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Two logic signs differing only in whether the horizontal stroke is single or doubled, and separating proof from truth.
A single turnstile is a claim about derivation: the formula on the right can be *proved* from the assumptions on the left, inside a stated formal system. A double turnstile is a claim about meaning: every interpretation that makes the assumptions true also makes the conclusion true. One is syntax and the other is semantics, and whether they coincide is the question soundness and completeness theorems exist to answer — so treating them as the same sign discards the entire distinction those theorems are about.
One means derivable, the other means true in every model.
| Turnstile | Double turnstile | |
|---|---|---|
| Horizontal stroke | Single. | Doubled. |
| What it claims | Derivability within a stated proof system. | Truth in every model satisfying the assumptions. |
| Depends on | Which rules of inference the system permits. | How the symbols are interpreted, not how they are manipulated. |
| Relationship (from Symbolwise's records) | Commonly confused with Double turnstile — One means derivable, the other means true in every model. | Commonly confused with Turnstile — One means true in every model, the other means derivable. |
Wikipedia · retrieved 2026-07-31
Tertiary reference. Used for what the notation means in ordinary mathematical practice and for attribution of its introduction, not for a claim that any body defines it.
Unicode Code Chart: Mathematical Operators (U+2200–U+22FF)
The Unicode Consortium · retrieved 2026-07-30
Authoritative for the character's code point, name and block only. Unicode assigns identity; it does not define what a symbol means in use.
Glossary of mathematical symbols
Wikipedia · retrieved 2026-07-31
Tertiary reference. Used for what the notation means in ordinary mathematical practice and for attribution of its introduction, not for a claim that any body defines it.
Alpha is a Greek letter and stands for a quantity — an angle, a coefficient, a significance level. The proportionality sign is an operator and stands between two quantities to say they keep a constant ratio. The drawings differ in that the proportionality sign's right-hand stroke runs out in an open curve rather than closing into the letter's tail.
Look at the left-hand edge. A straight flat back is an AND gate: the output is true only when every input is. A curved concave back is an OR gate: true when any input is. Everything else about the shapes is similar, which is why the back edge is worth checking deliberately rather than at a glance.
Look at what follows. A prefix never stands alone — it is glued to a unit symbol, as in kW or kg. A constant stands by itself in an equation. Careful physical writing subscripts the constant as k_B for exactly this reason, and the kelvin's capital K is a third claimant to the same letter.
Length is the tell. A crescendo hairpin is drawn long, below a stave, and spans the passage it applies to — its extent *is* the instruction. A less-than sign is one character wide and sits between two values. If it stretches, it is music; if it fits between two numbers, it is mathematics.