In a mathematical proof, the therefore sign (∴) is a symbol that is sometimes placed before a logical consequence, such as the conclusion of a syllogism.^{[citation needed]} The symbol consists of three dots placed in an upright triangle and is read therefore. It is Unicode character U+2234 and on some systems may be entered using ALT8756 (the decimal version of 2234). While it is not generally used in formal writing, it is often used in mathematics and shorthand.
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According to Cajori, A History of Mathematical Notations, the therefore sign was first used by Johann Rahn in 1659, in the original German edition of his book Teutsche Algebra.
The inverted form ∵, known as the because sign, is sometimes used as a shorthand form of "because". This is Unicode character U+2235.
The therefore sign is sometimes used as a substitute for an asterism [⁂].
To denote logical implication or entailment, various signs are used in mathematical logic: →, ⇒, ⊃, ⊢, ⊨. These symbols are then part of a mathematical formula, and are not considered to be punctuation. In contrast, the therefore sign is traditionally used as a punctuation mark, and does not form part of a formula.^{[1]}
The graphically identical sign ∴ serves as a Japanese map symbol on the maps of the Geographical Survey Institute of Japan, indicating a tea plantation. On other maps the sign, often with thicker dots, is sometimes used to signal the presence of a national monument or ruins.
The character ஃ in the Tamil script represents the āytam, a special sound of the Tamil language.
Some secret traditions use this sign for abbreviation instead of the usual period. This usage is much more frequent in French Freemasonry than in the English one.^{[2]}
Used in a syllogism:
It Would be proper to indicate a premise with the because sign! For example:
∵ All men are mortals. ∵ Socrates is a man. ∴ Socrates is a mortal.^{[3]}
For proper Punctuation, there is another Latin phrase with a slightly different meaning, and less common in usage. Quod erat faciendum is translated as "which was to have been done". This is usually shortened to Q.E.F. The expression quod erat faciendum is a translation of the Greek geometers' closing ὅπερ ἔδει ποιῆσαι (hoper edei poiēsai). Euclid used this phrase to close propositions which were not proofs of theorems, but constructions. For example, Euclid's first proposition shows how to construct an equilateral triangle given one side.
