The purpose of the programme in combinatory logic is to rework the mathematical foundations of computer science on a theory of pure thought. It begins from the idea that, if logic is to be the science of correctly dealing with thought-objects, the underlying theory must be in some sense a part of, or at least a preliminary to, its structure, ie, a protologic. From this idea a combinatory algebra is constructed, using a programmatic mixture of the classical axiomatic and set-theoretic approaches. This text shows that sufficiently rich combinatory algebras can serve as a platform from which to develop the algorithmic aspects of many areas in computer science, mathematics and their applications. It should be of interest to those studying universal algebra, logic and computer algebra.

Assisted by: K. Aberer, B. Amrhein, O. Gloor, M. v. Mohrenschildt, D. Otth, G. Schwärzler, T. Weibel