Turnstile (symbol)

In mathematical logic and computer science the symbol has taken the name turnstile because of its resemblance to a typical turnstile if viewed from above. It is also referred to as tee and is often read as "yields", "proves", "satisfies" or "entails". The symbol was first used by Gottlob Frege in his 1879 book on logic, Begriffsschrift.[1]

Per Martin-Löf (1996) analyzes the symbol thus: "...[T]he combination of Frege's Urteilsstrich, judgement stroke [ | ], and Inhaltsstrich, content stroke [—], came to be called the assertion sign."[2] Frege's notation for a judgement of some content A

can then be read

I know A is true.[3]

In the same vein, a conditional assertion

can be read as:

From P, I know that Q

In TeX, the turnstile symbol is obtained from the command \vdash. In Unicode, the turnstile symbol () is called right tack and is at code point U+22A2.[4] (Code point U+22A6 is named assertion sign ().) On a typewriter, a turnstile can be composed from a vertical bar (|) and a dash (–). In LaTeX there is a turnstile package which issues this sign in many ways, and is capable of putting labels below or above it, in the correct places.[5]

Interpretations

The turnstile represents a binary relation. It has several different interpretations in different contexts:

means that Q is derivable from P in the system.
Consistent with its use for derivability, a "⊢" followed by an expression without anything preceding it denotes a theorem, which is to say that the expression can be derived from the rules using an empty set of axioms. As such, the expression
means that Q is a theorem in the system.
means that S is provable from T.[7] This usage is demonstrated in the article on propositional calculus.

See also

Notes

References

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