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".

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.[1] (Code point U+22A6 is named assertion sign ().) 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.[2]

Interpretations

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

can then be read
I know A is true.[4]
In the same vein, a conditional assertion
can be read as:
From P, I know that Q
means that Q is derivable from P 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 Q is a theorem in the system.
means that S is provable from T.[6] 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 . One says that is a semantic consequence of , or , when all possible valuations in which is true, is also true. For propositional logic, it may be shown that semantic consequence and derivability are equivalent to one-another. That is, propositional logic is sound ( implies ) and complete ( implies )[7]

Similar graphemes

See also

Notes

  1. ^ "Unicode standard" (PDF).
  2. ^ "CTAN: /tex-archive/macros/latex/contrib/turnstile". ctan.org.
  3. ^ Martin-Löf 1996, pp. 6, 15
  4. ^ Martin-Löf 1996, p. 15
  5. ^ "Chapter 6, Formal Language Theory" (PDF).
  6. ^ Troelstra & Schwichtenberg 2000
  7. ^ Dirk van Dalen, Logic and Structure (1980), Springer, ISBN 3-540-20879-8. See Chapter 1, section 1.5.
  8. ^ "Peter Selinger, Lecture Notes on the Lambda Calculus" (PDF).
  9. ^ Schmidt 1994
  10. ^ "adjoint functor in nLab". ncatlab.org.
  11. ^ @FunctorFact (5 July 2016). "Functor Fact on Twitter" (Tweet) – via Twitter.
  12. ^ "A Dictionary of APL". www.jsoftware.com.
  13. ^ Iverson 1987
  14. ^ Stanley, Richard P. (1999). Enumerative Combinatorics. Vol. 2 (1st ed.). Cambridge: Cambridge University Press. p. 287. |volume= has extra text (help)
  15. ^ fx-92 Spéciale Collège Mode d'emploi (PDF). CASIO COMPUTER CO., LTD. 2015. p. 12.
  16. ^ "Remainder Calculations - Casio fx-92B User Manual [Page 13] | ManualsLib". www.manualslib.com. Retrieved 2020-12-24.

References