At least three distinct definitions of logical consequence have been proposed in many-valued logic, which coincide in the two-valued case, but come apart as soon as three truth values come into play. Those definitions include so-called *pure consequence* (the preservation of a common set of values from premisses to conclusion) as most familiar from the work of Lukasiewicz and Tarski (viz. Lukasiewicz 1920, Tarski 1930, Hajek 1998); so-called *mixed consequence* (the inclusion of the premises in some designated set implies the inclusion of the conclusion in some possibly distinct set, as considered in Malinowski 1990, Zardini 2008, Cobreros et al. 2012, Ripley 2013 among others); and *order-theoretic consequence* (the value of the conclusion should not be less than the infimum of the values of the premisses, as in Machina 1976).

While obvious conceptual and logical links exist between those definitions, the question of whether those schemes can be subsumed under a common and more abstract template is not as obvious as it might seem. In this paper, we examine whether those definitions together carve out a natural class of consequence relations. We respond positively by identifying three minimal constraints all such schemes satisfy, namely: truth-functionality, bivalence-compliance, and a constraint of value-monotonicity. We call *monotone truth-functional* the class of consequence relations satisfying those properties. Our main result is that the class of monotone truth-functional consequence relations for a many-valued logic coincides exactly with the class of mixed consequence relations and their conjunction -- therefore including pure consequence relations and the order-theoretic consequence. We also provide an enumeration of the set of monotone-truth-functional relations in the case of finite many-valued logics based on well-ordered truth values as well as on a particular class of partially ordered truth values. 

Finally, we discuss the significance of our result for the problem of characterizing what counts as a *good* consequence relation. On the one hand, our result may be seen as insufficiently permissive (because although nonreflexive and nontransitive relations are admitted, nonmonotone relations are not), and on the other as insufficiently restrictive (because a plethora of consequence relations remain when values are partially ordered). We put both problems in perspective and discuss the possibility of including further independent criteria.