Mathematical logic

253 Pages·2007·2.01 MB·English

by  Chiswell, Ian, Hodges, Wilfrid

Mathematical Logic
IAN CHISWELL and WILFRID HODGES
1
3
GreatClarendonStreet,OxfordOX26DP
OxfordUniversityPressisadepartmentoftheUniversityofOxford.
ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship,
andeducationbypublishingworldwidein
Oxford NewYork
Auckland CapeTown DaresSalaam HongKong Karachi
KualaLumpur Madrid Melbourne MexicoCity Nairobi
NewDelhi Shanghai Taipei Toronto
Withofficesin
Argentina Austria Brazil Chile CzechRepublic France Greece
Guatemala Hungary Italy Japan Poland Portugal Singapore
SouthKorea Switzerland Thailand Turkey Ukraine Vietnam
OxfordisaregisteredtrademarkofOxfordUniversityPress
intheUKandincertainothercountries
PublishedintheUnitedStates
byOxfordUniversityPressInc.,NewYork
(cid:1)c IanChiswellandWilfridHodges,2007
Themoralrightsoftheauthorhavebeenasserted
DatabaserightOxfordUniversityPress(maker)
Firstpublished2007
Allrightsreserved.Nopartofthispublicationmaybereproduced,
storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans,
withoutthepriorpermissioninwritingofOxfordUniversityPress,
orasexpresslypermittedbylaw,orundertermsagreedwiththeappropriate
reprographicsrightsorganization.Enquiriesconcerningreproduction
outsidethescopeoftheaboveshouldbesenttotheRightsDepartment,
OxfordUniversityPress,attheaddressabove
Youmustnotcirculatethisbookinanyotherbindingorcover
andyoumustimposethesameconditiononanyacquirer
BritishLibraryCataloguinginPublicationData
Dataavailable
LibraryofCongressCataloginginPublicationData
Dataavailable
TypesetbyNewgenImagingSystems(P)Ltd.,Chennai,India
PrintedinGreatBritain
onacid-freepaperby
BiddlesLtd.,King’sLynn,Norfolk
ISBN978–0–19–857100–1
ISBN978–0–19–921562–1(Pbk)
10 9 8 7 6 5 4 3 2 1
Preface
This course in Mathematical Logic reflects a third-year undergraduate module
that has been taught for a couple of decades at Queen Mary, University of
London. Both the authors have taught it (though never together). Many years
agothefirstauthorputtogetherasetoflecturenotesbroadlyrelatedtoDirkvan
Dalen’s excellent text Logic and Structure (Springer-Verlag, 1980). The present
text is based on those notes as a template, but everything has been rewritten
with some changes of perspective. Nearly all of the text, and a fair number of
the exercises, have been tested in the classroom by one or other of the authors.
The book covers a standard syllabus in propositional and predicate logic.
A teacher could use it to follow a geodesic path from truth tables to the Com-
pleteness Theorem. Teachers who are willing to follow our choice of examples
from diophantine arithmetic (and are prepared to take on trust Matiyasevich’s
analysisofdiophantinerelations)shouldfind,aswedid,thatGo¨del’sIncomplete-
ness Theorem and the undecidability of predicate logic fall out with almost no
extra work. Sometimes the course at Queen Mary has finished with some appli-
cations of the Compactness Theorem, and we have included this material too.
We aimed to meet the following conditions, probably not quite
compatible:
• The mathematics should be clean, direct and correct.
• As each notion is introduced, the students should be given something rele-
vantthattheycandowithit,preferablyatleastacalculation.(Forexample,
parsingtrees,besidessupportinganaccountofdenotationalsemantics,seem
to help students to make computations both in syntax and in semantics.)
• Appropriate links should be made to other areas in which mathematical
logic is becoming important, for example, computer science, linguistics and
cognitivescience(thoughwehavenotexploredlinkstophilosophicallogic).
