Universal Affirmative and Universal Negative are Contrary iff First Predicate is not Vacuous
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
- $\exists x: \map S x$
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
- 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