Particular Affirmative and Particular Negative are Subcontrary iff First Predicate is not Vacuous

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Theorem

Consider the categorical statements:

\(\ds \mathbf I:\)    The particular affirmative:      \(\ds \exists x:\) \(\ds \map S x \land \map P x \)      
\(\ds \mathbf O:\)    The particular negative:      \(\ds \exists x:\) \(\ds \map S x \land \neg \map P x \)      

Then:

$\mathbf I$ and $\mathbf O$ are subcontrary

if and only if:

$\exists x: \map S x$


Using the symbology of predicate logic:

$\exists x: \map S x \iff \paren {\paren {\exists x: \map S x \land \map P x} \lor \paren {\exists x: \map S x \land \neg \map P x} }$


Proof

Sufficient Condition

Let $\exists x: \map S x$.

Suppose $\mathbf I$ and $\mathbf O$ are both false.

As $\mathbf I$ is false, then by the Rule of Conjunction:

$\neg \map P x$

As $\mathbf O$ is false, then by the Rule of Conjunction:

$\neg \neg \map P x$

and so by Double Negation:

$\map P x$

It follows by Proof by Contradiction that $\mathbf I$ and $\mathbf O$ are not both false.

Thus, by definition, $\mathbf I$ and $\mathbf O$ are subcontrary statements.

$\Box$


Necessary Condition

Let $\mathbf I$ and $\mathbf O$ be subcontrary statements

Suppose $\neg \exists x: \map S x$.

From the definition of logical conjunction, it follows that:

$\neg \paren {\exists x: \map S x \land \map P x}$

Similarly from the definition of logical conjunction, it follows that:

$\neg \paren {\exists x: \map S x \land \neg \map P x}$

That is, both $\mathbf I$ and $\mathbf O$ are false.

So, by definition, $\mathbf I$ and $\mathbf O$ are not subcontrary.

It follows by Proof by Contradiction that $\exists x: \map S x$.

$\blacksquare$


Also defined as

Some sources gloss over the possibility of $\map S x$ being vacuous and merely report that the particular affirmative and particular negative are subcontrary.


Sources

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