  Modal quantum theory (MQT) is a "toy model" of quantum theory in which
amplitudes are elements of a general field. The theory predicts, not the
probabilities of a measurement result, but only whether or not a result is
possible. In this paper we review MQT and extend it to include mixed states,
generalized measurements and open system dynamics. Even though MQT does not
have density operators, superoperators or any concept of "positivity", we can
nevertheless establish a precise analogue to the usual representation theorem
for CP maps. We also embed MQT in a larger class of modal theories. We show
that the possibility assignments for separate measurements on a bipartite
system in MQT are always weakly consistent with some probability assignment
that respects the no-signalling principle.
