Universal Negative implies Particular Negative iff First Predicate is not Vacuous
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:
- $\map {\mathbf E} {S, P} \implies \map {\mathbf O} {S, P}$
- $\exists x: \map S x$
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
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $4$: Propositional Functions and Quantifiers: $4.1$: Singular Propositions and General Propositions