Verb

First, we'll need an extension for John and Mary. For the time being, let's take the following:

$\llbracket$ $N''_1$ and $N''_2$ $\rrbracket$ = $\{N''_1,N''_2\}$

Evaluation will then be something like:

$\llbracket$ John and Mary ate $\rrbracket$ = $ate'(\{john', mary'\})$

Under this approach any verb that can be used with an and-connected argument needs to know how to parse the set appropriately. We also are restricted in that we can't have a verb like ate$'$ sometimes be given entities, as in John ate, and sometimes sets. This requires us to consider all noun phrases as sets of entities.

Note that as John and Mary are noun phrases, the extension of John and Mary is not a set of entities but a set of sets of entities. That is,

$\llbracket John \: and \: Mary\rrbracket = \{\llbracket John\rrbracket, \llbracket Mary\rrbracket\} = \{\{john'\},\{mary'\}\}$

We would then evaluate a sentence of this form as:

$$\llbracket John \: and \: Mary \: ate \: pizza \: and \: cabbages\rrbracket = (\llbracket ate \: pizza \: and \: cabbages\rrbracket) (\llbracket John \: and \: Mary\rrbracket)$$

$$= ((\lambda x \lambda y . \: ate'(y,x)) (\{\llbracket Pizza\rrbrack... ...bbages\rrbracket\})) (\{\llbracket John\rrbracket, \llbracket Mary\rrbracket\})$$

$$= ((\lambda x \lambda y . \: ate'(y,x)) (\{\{pizza'\},\{cabbages'\}\})) (\{\{john'\},\{mary'\}\})$$

$$= (\lambda y . \: ate'(y,\{\{pizza'\},\{cabbages'\}\})) (\{\{john'\},\{mary'\}\})$$

$$= ate'(\{\{john'\},\{mary'\}\},\{\{pizza'\},\{cabbages'\}\})$$

This is all well and good unless we want the other interpretation of this sentence, that John ate pizza $\wedge$ Mary ate cabbages. There's no way for ate$'$ to put things together in the right way, as it is only being given sets and sets are by their nature unordered. We need something where ate$'$ can match up things properly, something like a list. So lets redefine the extension of and:

$\llbracket$ $N''_1$ and $N''_2$ $\rrbracket$ = $\langle N''_1,N''_2\rangle$

Now we can have each verb decide on its own how to parse its arguments. Some verbs will now have ugly extensions, but that's the domain of lexical semantics.