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:

$$((a\Rightarrow b)\wedge (d\Rightarrow c))\Rightarrow ((a\wedge d)\Rightarrow (b\wedge c))$$

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:

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

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.

2 Readings

• Sowa, John F. (2002), “Peirce’s Rules of Inference”, http://www.jfsowa.com/peirce/infrules.htmOnline.

3 Resources

• Dau, Frithjof (2008), http://web.archive.org/web/20070706192257/http://dr-dau.net/pc.shtmlComputer Animated Proof of Leibniz’s Praeclarum Theorema.

• Megill, Norman (2008), http://us.metamath.org/mpegif/prth.htmlPraeclarum Theorema @ http://us.metamath.org/mpegif/mmset.htmlMetamath Proof Explorer.