 This objection does not appreciate the role played by higher-order abstraction in compositional semantics, which is not to form sentences about higher-order functions. Rather, \(\lambda\)-abstraction is used as a heuristic tool to describe compositional contributions of expressions to larger syntactic environments (cf. Zimmermann 2021, sec. 2.1). Thus, e.g., the extension of a determiner is defined as its contribution to the truth value of a sentence in which it occurs (in subject position), which can be described in terms of the extensions of the nouns and verb phrases it combines with – and these extensions are themselves sets (by a similar reasoning). The abstract higher-order objects are thus merely convenient ways of describing the kinematics of compositionality and do not serve as the objects that the sentences of the language so described are about, or that its terms refer to. As a case in point, it can be shown that even though the (indirect) interpretation of the English fragment of Montague 1973 makes use of \(\lambda\)-abstraction over second-order variables, its expressive power is much weaker than higher-order type logic and does not even have the resources to formulate certain meaning postulates to which its lexical items abide (Zimmermann 1983). In fact, Hintikka’s alternative (game-theoretical) semantics fares no better once it is formulated in a compositional way (see Hodges 1997 or Caicedo et al. 2009).