The therefore sign ``$\therefore$ ' is used especially in handwritten mathematical text as a shorthand of the 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 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 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 $$\sin\alpha = \frac{1}{2} \; \land \; 0^\circ \le \alpha \le 90^\circ \;\, \Rightarrow \;\, \alpha = 30^\circ.$$