# Definition:Formal Language/Alphabet

## Definition

Let $\mathcal L$ be a formal language.

The **alphabet** $\mathcal A$ of $\mathcal L$ is a set of symbols from which collations in $\mathcal L$ may be constructed.

An **alphabet** consists of the following parts:

Depending on the specific nature of any particular formal language, these too may be subcategorized.

### Letter

A **letter** of a formal language is a more or less arbitrary symbol whose interpretation depends on the specific context.

In building a formal language, **letters** are considered to be the undefined terms of said language.

An important part of assigning semantics to a formal language is to provide an interpretation for its **letters**.

### Sign

A **sign** of a formal language $\mathcal L$ is a symbol whose primary purpose is to structure the language.

In building a formal language, **signs** form the hooks allowing the formal grammar to define the well-formed formulae of the formal language.

Common examples of **signs** are parentheses, "(" and ")", and the comma, ",".

The logical connectives are also **signs**.

**Signs** form part of the alphabet of a formal language.

Unlike the letters, they must be the same for each signature for the language.

### Primitive Symbol

Let $\mathcal A$ be the alphabet of a formal language $\mathcal L$.

The symbols which comprise $\mathcal A$ are called the **primitive symbols** of $\mathcal A$.

It is usual, during the development of a formal system, to introduce further symbols in order to abbreviate what would otherwise be unwieldy constructions.

Hence the distinction between these newly-introduced symbols and the **primitive symbols**.

## Also denoted as

Some sources use $\Sigma$ to denote an arbitrary alphabet.

## Sources

- 1988: Dominic Welsh:
*Codes and Cryptography*... (previous) ... (next): Notation: Alphabets and strings - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**alphabet** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**alphabet**