00A06-ThereforeSign.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{ThereforeSign}
\pmcreated{2013-03-22 17:55:51}
\pmmodified{2013-03-22 17:55:51}
\pmowner{pahio}{2872}
\pmmodifier{pahio}{2872}
\pmtitle{therefore sign}
\pmrecord{8}{40425}
\pmprivacy{1}
\pmauthor{pahio}{2872}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{00A06}
\pmclassification{msc}{00A05}
\pmrelated{RingsOfRationalNumbers}
\pmrelated{LogarithmicScale}

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\begin{document}
The {\em therefore sign} ``$\therefore$'' is used especially in handwritten mathematical text as a shorthand of the \PMlinkescapetext{word `therefore' or `thus', between some sentences} or relations:
$$S_1 \quad \therefore \; S_2$$
It expresses that $S_2$ has been inferred from $S_1$ or from $S_1$ and some preceding facts.  The sign is rather a punctuation mark than a symbol of logical implication.  Grammatically, it could be characterised a conclusive coordinating \PMlinkescapetext{conjunction}.  The usage of the symbol is not mathematically well-defined, and it often means `we can conclude in context' or `we can conclude from statements already shown or assumed to be true'.\\

For example, in determining an angle of a right triangle, one may write
$$\sin\alpha = \frac{1}{2} \quad \therefore \; \alpha = 30^\circ$$
Here, ``$\therefore$'' does not \PMlinkescapetext{represent} a proper implication ``$\Rightarrow$'', since the exact implication here would be
$$\sin\alpha = \frac{1}{2} \;\, \Leftarrow \;\, \alpha = 30^\circ.$$
To obtain a strict implication, we would need to introduce some of the context.
For instance, we know that, since $\alpha$ is an angle of a right triangle,
$0^\circ \le \alpha \le 90^\circ$,
so what we wrote could be interpreted as the implication
$$\sin\alpha = \frac{1}{2} \; \land \;
  0^\circ \le \alpha \le 90^\circ
  \;\, \Rightarrow \;\, \alpha = 30^\circ.$$
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\end{document}