Definition:Square of Opposition/Categorical Statements

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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.


Vacuous Terms

The traditional treatment of the categorical syllogism makes the assumption that no term is vacuous.

However, from the point of view of the full predicate logic, this assumption may not be valid.


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.


Also see


Sources