Propositional and predicate calculus: a model of argument

324 Pages·2005·2.44 MB·English

by  Goldrei

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DEREK GOLDREI
Propositional and Predicate Calculus: A Model
of Argument

Derek Goldrei
Propositional and
Predicate Calculus
A Model of Argument
123
Derek Goldrei, MA, MSc
Open University
Milton Keynes
UK
Mathematics Subject Classifi cation (2000): 03B05, 03B10,03C07
British Library Cataloguing in Publication Data
Goldrei, Derek
Propositional and predicate calculus: a model of argument
1. Propositional calculus 2. Predicate calculus
I. Title
511.3
ISBN 1852339217
Library of Congress Cataloging-in-Publication Data
Goldrei, Derek.
Propositional and predicate calculus: a model of argument/Derek Goldrei.
p. cm.
Includes bibliographical references and index.
ISBN 1-85233-921-7 (acid-free paper)
1. Propositional calculus. 2. Predicate calculus. I. Title.
QA9.3.G65 2005
511.3--dc22 2005040219
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ISBN-10: 1-85233-921-7
ISBN-13: 978-1-85233-921-0
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PREFACE
How to Use This Book
Thisbook isintendedto be usedbyyouforindependent study,withno other
reading or lectures etc., much along the lines of standard Open University
materials. There are plenty of exercises within the text which we would rec-
ommend you to attempt at that stage of your work. Almost all are intended
tobereasonablystraightforwardonthebasisofwhat’scomebeforeandmany
are accompanied by solutions – it’s worth reading these solutions as they of-
ten contain further teaching, but do try the exercises first without peeking,
to help you to engage with the material. Those exercises without solutions
might well be very suitable for any tutor to whom you have access to use as
the basis for any continuous assessment of this material, to help you check
that you are making reasonable progress. But beware! Some of the exercises
pose questions for which there is not always a clear-cut answer: these are
intendedto provokedebate! Inaddition there are further exerciseslocatedat The book is also peppered with
the end of most sections. These vary from further routine practice to rather notes in the margins, likethis!
Theyconsist ofcommentsmeant to
hard problems: it’s well worth reading through these exercises, even if you
beon thefringe of the main text,
don’t attempt them, as they often give an idea of some important ideas or
ratherthan thecore of the
results not in the earlier text. Again your tutor, if you have one, can guide teaching, for instance reminders
you through these. about ideas from earlier in the
book or particularly subjective
If you would like any further reading in logic textbooks, there are plenty of opinions of theauthor.
goodbooksavailablewhichuseessentiallythesamesystem,forinstancethose
by Enderton [12], Hamilton [18], Mendelson [25] and Cori and Lascar [7].
Acknowledgments
I would like to thank all those who have in some way helped me to write Plainly theblame for any errors
this book. My enthusiasm for the subject was fuelled by Robin Gandy, Paul and inadequacies of this book lies
entirely with me. But perhaps at
Bacsich, Jane Bridge, Angus Macintyre and Harold Simmons, when I stud-
some deep and subtlelevel, the
ied at the Universities of Oxford and Aberdeen. Anything worthwhile I have
fault lies with everyoneelse!
successfullylearntaboutteachingstemsfrommycolleaguesattheOpenUni-
versity and the network of mathematicians throughout the UK who support
theOpenUniversitybyworkingasAssociateLecturers,externalassessorsand
examiners. They have taught me so much. It has been particularly stimu-
lating writing this book alongside producing the Open University’s course on
Mathematical Logic (with a very different angle on the subject) with Alan
Pears, Alan Slomson, Alex Wilkie, Mary Jones, Roger Lowry, Jeff Paris and
Frances Williams. And it is a privilege to be part of a university which puts
somuchcareandeffortinto its teaching andthe supportof itsstudents. The
practicalities of producing this book owe much to my publishers, Stephanie
HardingandKarenBorthwickatSpringer; andto my colleaguesatthe Open
University who have done so much to provide me with a robust and attrac-
tiveLaTEXsystem: AlisonCadle,DavidClover,JonathanFine,BobMargolis
and Chris Rowley. And thanks to Springer, I have received much invaluable
advice on content from their copy-editor Stuart Gale and their anonymous,
v
Preface
very collegial, reviewers. I would also like to thank Michael Goldrei for his
work on the cover design.
Perhapsthemaininspirationforwritingthebookistheenthusiasmandtalent
for mathematical logic displayed by my old students at the Open University
andattheUniversityofOxford,especiallythoseofSomerville,St. Hugh’sand
MansfieldColleges. InparticularI’dliketothankthefollowingfortheircom-
mentsonpartsofthe book: Dimitris Azanias,DavidBlower,DuncanBlythe,
RosaClements,RhodriDavies,DavidElston,MichaelHopley,GerrardJones,
Eleni Kanellopoulou, Jakob Macke, Zelin Ozturk, Nicholas Thapen, Matt
Towers,Chris Wall, Garth Wilkinson, Rufus Willett and especially Margaret
Thomas.
This book is dedicated to all those whose arguments win me over, especially
Jennie, Michael, Judith and Irena.
vi
CONTENTS
1 Introduction 1
1.1 Outline of the book 1
1.2 Assumed knowledge 6
2 Propositions and truth assignments 17
2.1 Introduction 17
2.2 The construction of propositional formulas 19
2.3 The interpretation of propositional formulas 31
2.4 Logical equivalence 48
2.5 The expressive power of connectives 63
2.6 Logical consequence 74
3 Formal propositional calculus 85
3.1 Introduction 85
3.2 A formal system for propositional calculus 87
3.3 Soundness and completeness 100
3.4 Independence of axioms and alternative systems 119
4 Predicates and models 133
4.1 Introduction: basic ideas 133
4.2 First-order languages and their interpretation 140
4.3 Universally valid formulas and logical equivalence 163
4.4 Some axiom systems and their consequences 185
4.5 Substructures and Isomorphisms 208
5 Formal predicate calculus 217
5.1 Introduction 217
5.2 A formal system for predicate calculus 221
5.3 The soundness theorem 242
5.4 The equality axioms and non-normal structures 247
5.5 The completeness theorem 252
6 Some uses of compactness 265
6.1 Introduction: the compactness theorem 265
6.2 Finite axiomatizability 266
6.3 Some non-axiomatizable theories 272
6.4 The L¨owenheim–Skolem theorems 277
6.5 New models from old ones 289
6.6 Decidable theories 298
Bibliography 309
Index 311

