Universal Affirmative and Particular Negative are Contradictory
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Theorem
Consider the categorical statements:
\(\ds \mathbf A:\) | The universal affirmative: | \(\ds \forall x:\) | \(\ds \map S x \implies \map P x \) | ||||||
\(\ds \mathbf O:\) | The particular negative: | \(\ds \exists x:\) | \(\ds \map S x \land \neg \map P x \) |
Then $\mathbf A$ and $\mathbf O$ are contradictory.
Using the symbology of predicate logic:
- $\neg \paren {\paren {\forall x: \map S x \implies \map P x} \iff \paren {\exists x: \map S x \land \neg \map P x} }$
Proof
\(\ds \) | \(\) | \(\ds \mathbf A\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall x: \, \) | \(\ds \) | \(\) | \(\ds \map S x \implies \map P x\) | Definition of $\mathbf A$ | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall x: \, \) | \(\ds \) | \(\) | \(\ds \neg \paren {\map S x \land \neg \map P x}\) | Conditional is Equivalent to Negation of Conjunction with Negative | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \neg \exists x: \, \) | \(\ds \) | \(\) | \(\ds \map S x \land \neg \map P x\) | De Morgan's Laws: Denial of Existence | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds \neg \mathbf O\) | Definition of $\mathbf O$ |
The argument reverses:
\(\ds \) | \(\) | \(\ds \mathbf O\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \exists x: \, \) | \(\ds \) | \(\) | \(\ds \map S x \land \neg \map P x\) | Definition of $\mathbf O$ | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \exists x: \, \) | \(\ds \) | \(\) | \(\ds \neg \paren {\map S x \implies \map P x}\) | Conjunction with Negative is Equivalent to Negation of Conditional | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \neg \forall x: \, \) | \(\ds \) | \(\) | \(\ds \map S x \implies \map P x\) | De Morgan's Laws: Denial of Universality | |||||||||
\(\ds \leadsto \ \ \) | \(\ds \) | \(\) | \(\ds \neg \mathbf A\) | Definition of $\mathbf A$ |
The result follows by definition of contradictory.
$\blacksquare$
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