And an example derivation
Here is an example derivation for the sentence “Many monkeys mutter madly.”
I begin by presenting the lexical items, their semantic types, the category-type correspondance, and their lexical meanings.
Figure 1: Lexical items
Figure 2: Category-Type Correspondance
Figure 3: Meanings of lexical items
I now present the unique derivation of the desired sentence.
Figure 4: The derivation
Figure 5: The semantic interpretation
The resulting formula should be understood as ascribing the property of at the present time muttering and being mad (in the crazy sense) to many monkeys. This isn’t quite right, however: you can mutter crazily without being crazy. The problem of course is that we identified the meaning of the manner adjunct with simple predicate conjunction! More sophisticated treatments (see e.g. Thomason and Stalnaker) treat the adjunct as a predicate modifier, and thus don’t perform the reduction step in the figure. There are a number of ways to proceed. However, different adverbs have a different semantic behaviour, and it is unnatural to treat the head AdjV as the locus of the meaning. Instead, we might treat AdjV as semantically vacuous (not the identity function, but rather the backwards application function \(\lambda x,f.f\ x\)), and localize the meaning of the adjunct in the adjective madly itself, treating it as a function of type (et)et. But this is, while of fundamental semantic importance, orthogonal to our interface considerations here.