 What enables the derivation of contraction is the fact that our subscripts are sets. We do not keep track of how many times (more than once) that a hypothesis is used in its derivation. In order to reject contraction, we need a way of counting the number of uses of hypotheses. Thus natural deduction systems for contraction-free systems use “multisets” of relevance numerals instead of sets — these are structures in which the number of occurrences of a particular numeral counts, but the order in which they occurs does not. Even weaker systems can be constructed, which keep track also of the order in which hypotheses are used (see Read 1986 and Restall 2000).