Praeclarum Theorema

The “Splendid Theorem” of Leibniz

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Abstract

The praeclarum theorema, or splendid theorem, is a theorem of propositional calculus that was noted and named by G.W. Leibniz.

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:


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:

((a Rightarrow b) wedge (d Rightarrow c)) Rightarrow ((a wedge d) Rightarrow (b wedge 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:

 
 (1)

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

 
 (2)


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.

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