Table Of ContentCOMBINATORS, A.-TERMS AND PROOF THEORY
SYNTHESE LIBRARY
MONOGRAPHS ON EPISTEMOLOGY,
LOGIC, METHODOLOGY, PHILOSOPHY OF SCIENCE,
SOCIOLOGY OF SCIENCE AND OF KNOWLEDGE,
AND ON THE MATHEMATICAL METHODS OF
SOCIAL AND BEHAVIORAL SCIENCES
Editors:
DONALD DAVIDSON, Rockefeller University and Princeton University
JAAKKO HINTIKKA, Academy of Finland and Stanford University
GABRIEL NUCHELMANS, University of Leyden
WESLEY C. SALMON, Indiana University
SOREN STENLUND
COMBINATORS,
A-TERMS AND PROOF THEORY
D. REIDEL PUBLISHING COMPANY / DORDRECHT-HOLLAND
ISBN-13: 978-94-010-2915-5 e-ISBN-13:978-94-01 0-2913-1
001: 10.1007/978-94-010-2913-1
All Rights Reserved
Copyright © 1972 by D. Reidel Publishing Company, Dordrecht, Holland
Softcover reprint of the hardcover I st edition 1972
No part of this book may be reproduced in any form, by print, photoprint, microfilm,
or any other means, without written permission from the publisher
PREFACE
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. Soren Stenlund.
CONTENTS
PREFACE 5
CHAPTER 1. THE THEORY OF COMBINATORS AND THE A-CALCULUS 11
1. Introduction 11
2. Informal theory of combinators 17
2.1. Combinators and combinations 17
2.2. The combinators I, B, W, C, cit, 'I' 17
2.3. Extensionality 19
2.4. Alternative primitive combinators 20
2.5. Composite products, powers and deferred combinators 22
2.6. Truth-functions 23
3. Equality and reduction 26
3.2. Combinatory terms 27
3.3. IX-equality 28
3.4. Functional abstraction 28
3.5. Substitution 29
3.6. Weak reduction 30
3.7. Equality 33
3.8. Strong reduction 37
4. The A-calculus 41
4.1. A-terms 41
4.2. Substitution 42
4.3. Equality 42
4.4. Reduction 43
4.5. 6-reduction 45
4.6. Historical remarks 46
5. Equivalence of the A-calculus and the theory of combinators 47
8 CONTENTS
6. Set-theoretical interpretations of combinators 51
6.1. Introduction 51
6.2. Scott's models 53
6.3. Combinatorial completeness 55
6.4. A representation result 59
6.5. Limit spaces 65
7. Illative combinatory logic and the paradoxes 67
CHAPTER 2. THE CHURCH-ROSSER PROPERTY 73
1. Introduction 73
2. R-reductions 74
3. One-step reduction 76
4. Proof of main result 79
5. Generalization 84
6. Generalized weak reduction 86
CHAPTER 3. COMBINATORY ARITHMETIC 91
1. Introduction 91
2. Combinatory definability 92
3. Fixed-points and numeral sequences 101
4. Undecidability results 106
CHAPTER 4. COMPUTABLE FUNCTIONALS OF FINITE TYPE 109
1. Introduction 109
2. Finite types and terms of finite types 112
3. The equation calculus 114
4. The role of the induction rule 118
5. Soundness of the axioms 120
6. Defining axioms and uniqueness rules 123
7. Reduction rules 125
8. Computability and normal form 126
9. Interpretation of types and terms 134
CHAPTER 5. PROOFS IN THE THEORY OF SPECIES 143
1. Introduction 143
CONTENTS 9
2. Formulas, terms and types 144
3. A-terms and deductions 147
4. The equation calculus 151
5. Reduction and normal form 155
6. The strong normalization theorem 156
7. Interpretation of types and terms 165
BIBLIOGRAPHY 177
INDEX OF NAMES 181
INDEX OF SUBJECTS 182
CHAPTER 1
THE THEORY OF COMBINATORS AND
THE A-CALCULUS
I. INTRODUCTION
1.1. Combinatory logic started with Schonfinkel 1924. The main aim of
SchOnfinkel's paper was methodological: to reduce the primitive
logical notions to as few and definite notions as possible. Progress along
these lines had already been made by Frege, Russell and Whitehead and
by Sheffer, who reduced the propositional logical connectives to a single
one. Schonfinkel extended Sheffer's result to the predicate calculus and
proceeded further than that. He showed how to eliminate variables from
the predicate calculus by showing how the role of variables could be
taken over by two primitive operators and one primitive operation.
As Schonfinkel points out, the use of variables for expressing logical
properties is, in a sense, unnatural. By an assertion such as
we neither want to say something about 'p' or 'q' nor about something
denoted by those letters. What is asserted is that a certain relationship
holds between the logical constants -, (negation) and (implication).
-+
The variables 'p' and 'q' are used merely as a tool for communicating
a proposition. The notion of a variable is thus a linguistic concept rather
than a logical concept and, according to Schonfinkel, it is therefore not
appropriate for expressing the constant and universal laws of logic.
1.2. In order to avoid using variables, Schonfinkel developed a function
calculus, which is what we shall call the theory of combinators following
Curry 1930, who introduced combinators independently.
The notion of a function underlying this function calculus is the notion
12 COMBINATORS, A-TERMS AND PROOF THEORY
of a function in intension (in the terminology of Church 1941), which
means that a function is understood as a rule of correspondence, which,
when an object is given as an argument, yields another object as a value.
Such a rule of correspondence is usually expressed by a definition.
This is to be contrasted with a function in extension, which is determined
by the set of all pairs of an argument and the associated value.
The fact that the notion of a function in intension is the intended inter
pretation of Schonfinkel's function calculus is interesting apart from the
use he wants make of it. This is so because the original idea of a function
is certainly that of a rule of correspondence and it seems fair to say that
this notion of a function is the more general and logically fundamental
one. A function in extension can always be reduced to a rule of cor
respondence while the converse is not true. We can know a lot about a
rule of correspondence without complete knowledge of its arguments
and values.
In mathematics a domain of arguments and a range of values is usu
ally associated to a function and it makes usually no sense to apply the
function to an argument outside the domain. There are however certain
general kinds of functions or rules to which this remark does not apply.
Simple examples are the identity function and the constant functions
thought of as rules. They can be understood without knowledge of the
nature of their arguments. It even makes sense to apply such a function
to itself as an argument. The identity function and the constant function
tions are typical of the ones represented in Schonfinkel's function calculus.
As a matter of fact they were given special names.
1.3. The only primitive operation in the function calculus is application.
That is the operation of applying a function to its argument. If f is a
function and x a possible argument, then the result of the application
operation is denoted by
(fx).
In mathematics this is usually written f(x), but the notation (fx) is more
convenient in combinatory logic. (fx) is now an object of the function
calculus that may be applied to an argument y yielding
«(fx)y)
