The basic idea is to change the semantics of implication in such a way, that premises are always necessary for the concluded statement. In classical logic, as an example, a universally false formula always implies every other formula, and, conversely, everything will imply a tautology. In relevance logic, these statements would no longer be valid.

A calculus for relevance logic can be obtained from a Gentzen-style calculus by removing the weakening rules that allow for the introduction of arbitrary formulae on the right or left side of the sequents.