Definition:Square of Opposition

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Definition

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


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


This therefore illustrates the relations:

All $x$ 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 $\forall x: \Phi \left({x}\right)$ is vacuously true for such an empty universe, $\exists x: \Phi \left({x}\right)$ is not.

Thus $\exists x: \Phi \left({x}\right)$ is no longer subimplicant to $\forall x: \Phi \left({x}\right)$.

Similarly, as $\exists x: \neg \Phi \left({x}\right)$ is also false, it follows that $\exists x: \Phi \left({x}\right)$ and $\exists x: \neg \Phi \left({x}\right)$ are no longer subcontrary.


Categorical Statements

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


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


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