What is relevance logic?

We suggest two precise abstract definitions of the notion of ‘relevance logic’ which are both independent of any proof system or semantics. We show that according to the simpler one, R→¬ (the intensional fragment of R) is the minimal relevance logic, but R itself is not. In contrast, R and many other logics are relevance logics according to the second (more complicated) definition, while all fragments of linear logic are not.