Minimal logic

Minimal logic, or minimal calculus, is the symbolic logic system originally developed by Ingebrigt Johansson.

If we interpret Classical logic as a system of Natural deduction removing the rule of Double negative elimination results in Minimal logic. Then if we interpret Minimal logic as an Axiomatic system we can conservatively extend it with different Axioms resulting in a variety of Intermediate logics lying between Minimal logic and Classical logic. For example, adding the Ex falso quodlibet (EFQ, from a contradiction anything follows) results in the well-known Intuitionistic logic. These Intermediate logics all fall on a lattice with Classical logic at the "join" and Minimal logic at the "meet".