Relevant logic Glossary
Definition
Relevant logic is an alternative to the classical logical paradigm, based on the idea that the logical rules for introducing and eliminating connectives and quantifiers are all fine, but the underlying structure of what counts as a proof should be amended to avoid claiming that conclusions "follow from" premises which were not used in the derivations.
The most obvious feature of the resulting logic is that paradoxes of implication are no longer provable. Relevant logic is also paraconsistent, and has the feature that if A entails B in the propositional logic, then A and B share at least one atom.
Comments
See the first link below for an introduction.