# therefore sign

The therefore sign$\therefore$” is used especially in handwritten mathematical text as a shorthand of the 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 . 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 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, $0^{\circ}\leq\alpha\leq 90^{\circ}$, so what we wrote could be interpreted as the implication

 $\sin\alpha=\frac{1}{2}\;\land\;0^{\circ}\leq\alpha\leq 90^{\circ}\;\,% \Rightarrow\;\,\alpha=30^{\circ}.$
Title therefore sign ThereforeSign 2013-03-22 17:55:51 2013-03-22 17:55:51 pahio (2872) pahio (2872) 8 pahio (2872) Definition msc 00A06 msc 00A05 RingsOfRationalNumbers LogarithmicScale