# Definition:Symbol

## Definition

In its broadest possible sense:

In a narrower and more "mathematical" sense, a **symbol** is a sign of a particular shape to which is assigned a meaning, and is used to represent a concept or identify a particular object.

It is generally much more convenient to use a symbol than the plain speaking that it replaces, because it is invariably more compact. One character can replace a large number of words. As definitions become more complex, the symbols tend to convey more information -- but by the same coin, understanding exactly what a symbol means becomes more difficult.

Symbols may mean different things in different contexts. A symbol that means something in one context may mean something completely different in another. This is because the number of different concepts is greater than human ingenuity can create symbols for, and some of them naturally have to be used more than once.

This does not matter as long as, before we use any symbol, we define exactly what we mean by it. Some symbols are standard and rarely need defining, but in some contexts there are subtle differences to the *exact* meaning of a "standard" symbol. Therefore all fields of mathematics generally introduce themselves with a rash of definitions, many of which are symbols.

## Also known as

A **symbol** is sometimes known under its older term **ideogram**.

The term is in apposition to the term **phonogram**, which is a sign standing directly for a sequence of sounds that are then interpreted according to the natural language in which the **phonogram** is written.

Both an **ideogram** and a **phonogram** can be considered as **symbols**.

In certain contexts, the word **label** can be used.

## Also see

In formal systems, it is usual to name certain collections of **symbols** after the objects they represent.

For example, the language of predicate logic has predicate symbols and function symbols to represent predicates and functions, respectively.

## Historical Note

The use of **symbols** was one of the characteristics that initially distinguished symbolic logic from classical logic.

## Sources

- 1910: Alfred North Whitehead and Bertrand Russell:
*Principia Mathematica: Volume $\text { 1 }$*... (previous) ... (next): Chapter $\text{I}$: Preliminary Explanations of Ideas and Notations - 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous) ... (next): Chapter $\text I$ Introductory: $1$. Symbolic Logic and Classical Logic - 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $1$ Introduction: Logic and Language: $1.4$: Symbolic Logic - 1973: R.L. Wilder:
*Evolution of Mathematical Concepts*(Paperback ed.): Preface to Paperback Edition - 1979: John E. Hopcroft and Jeffrey D. Ullman:
*Introduction to Automata Theory, Languages, and Computation*... (next): Chapter $1$: Preliminaries: $1.1$ Strings, Alphabets and Languages - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{I}: 1$: The Logic of Statements $(1)$ - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.1$: Declarative sentences