The framework of strict-tolerant consequence championed by Cobreros, Egré, Ripley and van Rooij provides a novel setting that permits one to have a transparent truth predicate without abandoning classical logic. The semantics for this notion of consequence, due to van Rooij, employs the three-valued strong Kleene matrices. A second framework that has received renewed attention is the collection of weak Kleene matrices, which have frequently appeared in the context of so-called logics of nonsense, making the pairing of these topics a natural avenue for investigation. In this talk, I’ll discuss strict-tolerant consequence on the weak Kleene matrices, its corresponding proof theory, and its interpretation. I’ll also discuss how the resulting notion of consequence bears on several matters in philosophical logic, including the content-theoretic interpretation of bounds consequence, the semantic properties of paradoxical sentences in the Principia Mathematica, and debates concerning the logical analysis of category mistakes.
When speakers utter conflicting moral sentences (“X is wrong”/“X is not wrong”), it seems clear that they disagree. It has often been suggested that the fact that the speakers disagree gives us evidence for a claim about the semantics of the sentences they are uttering. Specifically, it has been suggested that the existence of the disagreement gives us reason to infer that there must be an incompatibility between the contents of these sentences (i.e., that it has to be the case that at least one of them is incorrect). This inference then plays a key role in a now-standard argument against certain theories in moral semantics. In this paper, we introduce new evidence that bears on this debate. We show that there are moral conflict cases in which people are inclined to say both (a) that the two speakers disagree and (b) that it is not the case at least one of them must be saying something incorrect. We then explore how we might understand such disagreements. As a proof of concept, we sketch an account of the concept of disagreement and an independently motivated theory of moral semantics which, together, explain the possibility of such cases.