1.4.3 Using Lambdas
Let's return to the sentence ``A woman walks''. According to our grammar, a determiner and a common noun can combine to form a noun phrase. Our semantic analysis couldn't be simpler: we will associate the NP node with the functional application that has the determiner representation as functor and the noun representation as argument. Structurally, this is of course the same thing that we did in Section 1.3.4. Only this time the semantic contributions of constituents are generally 
-expressions, and we will simply read the @-symbols as application markers. In fact it will turn out that the combination of functional application and 
-reduction is a method of such generality that we can even completely disregard the phrase-indices (such as 
and 
) that were part of our ``proto''-representations in Section 1.3.4.
Building a structured application...
As you can see from the picture, we use the -expression 
as our representation for the indefinite determiner ``a''. We'll take a closer look at this representation soon, after we've looked at how it does its job in the semantic construction process. But there's one thing that we have to remark already now. While the -bound variables in the examples we've seen so far were placeholders for missing constant symbols, 
and 
in our determiner-representation stand for missing predicates. The version of -calculus introduced here does not distinguish variables that stand for different kinds of missing information. Nevertheless we will stick to a convention of using lower case letters for variables that stand for missing constant symbols, and capital letters otherwise.
But now let's carry on with the analysis of the sentence ``A woman walks''. We have to incorporate the intransitive verb ``walks''. We assign it the representation 
. The following tree shows the final representation we obtain for the complete sentence:
The S node is associated with 
. We obtain this representation by a procedure analogous to that performed at the NP node. We associate the S node with the application that has the NP representation just obtained as functor, and the VP representation as argument.
...and reducing it.
Now instead of hand-tailoring lots of specially dedicated post-processing rules, we will simply -reduce the expression that we find at the S node as often as possible. We must follow its (bracketed) structure when we perform -reduction. So we start with reducing the application 
. We have to replace by 
, and drop the 
prefix. The whole representation then looks as follows:
Let's go on. This time we have two applications that can be reduced. We decide to get rid of the 
first. Replacing Q by 
we get:
Again we have the choice where to go on -reducing -- this time it should be obvious that our choice doesn't make any difference for the final result (in fact it never does. This property of -calculus is called confluence ). Thus let's -reduce twice. We have to replace both 
and 
by 
. Doing so finally gives us the desired:
Determiner
Finally, let's have a closer look at the determiner-representation we've been using. Remember it was 
. Why did we choose this expression? In a way, there isn't really an answer to this question, except simply: Because it works.
But then let's at least have a closer look at why it works. We know that a determiner must contribute a quantifier and the pattern of the quantification. Intuitively, indefinite determiners in natural language are used to indicate that there is something of a certain kind (epressed in the so-called restriction of the determiner), about which one is going to say that it also has some other property (expressed in the so-called scope of the determiner). In the sentence ``A woman walks'', the ``a'' indicates that there is something of a certain kind, namely ``woman'', and that this something also has a certain property, namely ``walk''.
So for the case of an indefinite determiner, we know that the quantifier in its first-order formalization has to be existential, and that the main connective within the quantification is a conjunction symbol. This is the principle behind formalizing indefinite determiners in first-order logic.
Now clever use of -bound variables in our determiner representation allows us to leave unspecified all but just these two aspects. All that is already ``filled in'' in the representation 
is the quantifier and a little bit about the internal structure of its scope, namely that the main connective is 
. The rest is ``left blank'', and this is indicated using variables.
The second important thing about a -expression is the order of its prefixes. This is where the role of syntactic structure comes in - and where explanation really doesn't go any further in the case of our determiner: We had to choose 
and not 
for the simple reason that phrases and sentences containing determiners are built up syntactically as they are. This holds with all generality: When deciding about the order of -prefixes of a meaning representation, one has to think of the right generalizations over the syntactic use of its natural language counterpart.