OUP accepted manuscript

Abstract

Several 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, 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, valuemonotonicity, 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.⇤ We are most grateful to two anonymous reviewers for very detailed comments that significantly improved our paper, and to Heinrich Wansing, our editor, for his editorial assistance and feedback. We are indebted to Denis Bonnay, Pablo Cobreros, Valentin Goranko, Heinrich Wansing for detailed comments and valuable suggestions, as well as to

Disjoint Logics

“…In [6] Chemlá et al present the notion of mixed consequence relation. Below, a standard is a set of truth values.…”

“…TS is the 3-valued q-matrix logics associated to the 3-element Kleene algebra. TS is discussed by, e.g., Cobreros et al in [7], and Chemlá et al in [6] in the context of the more general discussion of what represents a "proper" consequence relation between formulae. Moreover, it was also discussed by Malinowski in [25] as a tool to model empirical inference with the aid of the 3-valued Kleene algebra, and more recently was stressed by French in [14], in connection with the paradoxes of self-reference.…”

“…As Chemlá et al [6] define the notion, a logic can be permeable just because it is right to left permeable, though not left-to-right permeable  and neither universal nor trivial. sn is a logic like this.…”

“…Here is a quick answer to such a question. 6 It is common  and reasonable  to think that, in a valid inference  or at least in sound ones  premises provide reasons to accept the conclusions. Or, to put it more clearly, that to accept the premises provide reasons for accepting the conclusions.…”

“…Finally, in Section 7, we present some concluding remarks. 6 The details of how disjoint logics can be applied to this task will be defer to Section 4, after we have have presented the logics we will be exploring. 7 Though not all of these constraints are apt to be recovered by disjoint logics.…”