# Universal Affirmative and Negative are both False iff Particular Affirmative and Negative are both True

## Theorem

Consider the categorical statements:

 The universal affirmative: $\ds \map {\mathbf A} {S, P}:$ $\ds \forall x: \map S x$ $\ds \implies$ $\ds \map P x$ The universal negative: $\ds \map {\mathbf E} {S, P}:$ $\ds \forall x: \map S x$ $\ds \implies$ $\ds \neg \map P x$ The particular affirmative: $\ds \map {\mathbf I} {S, P}:$ $\ds \exists x: \map S x$ $\ds \land$ $\ds \map P x$ The particular negative: $\ds \map {\mathbf O} {S, P}:$ $\ds \exists x: \map S x$ $\ds \land$ $\ds \neg \map P x$

Then:

$\map {\mathbf A} {S, P}$ and $\map {\mathbf E} {S, P}$ are both false
$\map {\mathbf I} {S, P}$ and $\map {\mathbf O} {S, P}$ are both true.

## Proof

### Necessary Condition

Let $\map {\mathbf A} {S, P}$ and $\map {\mathbf E} {S, P}$ both be false.

 $\text {(1)}: \quad$ $\ds \neg \map {\mathbf A} {S, P}$ $\land$ $\ds \neg \map {\mathbf E} {S, P}$ $\text {(2)}: \quad$ $\ds \leadsto \ \$ $\ds \neg \map {\mathbf A} {S, P}$  $\ds$ Rule of Simplification: from $(1)$ $\text {(3)}: \quad$ $\ds \leadsto \ \$ $\ds \map {\mathbf O} {S, P}$  $\ds$ Universal Affirmative and Particular Negative are Contradictory: from $(2)$ $\text {(4)}: \quad$ $\ds \leadsto \ \$ $\ds \neg \map {\mathbf E} {S, P}$  $\ds$ Rule of Simplification: from $(1)$ $\text {(5)}: \quad$ $\ds \leadsto \ \$ $\ds \map {\mathbf I} {S, P}$  $\ds$ Particular Affirmative and Universal Negative are Contradictory: from $(4)$ $\text {(6)}: \quad$ $\ds \leadsto \ \$ $\ds \map {\mathbf I} {S, P}$ $\land$ $\ds \map {\mathbf O} {S, P}$ Rule of Conjunction: from $(5)$ and $(3)$

Hence $\map {\mathbf I} {S, P}$ and $\map {\mathbf O} {S, P}$ are both true.

$\Box$

### Sufficient Condition

Let $\map {\mathbf I} {S, P}$ and $\map {\mathbf O} {S, P}$ both be true.

 $\text {(1)}: \quad$ $\ds \map {\mathbf I} {S, P}$ $\land$ $\ds \map {\mathbf O} {S, P}$ $\text {(2)}: \quad$ $\ds \leadsto \ \$ $\ds \map {\mathbf I} {S, P}$  $\ds$ Rule of Simplification: from $(1)$ $\text {(3)}: \quad$ $\ds \leadsto \ \$ $\ds \neg \map {\mathbf E} {S, P}$  $\ds$ Particular Affirmative and Universal Negative are Contradictory: from $(2)$ $\text {(4)}: \quad$ $\ds \leadsto \ \$ $\ds \map {\mathbf O} {S, P}$  $\ds$ Rule of Simplification: from $(1)$ $\text {(5)}: \quad$ $\ds \leadsto \ \$ $\ds \neg \map {\mathbf A} {S, P}$  $\ds$ Universal Affirmative and Particular Negative are Contradictory: from $(4)$ $\text {(6)}: \quad$ $\ds \leadsto \ \$ $\ds \neg \map {\mathbf A} {S, P}$ $\land$ $\ds \neg \map {\mathbf E} {S, P}$ Rule of Conjunction: from $(5)$ and $(3)$

Hence $\map {\mathbf A} {S, P}$ and $\map {\mathbf E} {S, P}$ are both false.

$\blacksquare$