The Elements of Mathematical Logic

227 Pages·1950·35.614 MB·English

by  Paul C. Rosenbloom

The Elements of
Mathematical Logic
PAUL C. ROSENBLOOM
DEPARTMENT
1950
DOVER PUBLICATIONS, INC.
ASSOCIATE PROFESSOR, OF MATHEMATICS,
SYRACUSE UNIVERSITY
THE DOVER SERIES IN MATHEMATICS AND PHYSICS
w. prager,ConsultingEditor
>
FirstEdition
1950 J3V
PUBLICATIONS,
1780 BROADWAY, NEW 19, N. Y.
Printed& Boundinthe UnitedStatesofAmerica
COPYRIGHT
DOVER INC.
YORK
This book is intendedforreaders who, while mature mathe-
matically, have no knowledge of mathematical logic. We at-
tempttointroducethereader tothemostimportantapproaches
tothesubject, and, whereverpossiblewithin thelimitations of
space which we have set for ourselves, to give at least afew
nontrivialresultsillustrating each of theimportant methodsfor
attacking logical problems. Since Lewis' survey of symbolic
logic and Jorgensen's treatise on formal logic, both of
which are now obsolete, the onlywork of this naturehasbeen
theexcellentbook of Church, which is notsuitableforbeginners
and whichis not easilyaccessible. Thus thepresentbook differs
from those which confine themselves to the detailed develop-
ment of oneparticidar system offormal logic. We haveempha-
sizedinstead themodern tendencyof analyzing thestructureof
asystemasawhole. Wefeel thattoomanyauthorsin thisfield
haveoverlookedthenecessityofexhibiting thepowerof logical
methodsin non-trivial problems. Otherwise mathematical logic
is amereshorthandfor transcribingresults obtainedwithout its
aid,notatoolforresearch and discovery.
Thusin thechapteron thelogic ofclasses wehavea section
on thestructureand representation ofBoolean algebras,which
isappliedin thenextchaptertothestudyof deductivesystems.
In the thirdchapter wesketch the methods ofRussell, Quine,
Zermelo, Curry, and Church for the construction of logics of
propositional functions. Finally, we give a brief introduction
to the general syntax of language, with applications to unde-
cidabilityand incompleteness theorems.
Wehaveattemptedtomake theexposition aselementaryas
possible throughout. Nevertheless, those who are unfamiliar
with modern algebramay find it advisable to skip the proofs
in Chapter I,Section 3, on thefirst reading.
PREFACE
In thelast chapterweuse theprofoundandbeautiful ideasof
Post. Wehopethat oneby-product of thisbook will be amore
widespread recognition and appreciation of his work, which
amountstothecreation of anewbranch of mathematics of the
same fundamental importance as algebraand topology.
The connoisseur mayfind of some interest (1) theinsistence
onthedemonstrablepropertiesofaformal systemasacriterion
for its acceptability, (2) thesimpleproof of thecompletenessof
thetheoryofcombinators,* (3)thesimpleexplicitexampleof a
recursively unsolvable problem in elementary number theory,
(4) thefirst connectedexposition of alltheessentialstepsin the
proof of Church'stheorem ontherecursive unsolvability of the
decisionproblemfor therestricted function calculus.
Much of thematerial waspresentedin a coursegivenby the
author atLund University, Sweden,in the spring of 1948.
It is impossiblefor me to expressadequatelymy debttothe
late Professor H. B. Smith forhis constantkindness and gen-
erosity. I am grateful to Professors Churchman, Post, Curry,
McKinsey, Huntington, and Stone for their friendly encour-
agement when I was beginning my mathematical career. I
cannotrefrain from also thankingProfessors Cohenand Nagel,
since it was a misinterpretation of a footnote in their book
which led me to abandon chemistry for mathematics twelve
years ago! I thank Dover Publications, Inc. for its unfailing
courtesy and helpfulness during the preparation of this book.
Finally, I should like to express my gratitude to my beloved
wife, Elly, for providing the stimulus and the working condi-
tionswithout whichthe bookcould nothavebeenwritten.
