Turnstile (symbol)

In mathematical logic and computer science the symbol has taken the name turnstile because of its resemblance to a typical turnstile. It is also referred to as tee and is often read as "yields", "proves", "satisfies" or "entails".

Interpretations

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

In epistemology, Per Martin-Löf analyzes the symbol thus: "...he combination of Frege's Urteilsstrich, judgement stroke, and Inhaltsstrich, content stroke, came to be called the assertion sign." Frege's notation for a judgement of some content
In metalogic, the study of formal languages; the turnstile represents syntactic consequence. This is to say, that it shows that one string can be derived from another in a single step, according to the transformation rules of some given formal system. As such, the expression
In proof theory, the turnstile is used to denote "provability" or "derivability". For example, if is a formal theory and is a particular sentence in the language of the theory then
In sequent calculus, the turnstile is used to denote a sequent. A sequent asserts that, if all the antecedents are true, then at least one of the consequents must be true.
In the typed lambda calculus, the turnstile is used to separate typing assumptions from the typing judgment.
In category theory, a reversed turnstile, as in, is used to indicate that the functor is left adjoint to the functor. More rarely, a turnstile, as in, is used to indicate that the functor is right adjoint to the functor.
In APL the symbol is called "right tack" and represents the ambivalent right identity function where both ⊢ and ⊢ are. The reversed symbol "⊣" is called "left tack" and represents the analogous left identity where ⊣ is and ⊣ is.
In combinatorics, means that is a partition of the integer.
In Hewlett-Packard's HP-41C/CV/CX and HP-42S series of calculators, the symbol is called "Append character" and is used to indicate that the following characters will be appended to the alpha register rather than replacing the existing contents of the register. The symbol is also supported in a modified variant of the HP Roman-8 character set used by other HP calculators.
On the Casio fx-92 Collège 2D and fx-92+ Spéciale Collège calculators, the symbol represents the modulo operator; entering will produce an answer of, where is the quotient and is the remainder.
In model theory, means entails, every model of is a model of.

Typography

In TeX, the turnstile symbol is obtained from the command.
In Unicode, the turnstile symbol is called right tack and is at code point U+22A2.

* = turnstile
* = proves, implies, yields
* = reducible
* = reverse turnstile
* = non-theorem, does not yield
* ≡ 22A2⊢ 0338$̸

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.

Similar graphemes

꜔ Modifier Letter Mid Left-Stem Tone Bar
├ Box Drawings Light Vertical And Right
ㅏ Hangul Letter A
Ͱ Greek Capital Letter Heta
ͱ Greek Small Letter Heta
Ⱶ Latin Capital Letter Half H
ⱶ Latin Small Letter Half H⎬ Right Curly Bracket Middle Piece