# 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.

## 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.

 Title praeclarum theorema Canonical name PraeclarumTheorema Date of creation 2013-03-22 17:47:37 Last modified on 2013-03-22 17:47:37 Owner Jon Awbrey (15246) Last modified by Jon Awbrey (15246) Numerical id 16 Author Jon Awbrey (15246) Entry type Theorem Classification msc 03B70 Classification msc 03B35 Classification msc 03B22 Classification msc 03B05 Classification msc 03-03 Classification msc 01A45 Synonym splendid theorem Related topic OrderedGroup