Re: Richard J. Lipton • Logical Complexity Of Proofs
Re: Animated Logical Graphs • (35) (36) (37) (38) (39)

One way to see the difference between insight proofs and routine proofs is to pick a single example of a theorem in propositional calculus and prove it two ways, one more insightful and one more routine.

The praeclarum theorema, or splendid theorem, is a theorem of propositional calculus 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 theorem may be written as follows.

Using teletype parentheses for the logical negation of a proposition and simple concatenation for the logical conjunction of propositions and enables writing the theorem in the following in-line and lispish ways.

Inline

Lispish

Reference

• 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.

Praeclarum Theorema

• This Blog
• OEIS Wiki • (1) • (2) • (3)
• PlanetMath • Praeclarum Theorema
• Metamath Proof Explorer • Praeclarum Theorema
• Frithjof Dau • Computer Animated Proof of Leibniz’s Praeclarum Theorema

Resources

• Logic Syllabus
• Logical Graphs
• Cactus Language
• Futures Of Logical Graphs
• Minimal Negation Operators
• Survey of Theme One Program
• Survey of Animated Logical Graphs
• Propositional Equation Reasoning Systems
• How To Succeed In Proof Business Without Really Trying

Applications

• Applications of a Propositional Calculator • Constraint Satisfaction Problems
• Exploratory Qualitative Analysis of Sequential Observation Data

cc: Cybernetics • Ontolog • Peirce (1) (2) (3) (4) (5) (6) • Structural Modeling • Systems