In mathematical proof, the therefore sign (∴) is sometimes placed before a logical consequence, such as the conclusion of a syllogism. The symbol consists of three dots placed in an upright triangle and is read therefore. It is encoded at U+2234 ∴ therefore (HTML:
∴). While it is not generally used in formal writing, it is often used in mathematics and shorthand. It is complementary to U+2235 ∵ because (HTML:
According to Cajori, A History of Mathematical Notations, Johann Rahn used both the therefore and because signs to mean "therefore"; in the German edition of Teutsche Algebra (1659) the therefore sign was prevalent with the modern meaning, but in the 1668 English edition Rahn used the because sign more often to mean "therefore". Other authors in the eighteenth century also used three dots in a triangle shape to signify "therefore", but as with Rahn, there wasn't much in the way of consistency as to how the triangle was oriented; because with its current meaning appears to have originated in the nineteenth century. Also, in this century, the three-dot notation for therefore becomes very rare in continental Europe, but remains popular in the British Isles.
Example of use
Used in a syllogism:
- All gods are immortal.
- Zeus is a god.
- ∴ Zeus is immortal.
- ∴ x=5
It would also be proper to indicate a premise with the because sign. For example:
∵ All gods are immortals. ∵ Zeus is a god. ∴ Zeus is an immortal.
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.
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.
In Masonic traditions the symbol is used for abbreviation, instead of the usual period. For example "R∴W∴ John Smith" is an abbreviation for "Right Worshipful John Smith" (the term Right Worshipful indicates that Brother Smith is a Grand Lodge officer).
- "Cajori, vol. 1, p. 211". Archive.org. Retrieved 2013-10-05.
- Florian Cajori (2011) . A History of Mathematical Notations: Two Volumes in One. Cosimo, Inc. pp. 282–283. ISBN 978-1-61640-571-7.
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- Encyclopedia of Freemasonry Part 1 and Its Kindred Sciences Comprising the ..., Albert Gallatin Mackey, page 2, reprint in 2002, Kessinger Publishing, ISBN 0-7661-2650-1.