We may wonder about the status of logical accounts of the meaning of language. When does a particular proposal count as a theory? How do we judge a theory to be correct? What criteria can we use to decide whether one theory is âbetterâ than another? Implicitly, many accounts attribute a foundational status to set theory, and set-theoretic characterisations of possible worlds in particular. The goal of a semantic theory is then to find a translation of the phenomena of interest into a set-theoretic model. Such theories may be deemed to have âexplanatoryâ or âpredictiveâ power if a mapping can found into expressions of set-theory that have the appropriate behaviour by virtue of the rules of set-theory (for example Montague 1973; Montague1974). This can be contrasted with an approach in which we can help ourselves to ânewâ primitives and ontological categories, and devise logical rules and axioms that capture the appropriate inferential behaviour (as in Turner 1992). In general, this alternative approach can be criticised as being mere âdescriptivismâ, lacking predictive or explanatory power. Here we will seek to defend the axiomatic approach. Any formal account must assume some normative interpretation, but there is a sense in which such theories can provide a more honest characterisation (cf. Dummett 199). In contrast, the set-theoretic approach tends to conflate distinct ontological notions. Mapping a pattern of semantic behaviour into some pre-existing set-theoretic behaviour may lead to certain aspects of that behaviour being overlooked, or ignored (Chierchia & Turner 1988; Bealer 1982). Arguments about the explanatory and predictive power of set-theoretic interpretations can also be questioned (see Benacerraf 1965, for example). We aim to provide alternative notions for evaluating the quality of a formalisation, and the role of formal theory. Ultimately, claims about the methodological and conceptual inadequacies of axiomatic accounts compared to set-theoretic reductions must rely on criteria and assumptions that lie outside the domain of formal semantics as such.