Therefore sign
Encyclopedia
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
. The symbol consists of three dots placed in an upright triangle and is read therefore. It is encoded at . While it is not generally used in formal writing, it is often used in mathematics and shorthand. It is complementary to .
, 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.
:
It would also be proper to indicate a premise with the because sign. For example:
∵ All men are mortals. ∵ Socrates is a man. ∴ Socrates is a mortal.
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.
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 in the Tamil script
represents the āytam
, a special sound of the Tamil language
.
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).
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
, the therefore sign (∴) is a symbol that is sometimes placed before a logical consequence, such as the conclusion of a syllogism
Syllogism
A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...
. The symbol consists of three dots placed in an upright triangle and is read therefore. It is encoded at . While it is not generally used in formal writing, it is often used in mathematics and shorthand. It is complementary to .
History
According to CajoriFlorian Cajori
Florian Cajori was one of the most celebrated historians of mathematics in his day.- Biography :...
, A History of Mathematical Notations, the therefore sign was first used by Johann Rahn
Johann Rahn
Johann Rahn was a Swiss mathematician who is credited with the first use of the division symbol, ÷ and the therefore sign, ∴. The symbol is used in Teutsche Algebra, published in 1659.-References:...
in 1659, in the original German edition of his book Teutsche Algebra.
Example of use
Used in a syllogismSyllogism
A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...
:
- All men are mortal.
- Socrates is a man.
- Socrates is mortal.
- x+1=6
- x=5
It would also be proper to indicate a premise with the because sign. For example:
∵ All men are mortals. ∵ Socrates is a man. ∴ Socrates is a mortal.
Related signs
Shorthand
Shorthand is an abbreviated symbolic writing method that increases speed or brevity of writing as compared to a normal method of writing a language. The process of writing in shorthand is called stenography, from the Greek stenos and graphē or graphie...
form of "because". This is Unicode
Unicode
Unicode is a computing industry standard for the consistent encoding, representation and handling of text expressed in most of the world's writing systems...
character U+2235.
The therefore sign is sometimes used as a substitute for an asterism
Asterism (typography)
In typography, an asterism, from the Greek astēr , is a rarely used, and "nearly obsolete", symbol consisting of three asterisks placed in a triangle . It is used to, "indicate minor breaks in text," call attention to a passage, or to separate sub-chapters in a book...
.
To denote logical implication or entailment
Entailment
In logic, entailment is a relation between a set of sentences and a sentence. Let Γ be a set of one or more sentences; let S1 be the conjunction of the elements of Γ, and let S2 be a sentence: then, Γ entails S2 if and only if S1 and not-S2 are logically inconsistent...
, various signs are used in mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
: , , , , . 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
Ruins
Ruins are the remains of human-made architecture: structures that were once complete, as time went by, have fallen into a state of partial or complete disrepair, due to lack of maintenance or deliberate acts of destruction...
.
The character in the Tamil script
Tamil script
The Tamil script is a script that is used to write the Tamil language as well as other minority languages such as Badaga, Irulas, and Paniya...
represents the āytam
Aytam
Āytam is a special sound in the Tamil Language. It is represented in the Tamil script by the character ஃ. It is special in the senses of not being an independent sound and being archaic and employed only in idiomatic and fossilized words such as அஃது, இஃது etc.The sound āytam is mentioned in the...
, a special sound of the Tamil language
Tamil language
Tamil is a Dravidian language spoken predominantly by Tamil people of the Indian subcontinent. It has official status in the Indian state of Tamil Nadu and in the Indian union territory of Pondicherry. Tamil is also an official language of Sri Lanka and Singapore...
.
In Masonic traditions the symbol is used for abbreviation
Abbreviation
An abbreviation is a shortened form of a word or phrase. Usually, but not always, it consists of a letter or group of letters taken from the word or phrase...
, 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).