# Forces

## The Mathematical Symbol "Forces (⊩)"

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The ⊩ Symbol in Mathematics: Model Theoretic Entailment

In the diverse world of mathematical symbols, ⊩ stands out as a notation denoting "Forces" or "Entails" predominantly in the realms of model theory, set theory, and formal logic. It describes a particular relationship between structures and formulas or propositions. This lesson covers the primary usages of ⊩, complemented by two examples for each context.

## 1. Model Theory

In model theory, the ⊩ symbol is employed to express that a particular structure is a model of a given formula, or more broadly, that the formula is true in that structure.

• Example 1: If $$\mathcal{M}$$ is a structure and $$\phi$$ is a formula, the notation $$\mathcal{M} ⊩ \phi$$ conveys that $$\phi$$ is true in $$\mathcal{M}$$.
• Example 2: Given a structure $$\mathcal{N}$$ and a set of formulas $$\Sigma$$, if every formula in $$\Sigma$$ holds true in $$\mathcal{N}$$, one might express this as: $$\mathcal{N} ⊩ \Sigma$$.

## 2. Set Theory and Formal Logic

Within set theory, especially in the context of forcing in the study of models of set theory, and in formal logic, the ⊩ symbol depicts a relationship between a particular condition and a proposition.

• Example 1: In forcing, if $$p$$ is a condition and $$\Phi$$ is a proposition, the notation $$p ⊩ \Phi$$ signifies that the condition $$p$$ forces the proposition $$\Phi$$.
• Example 2: In a system of formal logic, if we have a set of axioms $$A$$ and a proposition $$\psi$$, and $$\psi$$ can be derived from $$A$$, this relationship can be denoted as: $$A ⊩ \psi$$.

In summary, the ⊩ symbol carves a unique niche in expressing model-theoretic entailment and the concept of forcing in set theory. Its presence in formal expressions brings clarity and specificity, underscoring the deep relationships between mathematical structures and propositions.

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## Codes for the ⊩ Symbol

 The Symbol ⊩ Alt Code Alt 8873 HTML Code ⊩ HTML Entity ⊩ CSS Code \22A9 Hex Code ⊩ Unicode U+22A9

## 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 8873. 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: &#8873;</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: "\22A9";}
</style>
<span>My symbol:</span>
My symbol: ⊩

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

HTML TextOutput
<b>My symbol: &#x22A9;</b>My symbol: ⊩
On the assumption that you already have your canvas and the context set up, use the Hex code in the format 0x22A9 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+22A9. 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
22A9
[Hold down Alt]
[Press x]

(The 22A9 turns into ⊩. Note that you can omit any leading zeros.)
In JavaScript, the syntax is \uXXXX. So, our example would be \u22A9. (Note that the format is 4 hexadecimal characters.)
JavaScript TextOutput
let str = "\u22A9"
document.write("My symbol: " + str)
My symbol: ⊩

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