all involve intensionality. The meaning (and thereby the truth value) of the attitudinal sentence (3.14) depends on the meaning (intension) of the subordinate clause, not just its truth value (extension). The meaning of (3.15) depends on the meaning of happy, but does not require happy to be a property of John or anything else. The meaning of (3.16) does not depend on the actual existence of a starship, but does depend on the meaning of that phrase. And fake beard in (3.17) refers to something other than an actual beard, though its meaning naturally depends on the meaning of beard.  A Montagovian analysis certainly would deal handily with such sentences. But again, we may ask how much of the expressive richness of Montague's type theory is really essential for computational linguistics. To begin with, sentences such as (3.14) are expressible in classical modal logics, without committing to higher types. On the other hand (3.16) resists a classical modal analysis, even more firmly than Montague's “John seeks a unicorn,” for which an approximate classical paraphrase is possible: “John tries (for him) to find a unicorn”. A modest concession to Montague, sufficient to handle (3.15)–(3.17), is to admit intensional predicate modifiers into our representational vocabulary. We can then treat look as a predicate modifier, so that look(happy) is a new predicate derived from the meaning of happy. Similarly we can treat design as a predicate modifier, if we are willing to treat a starship as a predicative phrase, as we would in “The Enterprise is a starship”. And finally, fake is quite naturally viewed as a predicate modifier, though unlike most nominal modifiers, it is not intersective (#John wore something that was a beard and was fake) or even subsective (#John wore a particular kind of beard). Note that this form of intensionality does not commit us to a higher-order logic—we are not quantifying over predicate extensions or intensions so far, only over individuals (aside from the need to allow for plural entities, as noted). The rather compelling case for intensional predicate modifiers in our semantic vocabulary reinforces the case made above (on the basis on extensional examples) for allowing predicate modification.