Life without types

At this point, we should remind ourselves that these intended meanings do not stop us from constructing seemingly nonsensical expressions such as $x x$.

At first sight, it seems strange that we should try to embark on a syntax for functions in which there is no notion of ``type''. However, untyped formalisms are familiar to us

• Set theory is untyped. Everything is a set. To do arithmetic, we define certain sets to denote numbers. Functions on numbers apply in principle to all sets but are ``meaningless'' in the wrong context.

• A computer's memory is intrinsically untyped. Consider the operation of dereferencing a pointer in C--that is, treat a memory location as a pointer and look up the location that it points to. This operation is performed without checking if the contents of the pointer are of the right ``type''--that is, a memory location legally available to this program. If the contents of the pointer are valid, the operation succeeds. However, if the pointer is invalid (for instance, if it is uninitialized), the operation fails (sometimes dramatically, as we all know!)

The point is that we have a syntax for representing data (sets, bit strings) and we have rules for manipulating these representations. Some representations are more meaningful than others, but the manipulation rules can be applied uniformly to both the meaningful and the meaningless representations. All we ask is that if the original data represents a meaningful value, then, after manipulation, it still represents a meaningful value.

A similar view is taken in the lambda calculus--for instance $M M'$ represents application if the structure of the expression $M$ corresponds to a ``function''. Otherwise, we don't care what this construction means (e.g., if $M M'$ is $x x$, as describe earlier).