Logical Frameworks for Truth and Abstraction: An Axiomatic Study

474 Pages·1996·13.16 MB·English

by  Andrea Cantini

LOGICAL FRAMEWORKS
FOR TRUTH
AND ABSTRACTION
An Axiomatic Study
STUDIES IN LOGIC
AND
THE FOUNDATIONS OF MATHEMATICS
VOLUME 135
Honorary Editor:
E SUPPES
Editors:
S. ABRAMSKY, London
S. ARTEMOV, Moscow
J. BARWISE, Stanford
H.J. KEISLER, Madison
A.S. TROELSTRA, Amsterdam
ELSEVIER
AMSTERDAM (cid:12)9L AUSANNE (cid:12)9N EW YORK (cid:12)9O XFORD (cid:12)9S HANNON (cid:12)9T OKYO
LOGICAL FRAMEWORKS
FOR TRUTH
AND AB STRACTION
An Axiomatic Study
Andrea CANTINI
Department of Philosophy
University of Florence
Florence, Italy
1996
ELSEVIER
AMSTERDAM (cid:12)9L AUSANNE (cid:12)9N EW YORK (cid:12)9O XFORD (cid:12)9S HANNON (cid:12)9T OKYO
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PREFACE
This book is concerned with logical systems, which are usually termed type-
free or self-referential and emerge from the traditional discussion on logical
and semantical paradoxes. We will consider non-set-theoretic frameworks,
where forms of type-free abstraction and self-referential truth can
consistently live together with an underlying theory of combinatory logic.
However, this is not a book on paradoxes; nor we aim at a grand logic
la Frege-Russell, inspired by a foundational program. We shall rather
investigate type-free systems, in order to show that"
(i) there are rich theories of self-application, involving both operations and
truth, which can serve as foundations for property theory and formal
semantics;
(ii) these theories give new outlooks on classical topics, such as inductive
definitions and predicative mathematics;
(iii) they are promising as far as applications are concerned.
This way of looking is justified by the history of the antinomies in our
century. In spite of isolated foundational and philosophical traditions, the
research arising from paradoxes has become progressively closer to the
mainstream of mathematical logic and it has received substantial impulse
during the last twenty years: a number of significant developments,
techniques and results have been cropping up through the work of several
logicians (see below for our main debts). Therefore a major aim of this book
is to attempt a unifying view of relevant research in the field, by dwelling
on connections with well-established logical knowledge and on applicable
theories and concepts.
However, the present work is far from being comprehensive. We do not
treat illative combinatory logic (with the exception of a system of Ch.VI,
investigated by Flagg and Myhill 1987), nor we deal with the Barwise-
Etchemendy approach to self-reference via non-well-founded sets. Another
significant direction, which is only touched upon in two sections of chapter
XIII, is the systematic development of the general theory of semi-inductive
definitions (in the sense of Herzberger, Gupta and others).
vI Preface
The project started some years ago, when Prof. A. S. Troelstra kindly
suggested an English translation of the author's monograph (Cantini 1983a)
about theories of partial operations and classifications in the sense of
Feferman (1974). The attempted translation soon shifted towards a
thorough expanded revision of the old text, and eventually gave rise to an
entirely new set of notes at the end of 1988. After a stop of almost two
years, these notes were taken up again, fully rewritten and reorganized. The
manuscript was submitted to the editor for final refereeing in October 1993.
The content and the results of the present version are disjoint from the
1983 monograph; they partly overlap with the 1988 notes, except for a
different choice of primitive notions and for the addition of Ch.VI, parts of
Ch.XIII and the epilogue. Ch.VIII offers a development of topics, contained
in the author's paper "Levels of Truth" (to appear in the Notre Dame
Journal of Formal Logic, 1995): we thank the Editors for granting the
permission of using parts of that paper in Ch.VIII of this book.
Acknowledgments. The present work owes a great deal to the writings of
several logicians, and even if I tried hard to make a complete list of my
debts in the text and in the reference list, I am sure that there are
omissions: I apologize for them.
As to the proper content of the book, pertaining to type-free abstraction
and self-referential truth, I would like to underline my intellectual debt
with the following papers (listed in alphabetical order): Aczel(1980),
Feferman (1974), (1984), (1991), Fitch(1948), (1967), Friedman and Sheard
(1987), Kripke (1975), Myhill (1984), Scott (1975).
Profl W. Buchholz offered an invaluable help in correcting errors of any
kind and in proposing technical improvements. I owe a special thank to
him, also because the topics I dealt with were not touching his main
research interests.
I am grateful to Prof. S. Feferman and to Profi G. Js for keeping me
informed over the years about their own research on type-free systems and
proof theory, and for important advice. J~iger's Ph.D. student, T. Strahm
made useful critical comments on the first chapter.
Dr. R. Giuntini and Dr. P. Minari undertook the final proof-reading of
chapters I-VIII and XII-XIV; I warmly thank them for a host of useful
remarks and corrections.
I am deeply indebted to Dr. A. P. Tonarelli for proof-reading the
remaining chapters and for eagle-eyed assistance in the unrewarding task of
preparing the final manuscript.
Preface vii
Of course, I must stress that I am fully responsible for all errors, to be
found in the whole work.
I am grateful to the Alexander-von-Humboldt Stiftung (Germany) for
granting me a "Wiederaufnahme" of a research fellowship at the Ludwig-
Maximilians-Universit~it M/inchen in Sommer Semester 1991, when the
present work was at a difficult stage.
Partial support to the present project was granted by the Italian
National Research Council (CNR)-and the Italian Ministry for University,
Scientific Research and Technology (MURST).
Last but not least, this work is dedicated to my children Giulia and
Francesco.
Firenze, April 1995
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CONTENTS
Preface
Contents IX
Introduction
PART A: COMBINATORS AND TRUTH 11
Introducing operations 13
The basic language 14
o
2. Operations I: general facts 15
3. Operations Ih elementary recursion theory 18
4A. The Church-Rosser theorem 22
4B. Term models 26
5. The graph model 28
6. An effective version of the extensional model D co 34
Appendix 39
II Extending operations with reflective truth 43
7. Extending combinatory algebras with truth 45
8. The theory of operations and reflective truth:
simple consequences 51
9A. Type-free abstraction, predicates and classes 55
9B. Operations on predicates and classes 59
10A. The fixed point theorem for predicates 63
10B. Applications to semantScs and recursion theory 68
11. Non-extensionality 73
Appendix I: a property theoretic definition of the
fixed point operator for predicates 76
Appendix Ih on the explicit abstraction theorem 77
Appendix III: independence of truth predicates
from the encoding of logical operators 80