Propositional and Predicate Calculus A Model of Argument

320 Pages·2005·3.22 MB·English

by  Derek Goldrei MA

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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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
1 Introduction
system, we shall need to make compromises and simplifications within our
model to produce something that is both mathematically tractable and pro-
vides useful insights into the actual and arguably much more complex world.
Here our model will be of truth and proof, and we shall investigate the con-
nection between them.
As we shall be modelling proofs from axioms, let us look at a set of axioms
for R that could be found in a standard textbook on real analysis.
AxiomsforR
The real numbers system consists of a set S, usually written as R, with Weare deliberately writing the set
binaryoperations+and·,unaryoperations−and−1,abinaryrelation as S rather than R to reinforce the
≤ (besides =, equality), and special elements written as 0 and 1, such point that theaxioms involveno
knowledge of what theobjects in S
that 0(cid:2)=1, which satisfy the following properties.
are. The axioms are often said to
1. For all x,y,z ∈S, x+(y+z)=(x+y)+z. axiomatize acomplete ordered field.
2. For all x∈S, x+0=0+x=x.
3. For each x∈S, x+(−x)=(−x)+x=0.
4. For all x,y ∈S, x+y =y+x.
5. For all x,y,z ∈S, x·(y·z)=(x·y)·z.
6. For all x∈S, x·1=1·x=x.
7. For all x∈S with x(cid:2)=0, x·x−1 =x−1·x=1.
8. For all x,y ∈S, x·y =y·x.
9. For all x,y,z ∈S, x·(y+z)=(x·y)+(x·z).
10. For all x∈S, x≤x.
11. For all x,y ∈S, if x≤y and y ≤x, then x=y.
12. For all x,y,z ∈S, if x≤y and y ≤z, then x≤z.
13. For all x,y ∈S, x≤y or y ≤x.
14. For all x,y,z ∈S, if x≤y, then x+z ≤y+z.
15. For all x,y,z ∈S, if x≤y and 0≤z, then x·z ≤y·z.
16. (Completenessaxiom)Anynon-emptysubsetAofSwhichisbounded ‘Bounded above’ and so on are
above has a least upper bound in S. definedin terms of more basic
terminology.
Withinthecontextofafirstcourseonrealanalysis,onewouldbuildupfurther
properties of the set S, starting with elementary properties, such as
for all x,y ∈S, if x+y =x+z, then y =z. This is sometimes grandly called
theleft cancellation law for +.
An argument proving this might run as follows. Suppose that x+y =x+z.
Then
(−x)+(x+y)=(−x)+(x+z).
By axiom 1, (−x)+(x+y) equals ((−x)+x)+y, which by axiom 3 equals
0+y, which by axiom2 equals y. Similarly (−x)+(x+z)=z, so the above
gives y =z, as required.
2
1.1 Outlineofthebook
It might be good for you to try proving a few elementary properties of S for
yourself, just to get a feeling of what sort of features of proofs our model
might have to take into account.
Exercise1.1
Give proofs purely from these properties of each of the following. (You may
of course prove and then exploit subsidiary results, or lemmas like the left
cancellation law for + above.)
(a) For all x∈S, (−(−x))=x.
(b) If x,y ∈S satisfy x·y =1, then y =x−1.
(c) For all x∈S, x·0=0.
(d) For all x∈S, (−1)·x=−x.
Solution
We shall give a solution only for part (a).
Take any x∈S. Then by axiom 3,
(−x)+x=0,
and by the same axiom used with the element −x instead of x,
(−x)+(−(−x))=0,
so that
(−x)+x=(−x)+(−(−x)).
By the left cancellation law for +, we can conclude that x = (−(−x)), or
equivalently (−(−x))=x, as required.
Here are just a few of the features in arguments of this sort which we shall
try to incorporate into our mathematical model of proof.
• Proofs consist of statements, like ‘(−x)+(x+y) equals ((−x)+x)+y’,
connected by justification for these statements.
• Thestatementsinaproofuseafairlylimitedtechnicallanguage,including
symbols like +, · and =.
• Proofsshouldbepresentedinawaythatallowsotherstofolloweachstep.
• Proofs should involve no properties of the set S not ultimately traceable
back to the axioms.
• Proofs should make no assumptions about the nature of the elements x,
y, 0, 1 and so on, or the operations +, · and −, other than what we are
told about the symbols by the axioms.
• Toproveastatementoftheform‘forallx∈S,xhassomeproperty’,take
