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

$\vdash \paren {\paren {p \implies q} \land \paren {r \implies s} } \implies \paren {\paren {p \land r} \implies \paren {q \land s} }$

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

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

John F. Sowa: Peirce's Rules of Inference: Online version (2002)

Frithjof Dau: Computer Animated Proof of Leibniz's Praeclarum Theorema (2008)

Norman Megill: Praeclarum Theorema @ Metamath Proof Explorer (2008)