Mathematical Logic : A course with exercises -- Part I -- Propositional Calculus, Boolean Algebras, Predicate Calculus, Completeness Theorems

360 Pages·2000·12.16 MB·English

by  Rene Cori

Mathematical Logic
A Course with Exercises
Part I: Propositional calculus,
Boolean algebras, Predicate calculus
Rene Cori and Daniel Lascar
Equipe de Logique Mathematique
Universite Paris VII
Translated by
Donald H. Pelletier
York University, Toronto
OXFORD
UNIVERSITY PRESS

Mathematical Logic
A Course with Exercises
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First published in French as Logique mathematique
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Foreword to the Original French edition
lean-Louis Krivine
In France, the discipline of logic has traditionally been ignored in university-level
scientific studies. This follows, undoubtedly, from the recent history of mathe
matics in our country which was dominated, for a long while, by the Bourbaki
school for whom logic was not, as we know, a strong point. Indeed, logic origi
nates from reflecting upon mathematical activity and the common gut-reaction of
the mathematician is to ask: 'What is all that good for? We are not philosophers
and it is surely not by cracking our skulls over modus ponens or the excluded
middle that we will resolve the great conjectures, or even the tiny ones ... ' Not
so fast!
A new ingredient, of some substance, has come to settle this somewhat byzantine
debate over the importance of logic: the explosion of computing into all areas of
economic and scientific life, whose shock wave finally reached the mathematicians
themselves.
And, little by little, one fact dawns on us: the theoretical basis for this nascent
science is nothing other than the subject of all this debate, mathematical logic.
It is true that certain areas of logic were put to use more quickly than oth
ers. Boolean algebra, of course, for the notions and study of circuits; recursive
ness, which is the study of functions that are computable by machine; Herbrand's
theorem, resolution and unification, which form the basis of 'logic programming'
(the language PROLOG); proof theory, and the diverse incarnations of the Com
pleteness theorem, which have proven themselves to be powerful analytical tools
for mature programming languages.
But, at the rate at which things are going, we can imagine that even those areas
that have remained completely 'pure', such as set theory, for example, will soon
see their turn arrive.
As it ought to be, the interaction is not one-way, far from it; a flow of ideas and
new, deep intuitions, arising from computer science, has come to animate all these
sectors of logic. This discipline is now one of the liveliest there is in mathematics
and it. is evolving very rapidly.
So there is no doubt about the utility and timeliness of a work devoted to a
general introduction to logic; this book meets its destiny. Derived from lectures
for the Diplome d'Etudes Approfondies (DEA) of Logic and the Foundations
of Computing at the University of Paris VII, it covers a vast panorama: Boolean
VI FOR E W 0 R D TOT H E 0 RIG I N A L F R ENe H EDT T [ 0 N
algebras, recursiveness, model theory, set theory, models of arithmetic and GCidel's
theorems.
The concept of model is at the core of this book, and for a very good reason
since it occupies a central place in logic: despite (or thanks to) its simple, and even
elementary, character, it illuminates all areas, even those that seem farthest from
it. How, for example, can one understand a consistency proof in set theory without
first mastering the concept of being a model of this theory? How can one truly
grasp GCidel's theorems without having some notion of non-standard models of
Peano arithmetic? The acquisition of these semantic notions is, I believe, the mark
of a proper training for a logician, at whatever level. R. Cori and D. Lascar know
this well and their text proceeds from beginning to end in this direction. Moreover,
they have overcome the risky challenge of blending all the necessary rigour with
clarity, pedagogical concern and refreshing readability.
We have here at our disposal a remarkable tool for teaching mathematical logic
and, in view of the growth in demand for this subject area, it should meet with a
marked success. This is, naturally, everything I wish for it.
Foreword to the English edition
Wilfrid Hodges
Sehoul of Mathematical Sciences
Queen Mary and Westfield College
University of London
There are two kinds of introduction to a subject. The first kind assumes that you
know nothing about the subject, and it tries to show you in broad brushstrokes
why the subject is worth studying and what its achievements are. The second kind
of introduction takes for granted that you know what the subject is about and
why you want to study it, and sets out to give you a reliable understanding of the
basics. Rene Cori and Daniel Lascar have written the second sort of introduction
to mathematical logic. The mark of the book is thoroughness and precision with a
light touch and no pedantry.
The volume in your hand, Part I, is a mathematical introduction to first-order
logic. This has been the staple diet of elementary logic courses for the last fifty
years, but the treatment here is deeper and more thorough than most elementary
logic courses. For example the authors prove the compactness theorem in a general
form that needs Zorn's Lemma. You certainly shouldn't delay reading it until you
know about Zorn's Lemma - the applications here are an excellent way of learning
how to use the lemma. In Part I there are not too many excitements - probably the
most exciting topic in the book is the Godel theory in Chapter 6 of Part II, unless
you share my enthusiasm for the model theory in Chapter 8. But there are plenty of
beautiful explanations, put together with the clarity and elegance that one expects
from the best French authors.
For English students the book is probably best suited to Masters' or fourth-year
undergraduate studies. The authors have included full solutions to the exercises;
this is one of the best ways that an author can check the adequacy of the definitions
and lemmas in the text, but it is also a great kindness to people who are studying
on their own, as a beginning research student may be. Some thirty-five years ago
I found I needed to teach myself logic, and this book would have fitted my needs
exactly. Of course the subject has moved on since then, and the authors have
included several things that were unknown when I was a student. For example
their chapter on proof theory, Chapter 4 in this volume, includes a well-integrated
section on the resolution calculus. They mention the connection with PROLOG;
but in fact you can also use this section as an introduction to the larger topic
VIlI FOREWORD TO THE ENGLISH EDITION
of unification and pattern-matching, which has wide ramifications in computer
science.
One other thing you should know. This book comes from the famous Equipe de
Logique Mathematique at the University of Paris, a research team that has had an
enormous influence on the development of mathematical logic and its links with
other branches of mathematics. Read it with confidence.