# 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

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

Here, “$\therefore $” does not a proper implication^{} “$\Rightarrow $”, since the exact implication here would be

$$\mathrm{sin}\alpha =\frac{1}{2}\Leftarrow \alpha ={30}^{\circ}.$$ |

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

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

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 |