Universal Affirmative and Universal Negative are Contrary iff First Predicate is not Vacuous

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Theorem

Consider the categorical statements:

\(\ds \mathbf A:\)    The universal affirmative:      \(\ds \forall x:\) \(\ds \map S x \implies \map P x \)      
\(\ds \mathbf E:\)    The universal negative:      \(\ds \forall x:\) \(\ds \map S x \implies \neg \map P x \)      

Then:

$\mathbf A$ and $\mathbf E$ are contrary

if and only if:

$\exists x: \map S x$


Using the symbology of predicate logic:

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


Proof

Sufficient Condition

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

Suppose $\mathbf A$ and $\mathbf E$ are both true.

As $\mathbf A$ is true, then by Modus Ponendo Ponens:

$\map P x$

As $\mathbf E$ is true, then by Modus Ponendo Ponens:

$\neg \map P x$

It follows by Proof by Contradiction that $\mathbf A$ and $\mathbf E$ are not both true.

Thus, by definition, $\mathbf A$ and $\mathbf E$ are contrary statements.

$\Box$


Necessary Condition

Let $\mathbf A$ and $\mathbf E$ be contrary statements.

Suppose:

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

that is, $\map S x$ is vacuous.

From Denial of Existence:

$\forall x: \neg \map S x \dashv \vdash \neg \exists x: \map S x$

it follows that $\forall x: \map S x$ is false.

From False Statement implies Every Statement:

$\forall x: \map S x \implies \map P x$

is true, and:

$\forall x: \map S x \implies \neg \map P x$

is also true.

Thus, by definition, $\mathbf A$ and $\mathbf E$ are not contrary statements.

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 universal affirmative and universal negative are contrary.


Sources