praeclarum theoremaPraeclarumTheorema2008-02-10 14:51:272009-04-30 19:50:35Theorem% this is the default PlanetMath preamble. as your knowledge
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The \textbf{praeclarum theorema}, or \textit{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:
\begin{quote}
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, \textit{Logical Papers}, p. 41).
\end{quote}
Expressed in contemporary logical notation, the praeclarum theorema (PT) may be written as follows:
\[ ((a \Rightarrow b) \land (d \Rightarrow c)) \Rightarrow ((a \land d) \Rightarrow (b \land c)) \]
Representing \PMlinkname{propositions}{PropositionalCalculus} as \PMlinkname{logical graphs}{LogicalGraph} under the \PMlinkname{existential interpretation}{LogicalGraphFormalDevelopment}, the praeclarum theorema is expressed by means of the following formal equation:
\begin{center}\begin{tabular}{cc}
\includegraphics[scale=0.8]{PraeclarumTheoremaFigure1} & (1) \\
\end{tabular}\end{center}
And here's a neat proof of that nice theorem.
\begin{center}\begin{tabular}{cc}
\includegraphics[scale=0.8]{PraeclarumTheoremaFigure2} & (2) \\
\end{tabular}\end{center}
\section{References}
\begin{itemize}
\item
Leibniz, Gottfried W. (1679--1686 ?), ``Addenda to the Specimen of the Universal Calculus", pp. 40--46 in G.H.R. Parkinson (ed., trans., 1966), \textit{Leibniz : Logical Papers}, Oxford University Press, London, UK.
\end{itemize}
\section{Readings}
\begin{itemize}
\item
Sowa, John F. (2002), ``Peirce's Rules of Inference", \PMlinkexternal{Online}{http://www.jfsowa.com/peirce/infrules.htm}.
\end{itemize}
\section{Resources}
\begin{itemize}
\item
Dau, Frithjof (2008), \PMlinkexternal{Computer Animated Proof of Leibniz's Praeclarum Theorema}{http://dr-dau.net/pc.shtml}.
\item
Megill, Norman (2008), \PMlinkexternal{Praeclarum Theorema}{http://us.metamath.org/mpegif/prth.html} @ \PMlinkexternal{Metamath Proof Explorer}{http://us.metamath.org/mpegif/mmset.html}.
\end{itemize}