October 11, 1949
PAUL C. ROSENBLOOM
Syracuse,NewYork
*Curryhas arrivedindependentlyatessentially thesamesimplification
ofthetheoryofcombinators.Thisappearedsincetheabovewaswrittenin
Synthase,Vol.VII, 1948^9, No.6-A,p.391-398.
IInnttrroodduuccttiioonn 1
CHAPTER I. THELOGIC OFCLASSES
Section 1. Informal introduction. Fundamental theorems 1
Section 2. Boolean algebraas adeductivescience 8
Section 3. Thestructure andrepresentationof Boolean
algebras 18
11. THE LOGIC
Section 1. Fundamentals 28
Section2. Alternativeformulations 31
Section3. Deductivesystems 44
Section 4. Manyvalued logics, modal logics, intuitionism 51
CHAPTER 111. THELOGIC OF PROPOSITIONAL FUNCTIONS
Section 1. Informal introduction 69
Section 2. The functional logic ofthe firstorder 74
Section3. Someveryexpressive languages 91
Section4. Combinatorylogics 109
Section5. The developmentof mathematics within an
objectlanguage 133
Section 6. Theparadoxes 144
Section 7. Theaxiom ofchoice 146
CONTENTS
CHAPTER OF PROPOSITIONS
CHAPTERIV. THE GENERAL SYNTAXOFLANGUAGE
Section 1. Basic concepts. Simple languages 152
Section 2. Production, canonical languages, extension,
and definition 157
Section3. Normal languages. Theorems of Post and
Godel 170
appendix 1. Canonical forms ofLx ,L 2, andL 182
z
appendix2. Algebraic approach to language. Church's
theorem 189
Bibliographical and OtherRemarks 194
Index 209
Inthisbook weshall studythelaws oflogicbymathematical
methods. This mayseemunfair,sincelogic isusedinconstruct-
ingmathematical proofs, and it might appearthat thestudy of
logic shouldcome before thestudy of mathematics. Suchapro-
cedureis,however,typical of science.Ouractualknowledge is a
narrow band of light flanked on both sides by darkness. We
may, on theone hand, goforward and developfurther thecon-
sequences ofknown principles. Orelse we maypressbackward
theobscurityin which thefoundations of scienceareenveloped.
Justbyusingmathematical methods, i.e. by workingwith ideo-
grams (symbolsfor ideas) instead of ordinary words (symbols
for sounds),wecanthrownewandimportantlight onthelogical
principles used in mathematics. This approachhas ledtomore
knowledge about logic in one century than had been obtained
from the death of Aristotle up to 1847, when Boole's master-
piecewas published.
Webegin withthesimplest branch ofthesubject, thelogic of
classes. After an informal introduction, in whichwederive the
propertiesofclassesbyafreeuseofnaiveintuition,weformulate
that theoryasadeductive science, thatis, asasciencein which
the assumptions are explicitly stated, and in which everything
else follows from theassumptions bymeans ofexplicitly stated
rules. The assumptions are stated in terms of certain notions
which are notanalyzedfurther and are takenasundefined. All
otherconcepts of thescience aredefined in termsofthese.
Wethenproceedtoastudyof thesystemasawhole. Thatis,
instead of developing moreand more consequences of the as-
sumptions, we trytofind general characteristics of the science
itself. Thisis typical of themodern tendencyto emphasize the
structure of a science, to derive theorems about the science,
rather thanto concentrateonthe detailedderivation of results
i
INTRODUCTION
II
within the science. This study of the structure of the logic of
classesculminates in ChapterI, Section3.
We then applythesame methodstothelogic ofpropositions.
In doing this, we uncover a striking similarity between this
scienceand thelogic of classes. It isprecisely throughformulat-
ing theselogics as deductivesciencesthatwesee thatboth are
special examplesof ageneraltheory.
The logic of propositions has beenthe subject of much con-
troversy amonglogicians and mathematicians. We discuss the
various alternative approaches whichhavebeen proposed.
We then try toconstruct general logical theories which are
adequatefor at leasta largepartof mathematics. Herewerun
into difficulties since theunreineduse of naiveintuitivereason-
ingleadstodevastatingparadoxes.Thuswemustseekatheory
which admits as much as possible of thereasoning intuitively
acceptedasvalid, but includes such restrictions astoevadethe
paradoxes. But a profound theorem of Godel shows that no
logical theory of avery general type can include methods of
reasoning strong enough for the proof of its ownconsistency.
