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 may be written as follows:

((a ⇒ b) ∧ (d ⇒ c)) ⇒ ((a ∧ d) ⇒ (b ∧ c))

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:



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



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

Praeclarum Theorema Proof Animation


  • 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.



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