1.4.5 The Moral

Our examples have shown that -calculus is ideal for semantic construction in two respects:

1. The process of combining two representations was perfectly uniform. We simply said which of the representations is the functor and which the argument, whereupon combination could be carried out by applying functor to argument and -converting. We didn't have to make any complicated considerations here.

2. The load of semantic analysis was carried by the lexicon: We used the -calculus to make missing information stipulations when we gave the meanings of the words in our sentences. For this task, we had to think accurately. But we could make our stipulations declaratively, without hacking them into the combination process.

Our observations are indeed perfectly general. Doing semantic construction with the help of -calculus, most of the work is done before the actual combination process.

What we have to do...

When giving a -abstraction for a lexical item, we have to make two kinds of decisions:

1. We have to locate gaps to be abstracted over in the partial formula for our lexical item. In other words, we have to decide where to put the -bound variables inside our abstraction. For example when giving the representation for the proper name ``Mary'' we decided to stipulate a missing functor. Thus we applied a -abstracted variable to .

2. We have to decide how to arrange the -prefixes. This is how we control in which order the arguments have to be supplied so that they end up in the right places after -reduction when our abstraction is applied. For example we chose the order when we gave the representation for the indefinite determiner ``a''. This means that we will first have to supply it with the argument for the restriction of the determiner, and then with the one for the scope.

...and how

Of course we are not totally free in these decisions. What constrains us is that we want to be able to combine the representations for the words in a sentence so that they can be fully -reduced to a well-formed first order formula. And not just some formula, but the one that captures the meaning of the sentence.

So when we design a -abstraction for a lexical item, we have to anticipate its potential use in semantic construction. We have to keep in mind which final semantic representations we want to build for sentences containing our lexical item, and how we want to build them. In order to decide what to abstract over, we must think about which pieces of semantic material will possibly be supplied from elsewhere during semantic construction. And in order to arrange our -prefixes, we must think about when and from where they will be supplied.

Summing up

The bottom line of all this is that devising lexical representations will be the tricky part when we give the semantics for a fragment of natural language using -calculus. But with some clever thinking, we can solve a lot of seemingly profound problems in a very streamlined manner.