  We generalize the notion of consequence relation standard in abstract
treatments of logic to accommodate intuitions of relevance. The guiding idea
follows the \emph{use criterion}, according to which in order for some premises
to have some conclusion(s) as consequence(s), the premises must each be
\emph{used} in some way to obtain the conclusion(s). This relevance intuition
turns out to require not just a failure of monotonicity, but also a move to
considering consequence relations as obtaining between \emph{multisets}. We
motivate and state basic definitions of relevant consequence relations, both in
single conclusion (asymmetric) and multiple conclusion (symmetric) settings, as
well as derivations and theories, guided by the use intuitions, and prove a
number of results indicating that the definitions capture the desired results
(at least in many cases).
