Emmanuel Chemla, Paul Egré, Benjamin Spector (2017). “Characterizing logical consequence in many-valued logic”. Journal of Logic and Computation. in press

Abstract: Several definitions of logical consequence have been proposed in many-valued logic, which co- incide in the two-valued case, but come apart as soon as three truth values come into play. Those definitions include so-called pure consequence, order-theoretic consequence, and mixed consequence. In this paper, we examine whether those definitions together carve out a natural class of consequence relations. We respond positively by identifying a small set of properties that we see instantiated in those various consequence relations, namely truth-relationality, value-monotonicity, validity-coherence, and a constraint of bivalence-compliance, provably replaceable by a structural requisite of non-triviality. Our main result is that the class of consequence relations satisfying those properties coincides exactly with the class of mixed consequence relations and their intersections, including pure consequence relations and order-theoretic consequence. We provide an enumeration of the set of those relations in finite many-valued logics of two extreme kinds: those in which truth values are well-ordered and those in which values between 0 and 1 are incomparable.