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'praeclarum theorema'
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| Title of object: |
praeclarum theorema |
| Canonical Name: |
PraeclarumTheorema |
| Type: |
Theorem |
| Created on: |
2008-02-10 14:51:27 |
| Modified on: |
2008-09-02 15:05:13 |
| Classification: |
msc:01A45, msc:03-03, msc:03B05, msc:03B22, msc:03B35, msc:03B70 |
| Synonyms: |
praeclarum theorema=splendid theorem |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
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Content:
\PMlinkescapeword{ad}
\PMlinkescapeword{AD}
\PMlinkescapeword{c}
\PMlinkescapeword{C}
\PMlinkescapeword{calculus}
\PMlinkescapeword{Calculus}
\PMlinkescapeword{qed}
\PMlinkescapeword{QED}
\PMlinkescapeword{reflect}
\PMlinkescapeword{Reflect}
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:
\[ ((a \Rightarrow b) \land (d \Rightarrow c)) \Rightarrow ((a \land d) \Rightarrow (b \land c)) \]
Under the existential interpretation of logical graphs, the praeclarum theorema is represented by means of the following formal equivalence or logical 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
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}
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