Double turnstile explained
In logic, the symbol ⊨, ⊧ or
is called the
double turnstile. It is often read as "
entails", "
models", "is a
semantic consequence of" or "is stronger than".
[1] It is closely related to the
turnstile symbol
, which has a single bar across the middle, and which denotes
syntactic consequence (in contrast to
semantic).
Meaning
The double turnstile is a binary relation. It has several different meanings in different contexts:
- To show semantic consequence, with a set of sentences on the left and a single sentence on the right, to denote that if every sentence on the left is true, the sentence on the right must be true, e.g.
. This usage is closely related to the single-barred
turnstile symbol which denotes syntactic consequence.
- To show satisfaction, with a model (or truth-structure) on the left and a set of sentences on the right, to denote that the structure is a model for (or satisfies) the set of sentences, e.g.
. This is typically done
inductively along with restricting the range of a
variable assignment, a function mapping each variable symbol to a value in
it might hold.
[2] - In this context, the semantic consequence in the previous list can be stated as "For a given model
, if
then
".
. which is to say that the expression
is a semantic consequence of the empty set.
- You can also use this symbol as follows: ⊭ to denote the statement 'does not entail'.
Typography
In TeX, the turnstile symbols ⊨ and
are obtained from the commands
\vDash and
\models respectively.
In Unicode it is encoded at, and the opposite of it is .
In LaTeX there is the turnstile package, which issues this sign in many ways, including the double turnstile, 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.
See also
References
- Book: Nederpelt, Rob . Logical Reasoning: A First Course . King's College Publications. 2004. 3rd revised . 62 . Chapter 7: Strengthening and weakening. 0-9543006-7-X.
- Open Logic Project, First-order logic (p.7). Accessed 4 January 2022.
