In recent work, Fox (2016) has argued, on the basis of both empirical and conceptual considerations, that relevance (the set of propositions relevant in an utterance context) is closed under speaker belief: if $\phi $ is relevant, then it’s also relevant whether the speaker believes $\phi $. We provide a formally explicit implementation of this idea and explore its theoretical consequences and empirical predictions. As Fox (2016) already observes, one consequence is that ignorance inferences (and scalar implicatures) can only be derived in grammar, via a covert belief operator of the sort proposed by Meyer (2013). We show, further, that the maxim of quantity no longer enriches the meaning of an utterance, per se, but rather acts as a filter on what can be relevant in an utterance context. In particular, certain alternatives (of certain utterances) are shown to be incapable of being relevant in any context where the maxim of quantity is active — a property we dub obligatory irrelevance. We show that the resulting system predicts a quite restricted range of interpretations for sentences with the scalar item some, as compared to both neo-Gricean (Geurts, 2010; Horn, 1972; Sauerland, 2004) and grammatical (Chierchia et al., 2012; Fox, 2007; Meyer, 2013) theories of scalar implicature, and we argue that these predictions seem largely on the right track.

Content words (e.g. nouns and adjectives) are generally connected: there are no gaps in their denotations; no noun means ‘table or shoe’ or ‘animal or house’. We explore a formulation of connectedness which is applicable to content and logical words alike, and which compares well with the classic notion of monotonicity for quantifiers. On a first inspection, logical words satisfy this generalized version of the connectedness property at least as well as content words do — that is, both in terms of what may be observed in the lexicons of natural languages (although our investigations remain modest in that respect) and in terms of acquisition biases (with an artificial rule learning experiment). This reduces the putative differences between content and logical words, as well as the associated challenges that these differences would pose, e.g., for learners.