1 INTRODUCTION
1.1 Outline of the book
Mathematicsaboundswiththeoreticalresults! Butonwhatbasisdowetrust
any of them? Normallywe rely on seeing some sortof justification for results
whichwe,orsomeonewefeelwecantrust,canscrutinizeandthenverify. The
justificationwillnormallyinvolvesomesortofargumentshowingthataresult
holds. This book is about a mathematical model of such arguments, rather
than the mathematics being argued about. We shall not attempt to cope
with the full range of mathematical arguments, but only look at a fragment
covered by what is called the predicate calculus. The predicate calculus is an
important part of logic, the science of reasoning and the laws of thought.
The sort of argument we shall try to model is one which starts from given
assumptions and moves by steps to a conclusion. For instance, in everyday
maths we might start from the assumption that f is a differentiable function
from the set R of real numbers to itself and conclude after various steps that
f is continuous. The book is written for readers familiar with this sort of
argument. So you will know that the steps involve things like: use of the If a step seems too large to be
assumptionthatf isdifferentiable; thedefinitionofthewords‘differentiable’, followed, we usually seek an
explanation using more elementary
‘function’ and ‘continuous’; facts about inequalities and arithmetic involving
steps.
real numbers; and forms of reasoning enabling us to infer each step from
previousassertionsintheargument. Anessentialfeatureofsuchanargument
is our ability to follow and agree on each step, almost in a mechanical way,
like a computer recognizing whether input data conforms to agreed rules.
Let’s expand on what we call facts about the set R of real numbers. The
modern approach to R is to describe it not in terms of its members, but
in terms of the properties its members have. We are usually given some of Weshall give such axioms and
their very basic properties, called axioms, from which one can infer more illustrate some of theseinferences
soon.
complicated properties.
WehavetwodifferentbutconnectedsortsofexpectationoftheaxiomsforR:
(i) that the axioms and any statements we infer from them are true state-
ments about R;
(ii) thatanystatementthatwefeelistrueaboutRcanbeinferredfromthese
axioms.
Investigation of the connection between these expectations is a major theme
of the book.
WehavephrasedourexpectationsaboveintermsofwhatistrueaboutRand
what can be inferred from axioms. In this book, we shall attempt to explain
on the one hand what is meant by a statement being true in a structure like
R and, on the other, what constitutes inference using acceptable forms of
reasoning. Establishing a connection between these very different concepts,
called the completeness theorem, is a major goal of the book.
We must reiterate that what the book covers is a mathematical model. Just
as with a mathematical model of, say, the motion of the planets in the solar
1