Minimal logic

Minimal logic, or minimal calculus, is a symbolic logic system originally developed by Ingebrigt Johansson under the name "Minimalkalkül". It is a paraconsistent logic weaker than intuitionistic logic that rejects the principle of explosion (ex falso quodlibet), as well as the law of the excluded middle. Intuitionistic logic, like most constructive logics, only rejects the law of excluded middle.

As such, neither of the following two derivations are valid for all propositions ${\displaystyle A}$ and ${\displaystyle B}$ in minimal logic:

${\displaystyle \vdash (B\lor \neg B)}$,

${\displaystyle (A\land \neg A)\vdash B}$.

In classical logic, also the ex falso law

${\displaystyle (A\land \neg A)\to B}$,

or equivalently ${\displaystyle \neg A\to (A\to B)}$, is valid. These do not automatically hold in minimal logic.

The name minimal logic has sometimes also been used to denote logic systems with a restricted number of connectives.