Turnstile (symbol) explained

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".

Interpretations

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

• In epistemology, Per Martin-Löf (1996) analyzes the

\vdash symbol thus: "...[T]he combination of Frege's German: Urteilsstrich, judgement stroke [ | ], and German: Inhaltsstrich, content stroke [—], came to be called the assertion sign." Frege's notation for a judgement of some content \vdash A

can then be read

I know is true.

In the same vein, a conditional assertion

P \vdash Q

can be read as:

From, I know that

• 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.^[1] As such, the expression

P \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 axioms. As such, the expression

\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 \vdash S

means that is provable 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

\models . One says that S is a semantic consequence of T , or T \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 \models and derivability \vdash are equivalent to one-another. That is, propositional logic is sound (\vdash implies \models ) and complete (\models implies \vdash )^[2]

• In sequent calculus, the turnstile is used to denote a sequent. A sequent

A _{1,...,A} _{m} \vdash B _{1,...,B} _{n} asserts that, if all the antecedents A _{1,...,A} _{m} are true, then at least one of the consequents B _{1,...,B} _{n} must be true.

• In the typed lambda calculus, the turnstile is used to separate typing assumptions from the typing judgment.^[3]
• In category theory, a reversed turnstile (

\dashv ), as in F \dashv G , is used to indicate that the functor is left adjoint to the functor .^[4] More rarely, a turnstile (\vdash ), as in G \vdash F , is used to indicate that the functor is right adjoint to the functor .^[5]

• 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 .^[6]
• In combinatorics,

λ \vdash n means that is a partition of the integer .^[7]

• 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,^[8] the symbol represents the modulo operator; entering

5 \vdash 2 will produce an answer of Q =2 ;R =1 , where is the quotient and is the remainder. On other Casio calculators (such as on the Belgian variants—the fx-92B Spéciale Collège and fx-92B Collège 2D calculators^[9] —where the decimal separator is represented as a dot instead of a comma), the modulo operator is represented by ÷R instead.

• In model theory,

\varphi \vdash \psi means \varphi entails \psi , every model of \varphi is a model of \psi .

Typography

In TeX, the turnstile symbol

\vdash is obtained from the command .

In Unicode, the turnstile symbol (⊢) is called right tack and is at code point U+22A2.^[10] (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.^[11]

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

See also

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

References

• Book: Frege . Gottlob . Gottlob Frege . Begriffsschrift: Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens . Halle . 1879.
• Book: Iverson . Kenneth . Kenneth E. Iverson . A Dictionary of APL . 1987.
• Martin-Löf . Per . Per Martin-Löf . On the meanings of the logical constants and the justifications of the logical laws . . 1 . 1 . 1996 . 11–60. (Lecture notes to a short course at Università degli Studi di Siena, April 1983.)
• Book: Schmidt . David . David A. Schmidt . The Structure of Typed Programming Languages . . 1994 . 0-262-19349-3.
• Book: Troelstra . A. S. . A. S. Troelstra . Schwichtenberg . H. . H. Schwichtenberg . Basic Proof Theory . 2nd . . 2000 . 978-0-521-77911-1.

Notes and References

1. Web site: Chapter 6, Formal Language Theory.
2. Dirk van Dalen, Logic and Structure (1980), Springer, . See Chapter 1, section 1.5.
3. Web site: Peter Selinger, Lecture Notes on the Lambda Calculus.
4. Web site: adjoint functor in nLab. ncatlab.org.
5. FunctorFact . 750413999649546245 . 5 July 2016 . Functor Fact on Twitter .
6. Web site: A Dictionary of APL. www.jsoftware.com.
7. Book: Stanley, Richard P. . 287 . Enumerative Combinatorics . 1st . 2 . Cambridge . Cambridge University Press . 1999 .
8. Book: fx-92 Spéciale Collège Mode d'emploi. Casio. 2015. 12.
9. Web site: Remainder Calculations - Casio fx-92B User Manual [Page 13] ManualsLib]. 2020-12-24. www.manualslib.com.
10. Web site: Unicode standard.
11. Web site: CTAN: /tex-archive/macros/latex/contrib/turnstile. ctan.org.

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