Introduction

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:

Expressed in contemporary logical notation, the praeclarum theorema may be written as follows:

Representing propositions in the language of logical graphs, and operating under the existential interpretation, the praeclarum theorema is expressed by means of the following formal equivalence or logical equation:

(1)

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

(2)

The steps of the proof are replayed in the following animation.

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.

Readings

• Jon Awbrey • Propositional Equation Reasoning Systems
• John F. Sowa • Peirce’s Rules of Inference

Resources

• Logical Graph @ InterSciWiki
• Praeclarum Theorema @ PlanetMath
• Frithjof Dau • Computer Animated Proof of Leibniz’s Praeclarum Theorema
• Norman Megill • Praeclarum Theorema @ Metamath Proof Explorer