Definition:Formal System

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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.