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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  is  and  is  , then  will be  .
This is a fine theorem, which is proved in this way:
is , therefore is  (by what precedes),
is , therefore is (again by what precedes),
is , and is , therefore is . Q.E.D.
(Leibniz, Logical Papers, p. 41).
Expressed in contemporary logical notation, the praeclarum theorema (PT) may be written as follows:
Representing propositions as logical graphs under the existential interpretation, the praeclarum theorema is expressed by means of the following formal equation:
![\includegraphics[scale=0.8]{PraeclarumTheoremaFigure1}](http://images.planetmath.org:8080/cache/objects/10255/l2h/img9.png "\includegraphics[scale=0.8]{PraeclarumTheoremaFigure1}") |
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And here's a neat proof of that nice theorem.
![\includegraphics[scale=0.8]{PraeclarumTheoremaFigure2}](http://images.planetmath.org:8080/cache/objects/10255/l2h/img10.png "\includegraphics[scale=0.8]{PraeclarumTheoremaFigure2}") |
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- 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.
- Sowa, John F. (2002), “Peirce's Rules of Inference", Online.
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