# Definition:Formal System

## Definition

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

Let $\mathcal F$ be a formal system consisting of a formal language with deductive apparatus $\mathcal D$.

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

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

## Also known as

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

- A
**logical system** - A
**logistic system** - A
**logical calculus** - A
**logic**

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.

## Also see

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

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

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**axiomatic system**