The Lambda Calculus
Abstract:
Towards the end of the last section we saw how to transfer as much information as possible about the syntactic structure of a sentence into a kind of proto-semantic representation. But we still completely lack a uniform way of combining the collected semantic material into well-formed first order formulas.In this section we will dicuss a mechanism that fits perfectly for this task. It will allow us to explicitely mark gaps in first-order formulas and give them names. This way we can state precisely how to build a complete first order formula out of separate parts. The mechanism we're talking about is called λ-calculus. For present purposes we shall view it as a notational extension of first order logic that allows us to bind variables using a new variable binding operator λ. Here is a simple λ-expression: λx.WOMAN(x) The prefix λx. binds the occurrence of x in WOMAN(x). That way it gives us a handle on this variable, which we can use to state how and when other symbols should be inserted for it.
Table of Contents
Lambda-Abstractionλ-expressions are formed out of ordinary first order formulas using the
λ-operator.
The MoralOur examples have shown that
λ-calculus is ideal for semantic construction in two respects.
What's next For the remainder of this lecture, the following version of the three tasks listed earlier (
» Three Tasks) will be put into practise.