Double turnstile (⊨)
⊨, the double turnstile, means “entails” or “models”. Γ ⊨ φ says that φ is true in every case where all of Γ are true; M ⊨ φ says that the structure M makes φ true.
Easily confused with
How to read it
The Stanford Encyclopedia of Philosophy writes Γ ⊨ θ for an argument that is semantically valid: in every interpretation that makes all the premises true, the conclusion is true too. θ is then a logical, semantic or model-theoretic consequence of Γ. The same sign between a structure and a sentence, M ⊨ φ, means M satisfies φ. forall x: Calgary uses ⊨ for entailment in its introductory logic and notes two special cases: ⊨ C with nothing on the left says C is true on every valuation, a tautology, and a slashed ⊭ denies the entailment. Unicode names the character TRUE (U+22A8), with the glosses “statement is true, valid”, “is a tautology” and “satisfies”, and has a separate character, ⊧ MODELS (U+22A7), with shorter bars.
Also searched as: ⊨, ⊭, entails symbol, models symbol, satisfies symbol, semantic consequence.
⊨ and ⊢
⊨ is about truth, the single ⊢ about proof. forall x: Calgary puts it as valuations against proofs, and stresses that ⊨ is not a symbol of the logic itself but of the metalanguage, the language used to talk about it. The arrow → is part of the logic, and the textbook compares the two in a section of its own, “⊨ versus →”.
Read next
- Next in Logic symbolsTop and bottom (⊤, ⊥)⊤ (top) means “true” and ⊥ (bottom) means “false” in logic: ⊤ is a statement that is always true, ⊥ one that is always false, a contradiction.
- The same shape in African flagsFlag of ZimbabweZimbabwe’s flag has seven stripes, green, gold, red and black at the centre, and a white triangle at the pole with a red star and the gold Zimbabwe Bird. The government says green is vegetation and agriculture, gold the mineral wealth, red the blood shed in the liberation war, black the black majority, the white triangle peace and the red star the nation’s aspirations; the bird is the national emblem.
More logic symbols
The same shape elsewhere
Browse symbols drawn with lines.
Sources
- forall x: Calgary. An Introduction to Formal Logic, P. D. Magnus, Tim Button, Robert Trueman and Richard Zach, forall x: Calgary (Fall 2025 edition), Open Logic Project, University of Calgary, Sec. 12.5 The double turnstile
- Classical Logic, Stewart Shapiro and Teresa Kouri Kissel, Stanford Encyclopedia of Philosophy, Metaphysics Research Lab, Stanford University, Sec. 4 Semantics
- Earliest Uses of Symbols of Set Theory and Logic, Jeff Miller, Earliest Uses of Various Mathematical Symbols, MacTutor History of Mathematics, University of St Andrews, Double turnstile
- Mathematical Operators: Range 2200–22FF (code chart), The Unicode Standard, Version 18.0, Unicode Consortium, Relations, 22A7–22A8
Researched, edited and approved by Rory Hansen before publishing. Last reviewed October 9, 2026. How we research
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