Turnstile (symbol)

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.^[1]

Per Martin-Löf (1996) analyzes the $\vdash$ 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 $A$

$\vdash A$

can be then be read

I know $A$ is true".^[3]

In the same vein, a conditional assertion

$P \vdash Q$

can be read as:

From $P$, I know that $Q$

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.^[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

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.^[6] 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 $T$.^[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 $F \dashv G$, is used to indicate that the functor $F$ is left adjoint to the functor $G$.
• 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, $\lambda \vdash n$ means that $\lambda$ is a partition (number theory) of the integer $n$. ^[12]

See also

• Double turnstile $\models$
• Sequent
• Sequent calculus
• List of logic symbols
• List of mathematical symbols

Notes

1. Frege 1879
2. Martin-Löf 1996, pp. 6,15
3. Martin-Löf 1996, p. 15
4. Unicode standard
5. http://www.ctan.org/tex-archive/macros/latex/contrib/turnstile
6. http://dingo.sbs.arizona.edu/~hammond/ling178-sp06/mathCh6.pdf
7. Troelstra & Schwichtenberg 2000
8. http://www.mscs.dal.ca/~selinger/papers/lambdanotes.pdf
9. Schmidt 1994
10. http://www.jsoftware.com/papers/APLDictionary.htm
11. Iverson 1987
12. p.287 of Stanley, Richard P.. Enumerative Combinatorics. 1st ed. Vol. 2. Cambridge: Cambridge University Press, 1999.

References

• Frege, Gottlob (1879). "Begriffsschrift: Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens". Halle.
• Iverson, Kenneth (1987). "A Dictionary of APL".
• Martin-Löf, Per (1996). "On the meanings of the logical constants and the justifications of the logical laws" (PDF). Nordic Journal of Philosophical Logic 1 (1): 11–60. Lecture notes to a short course at Università degli Studi di Siena, April 1983.
• Schmidt, David (1994). "The Structure of Typed Programming Languages". MIT Press. ISBN 0-262-19349-3.
• Troelstra, A. S.; Schwichtenberg, H. (2000). "Basic Proof Theory" (second ed.). Cambridge University Press. ISBN 978-0-521-77911-1.