 This in itself says nothing about the nature of such properties. As we saw in  §3.1,  Montague took them to be intensions as set-theoretically characterized in terms of possible worlds. Moreover, he took them to be typed, since, to avoid logical paradoxes, he relied on type theory. After Montague, these two assumptions have been typically taken for granted in natural language semantics, though with attempts to recover hyperintensionality somehow (Cresswell 1985) in order to capture the semantics of propositional attitudes verbs such as “believe”, affected by the phenomena regarding mental content hinted at in  §3.1.  However, the development of type-free property theories has suggested the radically different road of relying on them to provide logical forms for natural language semantics (Chierchia 1985; Chierchia &amp; Turner 1988; Orilia 2000b; Orilia &amp; Landini 2019). This allows one to capture straightforwardly natural language inferences that appear to presuppose type-freedom, as they feature quantifiers that bind simultaneously subject and predicate positions (recall the example of  §1.2).  Moreover, by endowing the selected type-free property theory with fine-grained identity conditions, one also accounts for propositional attitude verbs (Bealer 1989). Thus, we may say that this line makes a case for properties understood as untyped and highly fine-grained.