Table Of Contentogical abyrinths
L L
ogical abyrinths
L L
Raymond M. Smullyan
AKPeters,Ltd.
Wellesley,Massachusetts
CRC Press
Taylor & Francis Group
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Contents
Preface ix
I BeWise,Generalize! 1
1 TheLogicofLyingandTruth-Telling 3
2 MaleorFemale? 17
3 SilentKnightsandKnaves 21
4 MadorSane? 25
5 TheDifficultiesDouble! 31
6 AUnification 37
II BeWise,Symbolize! 41
7 BeginningPropositionalLogic 43
8 Liars,Truth-Tellers,andPropositionalLogic 55
9 VariableLiars 67
10 LogicalConnectivesandVariableLiars 73
11 TheTableauMethod 83
12 AllandSome 99
13 BeginningFirst-OrderLogic 111
v
vi Contents
III Infinity 133
14 TheNatureofInfinity 135
15 MathematicalInduction 157
16 GeneralizedInduction,Ko¨nig’sLemma,Compactness 173
IV FundamentalResultsinFirst-OrderLogic 189
17 FundamentalResultsinPropositionalLogic 191
18 First-OrderLogic: Completeness,Compactness,
Skolem-Lo¨wenheimTheorem 205
19 TheRegularityTheorem 217
V AxiomSystems 227
20 BeginningAxiomatics 229
21 MorePropositionalAxiomatics 245
22 AxiomSystemsforFirst-OrderLogic 267
VI MoreonFirst-OrderLogic 275
23 Craig’sInterpolationLemma 277
24 Robinson’sTheorem 285
25 Beth’sDefinabilityTheorem 291
26 AUnification 297
27 LookingAhead 309
References 321
Preface
This book serves as a bridge from all my previous recreational puz-
zle books to all my technical writings in the fascinating field of sym-
bolic logic. It starts out as a typical puzzle book and ends up with
highly significant results—significant for philosophy, mathematics and
computer science. Afterhavingread thisbook, you willhavetheknowl-
edge of a typical one-semester graduate course in symbolic logic, and
you will then have the preparation to read, not only all of my technical
writings, but much of the general literature in the field. I have writ-
ten this book for the many people who have asked me what symbolic
logic is all about and where to find a good introductory account of the
subject.
Good question: What is symbolic logic? It is also called “mathemat-
ical logic,” though the word “mathematics” here has a totally different
meaningthanthemathematicsusuallytaughtintheschools—arithmetic,
algebraandgeometry. Mathematicallogic,insteadofdealingspecifically
with things like numbers, lines and points, deals rather withpropositions
and arguments, but codified in symbolic forms to facilitate more precise
reasoning.
As valuable as symbolic logic is, I have learned from much teaching
experience that it is best to begin by devoting some time to informal rea-
soning. The logic of lying and truth-telling is admirably suited to this,
being both entertaining and highly instructive; so this is how we begin
in Part I of this book, in which you follow the remarkable journey of
the anthropologist Abercrombie through successively more labyrinthine
lands of more and more curious liars and truth-tellers of the most var-
iedandsundrysort,culminatinginonegranddiscoverythatgeneralizes
all the problems of the preceding adventures. This prepares the reader
for Part II, which begins the formal study of symbolic logic. Part II be-
gins with the subject known as propositional logic, shows how this logic
formalizes the reasoning of the earlier chapters, and then advances to
the main subject of this book, which is first-order logic, a subject that has
many important applications in philosophy, mathematics and computer
science.
vii
viii Preface
In Part III, we journey through the amazing labyrinths of infinity, a
concept that has stirred the imagination of mankind as much as, if not
morethan,anyothersubject.
In Part IV, we establish many of the most important results of first-
orderlogic.
One important purpose of mathematical logic is to make precise the
notion of a proof. Just what is meant by the word “proof”? One person
suggestedtomethataproofissimplyanargumentthatconvincessome-
body. Not bad, but the trouble is that different people are convinced by
different arguments, so the concept of proof in that sense is quite sub-
jective. For example, someone once told me that he could prove that a
certain political party was wrong. Well, I’m sure he could prove it to his
own satisfaction, but hardly to the satisfaction of those who belong to
that party! Can’t one, then, find a more objective definition of the word
“proof”? Well, so far no such definition has been found for the general
notionofproof,butsymboliclogichasdoneanadmirablejobinmaking
absolutely precise the notion of “proof” for given axiom systems! The
ancient Greeks had an excellent, albeit informal, concept of proof. Eu-
clid starts out with certain basic propositions called axioms, which are
sufficiently self-evident not to require any sort of proof, and then, by
using logic, deduces from these axioms propositions that are far from
self-evident. All this has been refined by modern symbolic logic in the
studyofformalaxiomatics,whichisthesubjectofPartV.
We then go on to further study of first-order logic, and, in the penul-
timate chapter of this book, we find one giant theorem that, in a truly
remarkable manner, generalizes just about all the earlier results! Indeed,
“generalization”isverymuchthehallmarkofthisbook.
Thefinalchapterisaguidetothosereaderswhowanttoknowmore
about mathematical logic and wish to find out what further literature
exists.
Thisbookcanbeusedasatextbookforaone-ortwo-semestercourse
in logic. A pre-publication copy has been successfully used in this man-
neratHarvard.
IwishtoexpressmydeepestthanksandappreciationtoDr. Malgosia
Askanas,myformerstudent,forherexperteditingofmymanuscript.
ElkaPark,NY
2007
- Part I -
e ise eneralize
B W , G !
