Turnstile (symbol)

Not to be confused with ㅏ, ト, or 卜.

In mathematical logic and computer science the symbol $\vdash$ 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.

In TeX, the turnstile symbol $\vdash$ is obtained from the command \vdash. In Unicode, the turnstile symbol (⊢) is called right tack and is at code point U+22A2. On a typewriter, a turnstile can be composed from a vertical bar (|) and a dash (–). In LaTeX there is the turnstile package, which issues this sign in many ways, and is capable of putting labels below or above it, in the correct places. The article A Tool for Logicians is a tutorial on using this package.

Interpretations

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

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

$P \vdash Q$

means that $Q$ is derivable from $P$ in the system.

Consistent with its use for derivability, a "$\vdash$" 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

$\vdash Q$

means that $Q$ is a theorem in the system.

• In proof theory, the turnstile is used to denote "provability". For example, if $T$ is a formal theory and $S$ is a particular sentence in the language of the theory then

$T \vdash S$

means that $S$ is provable from $T$. 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.
• In category theory, a reversed turnstile (⊣), as in $F \dashv G$, is used to indicate that the functor $F$ is left adjoint to the functor $G$.

See also

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