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 $a d$ will be $b c$ .
This is a fine theorem, which is proved in this way:
$a$ is $b$ , therefore $a d$ is $b d$ (by what precedes),
$d$ is $c$ , therefore $b d$ is $b c$ (again by what precedes),
$a d$ is $b d$ , and $b d$ is $b c$ , therefore $a d$ is $b c$ . 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