Constraining Merge with formal semantics

In the Minimalist Program, all syntactic structure is built using the operation Merge, which takes two objects α and β and combines them into a new object K = {γ, {α, β}}, where γ is the ‘label' of the set K; in traditional generative speak, we take γ to be whichever of α and β 'projects'. This Bare Phrase Structure claims to have an advantage over previous structurebuilding rules in avoiding all the stipulative machinery of X-bar Theory, but in this form, Merge is too underspecified to determine what the label γ should be. Rather than the cop-out of just equipping Merge with a list of ordered pairs prescribing what label to give every possible pair of objects, we (as good Minimalists) should want the label to follow in some non-arbitrary way from the properties of the two objects, such that by their nature they couldn't have been Merged differently. I'm going to suggest that this labelling can happen in a natural way by considering semantics à la Montague (1973); if we associate some semantic representation in the λ-calculus with each word in the lexicon, then the label γ will just be the λ-term we produce by functionally applying the λ-terms corresponding to α and β together. This means γ isn't actually either α or β (as Chomsky assumes), but some new object containing information about both α and β. In this view, the syntactic and semantic components of the grammar happen together in lockstep - being really the same thing - and the properties of our semantic representations define the ways they can be Merged together. When the derivation finishes, a semantic representation of the whole utterance can be read directly from the root node of the tree.