whose purpose is to illustrate the relations between the various categories of quantified statement.

$\begin{xy} <-10em,0em>*+{\forall x: \map \Phi x} = "A", <10em,0em>*+{\forall x: \neg \map \Phi x} = "E", <-10em,-20em>*+{\exists x: \map \Phi x} = "I", <10em,-20em>*+{\exists x: \neg \map \Phi x} = "O", "A";"E" **@{-} ?<*@{<} ?>*@{>} ?*!/^.8em/{\text{Contraries}}, "A";"I" **@{-} ?>*@{>} ?*!/^3.2em/{\text{Subimplicant}}, "A";"O" **@{-} ?<*@{<} ?>*@{>} ?*!/^4em/{\text{Contradictories}}, "I";"E" **@{-} ?<*@{<} ?>*@{>} ?*!/_4em/{\text{Contradictories}}, "I";"O" **@{-} ?<*@{<} ?>*@{>} ?*!/_.8em/{\text{Subcontraries}}, "E";"O" **@{-} ?>*@{>} ?*!/^-3.2em/{\text{Subimplicant}}, \end{xy}$

This therefore illustrates the relations:

All $x$ have $\Phi$ is contrary to All $x$ do not have $\Phi$
All $x$ have $\Phi$ is contradictory to Some $x$ do not have $\Phi$
Some $x$ have $\Phi$ is contradictory to All $x$ do not have $\Phi$
Some $x$ have $\Phi$ is subimplicant to All $x$ have $\Phi$
Some $x$ do not have $\Phi$ is subimplicant to All $x$ do not have $\Phi$
Some $x$ have $\Phi$ is subcontrary to Some $x$ do not have $\Phi$

where $x$ is an object and $\Phi$ is a property.

Vacuous Universe

Beware
Note that if the universe of discourse is empty, then the square of opposition no longer holds.

Although $\forall x: \map \Phi x$ is vacuously true for such an empty universe, $\exists x: \map \Phi x$ is not.

Thus $\exists x: \map \Phi x$ is no longer subimplicant to $\forall x: \map \Phi x$.

Similarly, as $\exists x: \neg \map \Phi x$ is also false, it follows that $\exists x: \map \Phi x$ and $\exists x: \neg \map \Phi x$ are no longer subcontrary.

Categorical Statements

The square of opposition can be extended to the context of categorical statements:

$\begin{xy} <-10em,0em>*+{\forall x: \map S x \implies \map P x} = "A", <10em,0em>*+{\forall x: \map S x \implies \neg \map P x} = "E", <-10em,-20em>*+{\exists x: \map S x \land \map P x} = "I", <10em,-20em>*+{\exists x: \map S x \land \neg \map P x} = "O", "A";"E" **@{-} ?<*@{<} ?>*@{>} ?*!/^.8em/{\text{Contraries}}, "A";"I" **@{-} ?>*@{>} ?*!/^3.2em/{\text{Subimplicant}}, "A";"O" **@{-} ?<*@{<} ?>*@{>} ?*!/^4em/{\text{Contradictories}}, "I";"E" **@{-} ?<*@{<} ?>*@{>} ?*!/_4em/{\text{Contradictories}}, "I";"O" **@{-} ?<*@{<} ?>*@{>} ?*!/_.8em/{\text{Subcontraries}}, "E";"O" **@{-} ?>*@{>} ?*!/^-3.2em/{\text{Subimplicant}}, \end{xy}$