The Mathematical Symbol "Right Tack (⊢)"

The "Right Tack" Symbol (⊢)

The ⊢ symbol, commonly referred to as the "Right Tack" or "Turnstile", is a symbol of considerable significance in logic, mathematics, and computer science. It is traditionally used to represent entailment or provability.

Usage of ⊢

The Right Tack symbol is often employed in various contexts, especially in the fields of formal logic and mathematical proof theory.

Example 1: Logical Entailment

In propositional logic, if \( A \) and \( B \) are statements, the expression \( A ⊢ B \) means that \( B \) can be logically inferred from \( A \). It signifies that if \( A \) is true, then \( B \) must also be true.

Example 2: In Proof Theory

In proof theory, the turnstile symbol denotes provability. The expression \( \Gamma ⊢ \phi \) means that the formula \( \phi \) can be derived or proven from the set of formulas \( \Gamma \).

Significance in Mathematics and Logic

The ⊢ symbol has foundational importance in several areas:

  • Formal Logic: It represents the relationship between hypotheses and conclusions in logical deductions.
  • Proof Theory: The symbol is a staple in the notation of sequent calculus, a formal system for reasoning about proofs.
  • Model Theory: It differentiates between syntactic provability and semantic entailment, often using a dual symbol (⊧).
  • Computer Science: In formal methods, particularly in type theory and formal semantics, the turnstile is frequently used to specify the type or behavior of expressions or programs.

In summary, the ⊢ symbol is central to discussions and formalisms in logic and formal systems. It succinctly captures the essence of provability and entailment, concepts fundamental to rigorous mathematical and logical reasoning.

Mathematical symbol 'Right Tack'

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Codes for the ⊢ Symbol

The Symbol⊢
Alt CodeAlt 8866
HTML Code⊢
HTML Entity⊢
CSS Code\22A2
Hex Code⊢
UnicodeU+22A2

How To Insert the ⊢ Symbol

(Method 1) Copy and paste the symbol.

The easiest way to get the ⊢ symbol is to copy and paste it into your document.

Bear in mind that this is a UTF-8 encoded character. It must be encoded as UTF-8 at all stages (copying, replacing, editing, pasting), otherwise it will render as random characters or the dreaded �.

(Method 2) Use the "Alt Code."

If you have a keyboard with a numeric pad, you can use this method. Simply hold down the Alt key and type 8866. When you lift the Alt key, the symbol appears. ("Num Lock" must be on.)

(Method 3) Use the HTML Decimal Code (for webpages).

HTML TextOutput
<b>My symbol: &#8866;</b>My symbol: ⊢

(Method 4) Use the HTML Entity Code (for webpages).

HTML TextOutput
<b>My symbol: &vdash;</b>My symbol: ⊢

(Method 5) Use the CSS Code (for webpages).

CSS and HTML TextOutput
<style>
span:after {
content: "\22A2";}
</style>
<span>My symbol:</span> 		My symbol: ⊢

(Method 6) Use the HTML Hex Code (for webpages and HTML canvas).

HTML TextOutput
<b>My symbol: &#x22A2;</b>My symbol: ⊢
On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x22A2 to place the ⊢ symbol on your canvas. For example:
JavaScript Text
const x = "0x"+"E9"
ctx.fillText(String.fromCodePoint(x), 5, 5);
Output

⊢

(Method 7) Use the Unicode (for various, e.g. Microsoft Office, JavaScript, Perl).

The Unicode for ⊢ is U+22A2. The important part is the hexadecimal number after the U+, which is used in various formats. For example, in Microsoft Office applications (e.g. Word, PowerPoint), do the following:
TypeOutput
22A2
[Hold down Alt]
[Press x]		⊢
(The 22A2 turns into ⊢. Note that you can omit any leading zeros.)
In JavaScript, the syntax is \uXXXX. So, our example would be \u22A2. (Note that the format is 4 hexadecimal characters.)
JavaScript TextOutput
let str = "\u22A2"
document.write("My symbol: " + str)		My symbol: ⊢