Turnstile (symbol)

 ( Mathematical Symbols ) 

In mathematical logic and computer science the symbol ⊢ ($\backslash vdash$) 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".

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

• In epistemology, Per Martin-Löf (1996) analyzes the $\backslash vdash$ symbol thus: "...The 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

:$\backslash vdash\; A$

can then be read

: I know is true.

In the same vein, a conditional assertion

:$P\; \backslash vdash\; Q$

can be read as:

: From , I know that

• In metalogic, the study of ; the turnstile represents syntactic consequence (or "derivability"). This is to say, that it shows that one string can be formal proof from another in a single step, according to the transformation rules (i.e. the syntax) of some given formal system. As such, the expression

:$P\; \backslash vdash\; Q$

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 . As such, the expression

:$\backslash vdash\; Q$

means that is a theorem in the system.

• 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

:$T\; \backslash vdash\; S$

means that is formal proof from . This usage is demonstrated in the article on propositional calculus. The syntactic consequence of provability should be contrasted to semantic consequence, denoted by the double turnstile symbol $\backslash models$. One says that $S$ is a semantic consequence of $T$, or $T\; \backslash models\; S$, when all possible valuations in which $T$ is true, $S$ is also true. For propositional logic, it may be shown that semantic consequence $\backslash models$ and derivability $\backslash vdash$ are equivalent to one-another. That is, propositional logic is sound ($\backslash vdash$ implies $\backslash models$) and complete ($\backslash models$ implies $\backslash vdash$)Dirk van Dalen, Logic and Structure (1980), Springer, . See Chapter 1, section 1.5.

• In sequent calculus, the turnstile is used to denote a sequent. A sequent $A\_1,\backslash ,\backslash dots,A\_m\; \backslash ,\backslash vdash\backslash ,\; B\_1,\backslash ,\backslash dots,B\_n$ asserts that, if all the antecedents $A\_1,\backslash ,\backslash dots,A\_m$ are true, then at least one of the consequents $B\_1,\backslash ,\backslash dots,B\_n$ 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 ($\backslash dashv$), as in $F\; \backslash dashv\; G$, is used to indicate that the functor is adjoint functors to the functor . More rarely, a turnstile ($\backslash vdash$), as in $G\; \backslash vdash\; F$, is used to indicate that the functor is adjoint functors 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, $\backslash lambda\; \backslash vdash\; n$ means that is a partition of the integer .
• In Hewlett-Packard's HP-41C/CV/CX and HP-42S series of calculators, the symbol (at code point 127 in the FOCAL character set) 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 (at code point 148) 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 operation operator; entering $5\backslash vdash2$ will produce an answer of $Q=2;R=1$, where is the quotient and is the remainder.
• In model theory, $\backslash varphi\; \backslash vdash\; \backslash psi$ means $\backslash varphi$ entails $\backslash psi$, every model of $\backslash varphi$ is a model of $\backslash psi$.

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Typography In TeX, the turnstile symbol $\backslash vdash$ is obtained from the command .

In Unicode, the turnstile symbol (⊢) is called right tack and is at code point U+22A2. (Code point U+22A6 is named assertion sign (⊦).)

• = 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.

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Similar graphemes

• ꜔ (U+A714) Modifier Letter Mid Left-Stem Tone Bar
• ├ (U+251C) Box Drawings Light Vertical And Right
• ㅏ (U+314F) Hangul Letter A
• Ͱ (U+0370) Greek Capital Letter Heta
• ͱ (U+0371) Greek Small Letter Heta
• Ⱶ (U+2C75) Latin Capital Letter Half H
• ⱶ (U+2C76) Latin Small Letter Half H
• ⎬ (U+23AC) Right Curly Bracket Middle Piece

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See also

• Double turnstile
• List of logic symbols
• List of mathematical symbols

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Notes

• (Lecture notes to a short course at Università degli Studi di Siena, April 1983.)
• (1994). 9780262193498, MIT Press. ISBN 9780262193498
• (2025). 9780521779111, Cambridge University Press. ISBN 9780521779111

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