Implementation

We will need two senses of and to deal with our cases properly:

Dual: $(\lambda x \lambda y \lambda Q . \: Q(x) \wedge Q(y))$

Reflexive: $(\lambda x \lambda y \lambda Q . \: Q(x,y) \wedge Q(y,x)$

We can then evaluate most of our types of sentences:

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

$$= (\lambda Q . \: Q(john') \wedge Q(mary')) (\lambda x . \: ate'(x,pizza'))$$

$$= ate'(john',pizza') \wedge ate'(mary',pizza')$$

$$\llbracket John \: and \: Mary \: kissed\rrbracket = (\llbracket John \: and \: Mary\rrbracket) (\llbracket kissed\rrbracket)$$

$$= (\lambda Q . \: Q(john',mary') \wedge Q(mary',john')) (\lambda x \lambda y . \: kissed'(x,y))$$

$$= kissed'(john',mary') \wedge kissed'(mary',john')$$

$$\llbracket John \: and \: Mary \: collided\rrbracket = (\llbracket John \: and \: Mary\rrbracket) (\llbracket collided\rrbracket)$$

$$= (\lambda Q . \: Q(john',mary') \wedge Q(mary',john')) (\lambda x \lambda y . \: collided'(x,y))$$

$$= collided'(john',mary') \wedge collided'(mary',john')$$

$$= collided'(john',mary')$$

What we can't yet properly evaluate are sentences with multiple ands. Observe what happens if we try:

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

$$= (\lambda Q . \: Q(john') \wedge Q(mary'))$$

$$((\lambda P . \: P(pizza') \wedge P(cabbages')) (\lambda x \lambda y . \: ate'(y,x)))$$

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

Now we have a problem. We can't apply our function because the types don't match properly. We need some way to deal with the boolean and in function application. We could say either of the following:

$(\lambda Q . \: Q(a) \wedge Q(b))(\lambda x .\:C(x) \wedge D(x)) = C(a) \wedge D(a) \wedge C(b) \wedge D(b)$

$(\lambda Q . \: Q(a) \wedge Q(b))(\lambda x .\:C(x) \wedge D(x)) = C(a) \wedge D(b)$

This confusion is actually quite useful, as we noted previously that sentences of this form have two interpretations. The ambiguity we noted earlier can be ascribed to the choice between the two different modes of combination.

As we have ambiguity rising from this choice we might also expect it over the choice of which sense of and to use? First note that because the reflexive and only works in subject position and when there are no objects. Verbs used with this form of and, however, are always transitive. So we have a rule:

Whenever there is a transitive verb with no objects but with an and-connected noun phrase for the subject, the reflexive and is used, while otherwise the dual and is used.