• We try to take into account the needs of students and teachers who prefer
a formal treatment, as well as those who prefer an intuitive one.
WeusetheHintikkamodelconstructionratherthanthemoretraditionalHenkin-
Rasiowa-Sikorskione. Wedothisbecauseitismorehands-on: itallowsustoset
uptheconstructionbydecidingwhatneedstobedoneandthendoingit, rather
than checking that a piece of magic does the work for us.
We do not assume that our students have studied any logic before (though
in practice most will at least have seen a truth table). Until the more specialist
vi Preface
matter near the end of the book, the set theory is very light, and we aim to
explain any symbolism that might cause puzzlement. There are several proofs
by induction and definitions by recursion; we aim to set these out in a format
that students can copy even if they are not confident with the underlying ideas.
Other lecturers have taught the Queen Mary module. Two who have cer-
tainlyinfluencedus(thoughtheywerenotdirectlyinvolvedinthewritingofthis
book) were Stephen Donkin and Thomas Mu¨ller—our thanks to them. We also
thank Lev Beklemishev, Ina Ehrenfeucht, Jaakko Hintikka, Yuri Matiyasevich
and Zbigniew Ras for their kind help and permissions with the photographs of
Anatoliˇı Mal’tsev, Alfred Tarski, Hintikka, Matiyasevich and Helena Rasiowa
respectively. Every reasonable effort has been made to acknowledge copyright
where appropriate. If notified, the publisher will be pleased to rectify any errors
or omissions at the earliest opportunity.
We have set up a web page at
www.maths.qmul.ac.uk/∼wilfrid/mathlogic.html
for errata and addenda to this text.
Ian Chiswell
Wilfrid Hodges
School of Mathematical Sciences
Queen Mary, University of London
August 2006
Contents
1 Prelude 1
1.1 What is mathematics? 1
1.2 Pronunciation guide 3
2 Informal natural deduction 5
2.1 Proofs and sequents 6
2.2 Arguments introducing ‘and’ 9
2.3 Arguments eliminating ‘and’ 14
2.4 Arguments using ‘if’ 16
2.5 Arguments using ‘if and only if’ 22
2.6 Arguments using ‘not’ 24
2.7 Arguments using ‘or’ 27
3 Propositional logic 31
3.1 LP, the language of propositions 32
3.2 Parsing trees 38
3.3 Propositional formulas 45
3.4 Propositional natural deduction 53
3.5 Truth tables 62
3.6 Logical equivalence 69
3.7 Substitution 72
3.8 Disjunctive and conjunctive normal forms 78
3.9 Soundness for propositional logic 85
3.10 Completeness for propositional logic 89
4 First interlude: Wason’s selection task 97
5 Quantifier-free logic 101
5.1 Terms 101
5.2 Relations and functions 105
5.3 The language of first-order logic 111
5.4 Proof rules for equality 121
5.5 Interpreting signatures 128
5.6 Closed terms and sentences 134
5.7 Satisfaction 139
viii Contents
5.8 Diophantine sets and relations 143
5.9 Soundness for qf sentences 148
5.10 Adequacy and completeness for qf sentences 150
6 Second interlude: the Linda problem 157
7 First-order logic 159
7.1 Quantifiers 159
7.2 Scope and freedom 163
7.3 Semantics of first-order logic 169
7.4 Natural deduction for first-order logic 177
7.5 Proof and truth in arithmetic 186
7.6 Soundness and completeness for first-order logic 189
7.7 First-order theories 194
7.8 Cardinality 199
7.9 Things that first-order logic cannot do 206
8 Postlude 213
AppendixA The natural deduction rules 217
AppendixB Denotational semantics 223
AppendixC Solutions to some exercises 229
Index 245
1 Prelude
1.1 What is mathematics?
Euclid Egypt, c. 325–265 bc.
For Euclid, mathematics consists of proofs and
constructions.
Al-Khw¯arizm¯ı Baghdad, c. 780–850.
