Universal Negative implies Particular Negative iff First Predicate is not Vacuous

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Theorem

Consider the categorical statements:

\(\ds \map {\mathbf E} {S, P}:\)    The universal negative:      \(\ds \forall x:\) \(\ds \map S x \implies \neg \map P x \)      
\(\ds \map {\mathbf O} {S, P}:\)    The particular negative:      \(\ds \exists x:\) \(\ds \map S x \land \neg \map P x \)      

Then:

$\map {\mathbf E} {S, P} \implies \map {\mathbf O} {S, P}$

if and only if:

$\exists x: \map S x$


Using the symbology of predicate logic:

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


Proof

Sufficient Condition

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

Let $\map {\mathbf E} {S, P}$ be true.

As $\map {\mathbf E} {S, P}$ is true, then by Modus Ponendo Ponens:

$\neg \map P x$

From the Rule of Conjunction:

$\map S x \land \neg \map P x$

Thus $\map {\mathbf O} {S, P}$ holds.

So by the Rule of Implication:

$\map {\mathbf E} {S, P} \implies \map {\mathbf O} {S, P}$

$\Box$


Necessary Condition

Let $\map {\mathbf E} {S, P} \implies \map {\mathbf O} {S, P}$.

Aiming for a contradiction, suppose:

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

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

From De Morgan's Laws: 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:

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

is true.

So $\map {\mathbf E} {S, P}$ holds.


Again, $\neg \exists x: \map S x$.

Then by the Rule of Conjunction:

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

That is, $\map {\mathbf O} {S, P}$ does not hold.

So $\map {\mathbf E} {S, P}$ is true and $\map {\mathbf O} {S, P}$ is false.

This contradicts $\map {\mathbf E} {S, P} \implies \map {\mathbf O} {S, P}$ by definition of implication.

Thus $\exists x: \map S x$ must hold.

$\blacksquare$


Also defined as

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


Sources