# Praeclarum Theorema

## Theorem

### Formulation 1

$\left({p \implies q}\right) \land \left({r \implies s}\right) \vdash \left({p \land r}\right) \implies \left({q \land s}\right)$

### Formulation 2

$\vdash \left({\left({p \implies q}\right) \land \left({r \implies s}\right)}\right) \implies \left({\left({p \land r}\right) \implies \left({q \land s}\right)}\right)$

## Leibniz' Proof

The praeclarum theorema was noted and named by Leibniz, who stated and proved it in the following manner:

If $a$ is $b$ and $d$ is $c$, then $ad$ will be $bc$.
This is a fine theorem, which is proved in this way:
$a$ is $b$, therefore $ad$ is $bd$ (by what precedes),
$d$ is $c$, therefore $bd$ is $bc$ (again by what precedes),
$ad$ is $bd$, and $bd$ is $bc$, therefore $ad$ is $bc$.
Q.E.D.
(Leibniz, Logical Papers, p. 41).

## Linguistic Note

Praeclarum Theorema is Latin for splendid theorem.

It was so named by Gottfried Wilhelm von Leibniz.

## Also see

Compare the Constructive Dilemma, which is similar in appearance.

## Sources

• Gottfried W Leibniz: Addenda to the Specimen of the Universal Calculus (1679 – 1686): pp. $40 - 46$