Suszko's problem is the problem of finding the minimal
number of truth values needed to semantically characterize a syntactic
consequence relation. Suszko proved that every Tarskian consequence
relation can be characterized using only two truth values. Malinowski
showed that this number can equal three if some of Tarski's structural
constraints are relaxed. By so doing, Malinowski introduced a case of
so-called mixed consequence (following Cobreros et al. 2012's
terminology), allowing the notion of a designated value to vary
between the premises and the conclusions of an argument. In this paper
we give a more systematic perspective on Suszko's problem and on the
characterization of mixed consequence relations more generally. First,
we prove general representation theorems relating structural
properties of a consequence relation to their semantic counterparts.
Based on those we derive and strengthen maximum-rank results proved
recently by French and Ripley (2017), and by Blasio, Wansing and
Marcos (2017) in a different setting for logics with various
structural properties (reflexivity, transitivity, none, or both). We
use those results to discuss the foundational problem of what to admit
as a bone fide consequence relation in logic.