The aim of this monograph is to present some of the basic ideas and results in pure combinatory logic and their applications to some topics in proof theory, and also to present some work of my own. Some of the material in chapter 1 and 3 has already appeared in my notes Introduction to Combinatory Logic. It appears here in revised form since the presen­ tation in my notes is inaccurate in several respects. I would like to express my gratitude to Stig Kanger for his invalu­ able advice and encouragement and also for his assistance in a wide variety of matters concerned with my study in Uppsala. I am also in­ debted to Per Martin-USf for many valuable and instructive conversa­ tions. As will be seen in chapter 4 and 5, I also owe much to the work of Dag Prawitz and W. W. Tait. My thanks also to Craig McKay who read the manuscript and made valuable suggestions. I want, however, to emphasize that the shortcomings that no doubt can be found, are my sole responsibility. Uppsala, February 1972.

Inhalt

1. The Theory of Combinators and the ?-Calculus.- 1. Introduction.- 2. Informal theory of combinators.- 2.1. Combinators and combinations.- 2.2. The combinators I, B, W, C, ?, ?.- 2.3. Extensionality.- 2.4. Alternative primitive combinators.- 2.5. Composite products, powers and deferred combinators.- 2.6. Truth-functions.- 3. Equality and reduction.- 3.2. Combinatory terms.- 3.3. ?-equality.- 3.4. Functional abstraction.- 3.5. Substitution.- 3.6. Weak reduction.- 3.7. Equality.- 3.8. Strong reduction.- 4. The ?-calculus.- 4.1. ?-terms.- 4.2. Substitution.- 4.3. Equality.- 4.4. Reduction.- 4.5. ?-reduction.- 4.6. Historical remarks.- 5. Equivalence of the ?-calculus and the theory of combinators.- 6. Set-theoretical interpretations of combinators.- 6.1. Introduction.- 6.2. Scott's models.- 6.3. Combinatorial completeness.- 6.4. A representation result.- 6.5. Limit spaces.- 7. Illative combinatory logic and the paradoxes.- 2. The Church-Rosser Property.- 1. Introduction.- 2. R-reductions.- 3. One-step reduction.- 4. Proof of main result.- 5. Generalization.- 6. Generalized weak reduction.- 3. Combinatory Arithmetic.- 1. Introduction.- 2. Combinatory definability.- 3. Fixed-points and numeral sequences.- 4. Undecidability results.- 4. Computable Functionals of Finite Type.- 1. Introduction.- 2. Finite types and terms of finite types.- 3. The equation calculus.- 4. The role of the induction rule.- 5. Soundness of the axioms.- 6. Defining axioms and uniqueness rules.- 7. Reduction rules.- 8. Computability and normal form.- 9. Interpretation of types and terms.- 5. Proofs in the Theory of Species.- 1. Introduction.- 2. Formulas, terms and types.- 3. A-terms and deductions.- 4. The equation calculus.- 5. Reduction and normal form.- 6. The strong normalization theorem.- 7. Interpretation of types and terms.- Index of Names.- Index of Subjects.