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: \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}$

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.

Also see

 * Universal Affirmative and Particular Negative are Contradictory
 * Particular Affirmative and Universal Negative are Contradictory
 * Universal Affirmative and Universal Negative are Contrary iff First Predicate is not Vacuous
 * Particular Affirmative and Particular Negative are Subcontrary iff First Predicate is not Vacuous
 * Universal Affirmative implies Particular Affirmative iff First Predicate is not Vacuous
 * Universal Negative implies Particular Negative iff First Predicate is not Vacuous