This book is not about type theories in general but about one very neat and special system called “TA” for “type-assignment”. Its types contain type-variables and arrows but nothing else, and its terms are built by λ-abstraction and application from term-variables and nothing else. Its expressive power is close to that of the system called simple type theory that originated with Alonzo Church.

TA is polymorphic in the sense that a term can have more than one type, indeed an infinite number of types. On the other hand the system has no ∀-types and hence it is weaker than the strong polymorphic theories in current use in logic and programming. However, it lies at the core of nearly every one of them and its properties are so distinctive and even enjoyable that I believe the system is worth isolating and studying on its own. That is the aim of this book. In it I hope to try to pass on to the reader the pleasure the system's properties have given me.

TA is also an excellent training ground for learning the techniques of type-theory as a whole. Its methods and algorithms are not trivial but the main lines of most of them become clear once the basic concepts have been understood. Many ideas that are complicated and tedious to formulate for stronger type-theories, and many complex techniques for analysing structures in these theories, appear in TA in a very clean and neat stripped-down form.