therefore sign

The therefore sign “$\therefore$” is used especially in handwritten mathematical text as a shorthand of the or relations:

$$S_1\mathit{\hspace{1em}}\therefore S_2$$

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

$$\sin \alpha =\frac{1}{2}\mathit{\hspace{1em}}\therefore \alpha ={30}^{\circ }$$

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, $0^{\circ }\le \alpha \le {90}^{\circ }$, so what we wrote could be interpreted as the implication

$$\sin \alpha =\frac{1}{2}\wedge {\mathrm{\hspace{0.33em}0}}^{\circ }\le \alpha \le {90}^{\circ }\Rightarrow \alpha ={30}^{\circ }.$$

Title	therefore sign
Canonical name	ThereforeSign
Date of creation	2013-03-22 17:55:51
Last modified on	2013-03-22 17:55:51
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	8
Author	pahio (2872)
Entry type	Definition
Classification	msc 00A06
Classification	msc 00A05
Related topic	RingsOfRationalNumbers
Related topic	LogarithmicScale