a typicalx∈S, showthatit hasthe property,andconclude the property
holds for all x.
• To prove a statement of the form ‘if something then something else’,
assume the first ‘something’ and prove the ‘something else’.
Our list barely scrapes the surface of the sort of reasoning that is employed,
even in simple arguments like our solution to Exercise 1.1(a) or our proof of
3
1 Introduction
theleftcancellationlawfor+. Forinstance,thereareahostofwaysinwhich
we use =, the equality symbol, like the step from
x+y =x+z
to
(−x)+(x+y)=(−x)+(x+z)
in our proof of the left cancellation law for +, and at least some of these
should be covered within our model.
Our model of proof will ultimately cope with all the features needed in these
proofs. We shall look at a formal symbolic language from which statements
like ‘for all x, (−1)·x = −x’ can be constructed and at a formal system
for proofs which can handle such statements and, indeed, for this particular
example, derive it from some of the axioms given above. But a limitation
of this model is that the formal system will not be able to derive all the
statements one can derive in everyday maths about R from the axioms we
have given, as for important technical reasons it will not be able to handle
the completeness axiom.
The axioms for R are not entirely typical of the way axioms are used in The propertechnical description is
modern maths, in that they have the following special property: any two that the two sets with their
operations areisomorphic. Another
sets, equipped with suitable operations and relations, which both satisfy the
important axiom system with this
axioms,areessentiallythe same,sothatthe axiomstie down the one system.
property is Peano’s axioms for the
An example of a set of axioms for which there are many essentially different natural numbers,which we shall
sets satisfying the axioms is obtained by taking just the first three axioms of meet in Chapter 6.
those given for R, which axiomatize the theory of groups, a group being the
The first nine axioms are axioms
name for a set and suitable operation matching the + symbol for which all for the theory of fields. While we
threeaxiomsaretrue. Ourformalproofsystemwillbeabletocopewiththese hope that you haveencountered
axioms. Just as for the axioms for R, we shall expect a connection between thetheoriesofgroupsandfields,we
are not relying on this experience
thestatementswecanderivefromtheseaxiomsandthestatementswhichare
in thisbook, but will give you a
trueinallgroups. Importantly,weshallexplainwhatwemeanbyastatement
brief background in Chapter 4.
being true in a groupas something more than that it can be provedfrom the
axioms, so that the major result, the completeness theorem, connecting the
notions of truth and proof, genuinely connects different notions.
In representing mathematical statements within a formal symbolic language,
it will be important to have strict construction rules to delineate the strings
of symbols which are to be considered as statements, for instance to exclude
the equivalent of expressions like 8)−×5))7+ which signify nothing mean-
ingful in everyday maths. Likewise the rules have to provide expressions for
which, when meanings are given to the symbols, there is an unambiguous in-
terpretation. For instance, we want to avoid the analogue of expressions like
1+2×3innormalmaths,forwhichitisunclearwhetherismeant(1+2)×3
or 1+(2×3). A key feature of our formal language, and indeed the formal
proof system for handling statements in it, is what we shall describe as its
mechanical nature. The rules governing which strings of symbols are formal
statements and which combinations of statements constitute a formal proof
will all be ones that a computer could be programmed to check, with no un- SeeDavis [10] for a veryreadable
derstanding of any intended meanings of the symbols or rules. This aspect account interweaving thehistory of
logic with thedevelopment of the
of our undertaking not only predates the development of modern computers,
modern computer.
but has had a strong influence on both their development and how they are
4