# Definition:Formal System

## Definition

A **formal system** is a formal language $\LL$ together with a deductive apparatus for $\LL$.

Let $\FF$ be a **formal system** consisting of a formal language with deductive apparatus $\DD$.

By applying the formal grammar of $\LL$, one constructs well-formed formulae in $\LL$.

Of such a well-formed formula, one can then use the deductive apparatus $\DD$ to determine whether or not it is a theorem in $\FF$.

## Also known as

A **formal system** is also known as:

particularly in sources where the main application of **formal systems** lies in symbolic logic.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, these terms are discouraged because they provoke false conclusions about the scope of the term **formal system**.

Some sources use the term **axiomatic system**, particularly when applying this technique to specific fields of mathematics.

Some sources use the term **logistic system**, but this strictly speaking is a **formal system** which contains only axioms of logic.

## Also see

- Definition:Symbolic Logic, which is an important field of application for
**formal systems**.

- Results about
**formal systems**can be found**here**.

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**formal system (formal theory)** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**formal system (formal theory)** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**axiomatic system**