# praeclarum theorema

The praeclarum theorema, or splendid theorem, is a theorem of propositional calculus that was noted and named by G.W. 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).

Expressed in contemporary logical notation, the praeclarum theorema (PT) may be written as follows:

 $((a\Rightarrow b)\land(d\Rightarrow c))\Rightarrow((a\land d)\Rightarrow(b% \land c))$

Representing propositions (http://planetmath.org/PropositionalCalculus) as logical graphs (http://planetmath.org/LogicalGraph) under the existential interpretation (http://planetmath.org/LogicalGraphFormalDevelopment), the praeclarum theorema is expressed by means of the following formal equation:

 (1)

And here’s a neat proof of that nice theorem.

 (2)

## 1 References

• Leibniz, Gottfried W. (1679–1686 ?), “Addenda to the Specimen of the Universal Calculus”, pp. 40–46 in G.H.R. Parkinson (ed., trans., 1966), Leibniz : Logical Papers, Oxford University Press, London, UK.