Praeclarum Theorema

History
The praeclarum theorema, or splendid theorem, is a theorem of classical 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:

$$((a \Rightarrow b) \land (d \Rightarrow c)) \Rightarrow ((a \land d) \Rightarrow (b \land c))$$

Context : Classical Propositional Calculus
Classical Propositional Calculus

Statement
$$((a \Rightarrow b) \land (d \Rightarrow c)) \Rightarrow ((a \land d) \Rightarrow (b \land c))$$

Proof
See below.

Context : Logical Graphs
Logical Graph

Statement
Representing propositions as logical graphs under the existential interpretation, the praeclarum theorema is expressed by means of the following formal equation:

Proof
And here's a neat proof of that nice theorem.

Readings

 * Sowa, John F. (2002), "Peirce's Rules of Inference", Online.

Resources

 * Dau, Frithjof (2008), Computer Animated Proof of Leibniz's Praeclarum Theorema.


 * Megill, Norman (2008), Praeclarum Theorema @ Metamath Proof Explorer.