For Al-Khwa¯rizm¯ı, mathematics consists of
calculations.
2 Prelude
G. W. Leibniz Germany, 1646–1716.
According to Leibniz, we can calculate whether a
proof is correct. This will need a suitable language
(a universal characteristic) for writing proofs.
Gottlob Frege Germany, 1848–1925.
Frege invented a universal characteristic. He called
it Concept-script (Begriffsschrift).
Gerhard Gentzen Germany, 1909–1945.
Gentzen’s system of natural deduction allows us to
write proofs in a way that is mathematically
natural.
Prelude 3
1.2 Pronunciation guide
Togetcontrolofabranchofmathematics, youneedtobeabletospeakit. Here
aresomesymbolsthatyouwillprobablyneedtopronounce,withsomesuggested
pronunciations:
⊥ ‘absurdity’
(cid:3) ‘turnstile’
|= ‘models’
∀ ‘for all’
∃ ‘there is’
tA ‘the interpretation of t in A’
|=A φ ‘A is a model of φ’
≈ ‘has the same cardinality as’
≺ ‘has smaller cardinality than’
The expression ‘x (cid:8)→ y’ is read as ‘x maps to y’, and is used for describing
functions. For example, ‘x (cid:8)→ x2’ describes the function ‘square’, and ‘n (cid:8)→
n+2’describesthefunction‘plustwo’. Thisnotationisalwaysashorthand; the
surrounding context must make clear where the x or n comes from.
The notation ‘A ⇒ B’ is shorthand for ‘If A then B’, or ‘A implies B’,
or sometimes ‘the implication from A to B’, as best suits the context. Do not
confuse it with the notation ‘→’. From Chapter 3 onwards, the symbol ‘→’ is
not shorthand; it is an expression of our formal languages. The safest way of
reading it is probably just ‘arrow’ (though in Chapters 2 and 3 we will discuss
its translation into English).
Thenotation‘N’canbereadas‘thesetofnaturalnumbers’oras‘thenatural
number structure’, whichever makes better sense in context. (See Example 5.5.1
for the natural number structure. Note that our natural numbers are 0,1,2,...,
starting at 0 rather than 1.)
The following rough pronunciations of personal names may help, though
they are no substitute for guidance from a native speaker:
Frege: FRAY-ga Peirce: PURSE
Hintikka: HIN-ticka Helena Rasiowa: he-LAY-na
Leibniz: LIBE-nits ra-SHOW-va
L(cid:2) o´s: WASH Scholz: SHOLTS
L(cid:2) ukasiewicz: woo-ka-SHAY-vitch Dana Scott: DAY-na SCOTT
Matiyasevich: ma-ti-ya-SAY-vitch Sikorski: shi-COR-ski
Giuseppe Peano: ju-SEP-pe Van Dalen: fan DAH-len
pay-AH-no Zermelo: tser-MAY-low
2 Informal natural deduction
In this course we shall study some ways of proving statements. Of course not
every statement can be proved; so we need to analyse the statements before we
prove them. Within propositional logic we analyse complex statements down
into shorter statements. Later chapters will analyse statements into smaller
expressions too, but the smaller expressions need not be statements.
What is a statement? Here is a test. A string S of one or more words or
symbols is a statement if it makes sense to put S in place of the ‘...’ in the
question
Is it true that ...?
For example, it makes sense to ask any of the questions
Is it true that π is rational?
Is it true that differentiable functions are continuous?
Is it true that f(x)>g(y)?
So all of the following are statements:
π is rational.
Differentiable functions are continuous.
f(x)>g(y).
For this test it does not matter that the answers to the three questions are
different:
No.
Yes.
It depends on what f, g, x and y are.
On the other hand, none of the following questions make sense:
Is it true that π?
Is it true that Pythagoras’ Theorem?
Is it true that 3+cosθ?
So none of the expressions ‘π’, ‘Pythagoras’ Theorem’ and ‘3 + cosθ’ is a
statement.