Particular Affirmative and Universal Negative are Contradictory

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Theorem

Consider the categorical statements:

\(\ds \mathbf I:\)    The particular affirmative:      \(\ds \exists x:\) \(\ds \map S x \land \map P x \)      
\(\ds \mathbf E:\)    The universal negative:      \(\ds \forall x:\) \(\ds \map S x \implies \neg \map P x \)      

Then $\mathbf I$ and $\mathbf E$ are contradictory.


Using the symbology of predicate logic:

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


Proof

\(\ds \) \(\) \(\ds \mathbf E\)
\(\ds \therefore \ \ \) \(\ds \forall x:\) \(\) \(\ds \map S x \implies \neg \map P x\) Definition of $\mathbf E$
\(\ds \therefore \ \ \) \(\ds \forall x:\) \(\) \(\ds \neg \paren {\map S x \land \map P x}\) Modus Ponendo Tollens
\(\ds \therefore \ \ \) \(\ds \neg \exists x:\) \(\) \(\ds \map S x \land \map P x\) De Morgan's Laws: Denial of Existence
\(\ds \therefore \ \ \) \(\ds \) \(\) \(\ds \neg \mathbf I\) Definition of $\mathbf I$


The argument reverses:

\(\ds \) \(\) \(\ds \mathbf I\)
\(\ds \therefore \ \ \) \(\ds \exists x:\) \(\) \(\ds \map S x \land \map P x\) Definition of $\mathbf I$
\(\ds \therefore \ \ \) \(\ds \exists x:\) \(\) \(\ds \neg \paren {\map S x \implies \neg \map P x}\) Conjunction is Equivalent to Negation of Conditional of Negative
\(\ds \therefore \ \ \) \(\ds \neg \forall x:\) \(\) \(\ds \map S x \implies \neg \map P x\) De Morgan's Laws: Denial of Universality
\(\ds \therefore \ \ \) \(\ds \) \(\) \(\ds \neg \mathbf E\) Definition of $\mathbf E$


The result follows by definition of contradictory.

$\blacksquare$


Sources

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