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

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