Let \(A=(X,O)\) be a hypergraph, where \(X\) is a non-empty set and \(O\) is a covering of \(X\) by nonempty subsets of \(X\). By a result of T. A. Cook, the space \(J(A)\) of Jordan weights on \(A\) is an ordered Banach space under the base-norm induced by the base consisting of the probability weights on \(A\). The authors show that the cone of positive normal functionals on \(J(A)^*\) may be substantially larger than the cone of positive weights. For a semiclassical hypergraph (in the sense that \(O\) is a partition of \(X\)), the two cones coincide if and only if the number of elements of \(O\) of cardinality \(\geq 2\) is finite.