Minimal logic

Minimal logic, or minimal calculus, is a symbolic logic system originally developed by Ingebrigt Johansson.^[1] It is an intuitionistic and paraconsistent logic, that rejects both the law of the excluded middle as well as the principle of explosion (ex falso quodlibet), and therefore holding neither of the following two derivations as valid:

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

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

where ${\displaystyle A}$ and ${\displaystyle B}$ are any propositions. Most constructive logics only reject the former, the law of excluded middle. In classical logic, the ex falso laws

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

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

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

as well as their variants with ${\displaystyle A}$ and ${\displaystyle \neg A}$ switched, are equivalent to each other and valid. Minimal logic also rejects those principles.

Note that the name has sometimes also been used to denote logic systems with a restricted number of connectives.