Turnstile (symbol)

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Not to be confused with , or .

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

can be then be read

I know is true".[3]

In the same vein, a conditional assertion

can be read as:

From , I know that

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] 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[edit]

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

means that is derivable from 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 is a theorem in the system.
  • In proof theory, the turnstile is used to denote "provability". For example, if is a formal theory and is a particular sentence in the language of the theory then
means that is provable from .[7] This usage is demonstrated in the article on propositional calculus.
  • In the typed lambda calculus, the turnstile is used to separate typing assumptions from the typing judgment.[8][9]
  • In category theory, a reversed turnstile (), as in , is used to indicate that the functor is left adjoint to the functor .
  • In APL the symbol is called "right tack" and represents the ambivalent right identity function where both X⊢Y and ⊢Y are Y. The reversed symbol "⊣" is called "left tack" and represents the analogous left identity where X⊣Y is X and ⊣Y is Y.[10][11]
  • In combinatorics, means that is a partition (number theory) of the integer . [12]

See also[edit]

Notes[edit]

References[edit]