 Second, in the S′-rule, the subject is assumed to be a second-order predicate that is applied to the denotation of the VP (a monadic predicate) to yield a sentence intension, whereas we would ordinarily think of the subject-predicate semantics as being the other way around, with the VP-denotation being applied to the subject. But Montague's contention was that his treatment was the proper one, because it allows all types of subjects—pronouns, names, and quantified NPs—to be handled uniformly. In other words, an NP always denotes a second-order property, or (roughly speaking) a set of first-order properties (see also Lewis 1970). So for example, Thetis denotes the set of all properties that Thetis (a certain contextually determined individual with that name) has; (more exactly, in the present formulation Thetis denotes a function from properties to sentence intensions, where the intension obtained for a particular property yields truth in worlds where the entity referred to has that property); some woman denotes the union of all properties possessed by at least one woman; and every woman denotes the set of properties shared by all women.  Accordingly, the S′-rule yields a sentence intension that is true at a given world just in case the second-order property denoted by the subject maps the property denoted by the VP to such a truth-yielding intension.