praeclarum theorema

The praeclarum theorema, or splendid theorem, is a theorem of propositional calculusMathworldPlanetmath 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:


Representing propositionsPlanetmathPlanetmath ( as logical graphs ( under the existential interpretationPlanetmathPlanetmath (, 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 UniversalPlanetmathPlanetmath 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 InferenceMathworldPlanetmath”,

3 Resources

  • Dau, Frithjof (2008), Animated Proof of Leibniz’s Praeclarum Theorema.

  • Megill, Norman (2008), Theorema @ 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