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.