Indeed, in any system of logic of this general type, there are
propositions which can be proved by an argumentoutside the
systembut whichcannotbeprovedwithinthesystem.Thusno
formal logical system ofthis type, whichincludesall adequate
logicssofar proposed,can contain allvalid modesofreasoning.
Allthatwe can hopeforis strongerand strongersystems which
are adequatefor more and more powerful arguments, or else
some systemradically different from anything sofar proposed.
Inordertoarrive atsuchresultsasGodel's,itis necessaryfor
us to scrutinize our tools more closely. In a deductive science
theundefinedtermsaredenotedbycertain symbols, whichmay
be blobs ofprinter's ink, speech sounds, printed marks repre-
senting thelatter, etc. Thepropositions of thissciencearecom-
municated by means ofthesesigns. These signs, togetherwith
the rules governing their use and combination, constitute a
language for stating relationships within the science. This is
called the object language. In an exposition of the science the
assumptions mustbecommunicated in alanguagewhosemean-
111
ingis alreadyassumedtobeknown, sayEnglish. Thisis called
thesyntax language. We use the object languageto talk within
thescience and the syntaxlanguage to talk about the science.
In ordinaryusage the confusion betweenthe two leads to no
difficulty,but whenthescienceunderconsideration islogic itself
wemust lean overbackwards toavoidunclarity.
Theprimitive signs of theobjectlanguagearecalledits alpha-
bet. Certaincombinations of thesesigns maybe assignedmean-
ings.Suchcombinations areoftencalled wordsorsentences. If a
certain combination of signs denotesan object, then this com-
bination will be anamefor that object. In speakingabout the
objectweuseanameforit. Thus "Deweysmiled" is asentence
wherein we mention the man Dewey by using his name, the
word "Dewey."Whenwearetalking aboutanameorasymbol,
it is convenienttouse a specimenenclosed in quotation marks
asanameofthenameorsymbol. Thus," "Dewey""isanameof
"Dewey,"which is, in turn, anameof Dewey, whois aman.
Again, onp. 2,25thlinefrom thebottom, wearespeakingabout
anameof theuniversalclass, whileonthenextlineitisthenull
class itself which is mentioned. To avoid the use of names of
names of namesand thelike, weshall also use suchphrases as
"the letter—" or "the sign —" as names of the symbols of
which specimensareexhibited. Itis oftenoverlookedthat while
we cannot put a man on the printed page and are thusforced
to use a name when writing about him, we do have greater
resources when we wish to write about symbols.
Inparticular, asentenceis aname of aproposition. We shall
say that the sentenceexpresses the proposition, and we shall
often use "statement" as a synonymfor "sentence." We shall
often use the phrase "the proposition that p" to indicate the
proposition expressedby "p."Carefulattentiontothesematters
helpsin discussing ticklish questions.
We arethusled, inchapter IV,tothemathematical analysisof
language. Whereas in the previous chapters our attention is
centered on therelationships expressedby the objectlanguage,
in thelast chapterwefocus our attentionon thestructureof the
languageapartfrom its meaning.The former process is some-
IV
times called thesemantical study of language,i.e. thestudy of
themeaningsexpressedby thelanguage,whilethelatter is often
called the syntactical study of the language. The methods we
use were developedespecially by Post. Wefind in this chapter
that certain classes of languages, which include practically all
languages whichhavebeen preciselyformulated, canbe singled
out and possessimportant common properties. It is exactlythe
mathematical method ofabstractingfrom thespecialfeatures of
particular languages which enables us to prove rigorously a
numberofprofound generaltruths,wheremetaphysicianswould
argueback andforthfor centurieswithout everreaching acon-
clusion whichcould be tested.
Mathematicallogicis, then,nomereshorthandfor expressing
in ideograms what hasalreadybeen discoveredbyreasoning in
ordinarylanguage. It is,rather, apowerfuland versatile toolfor
solvingproblemswhichare inaccessibleto other methods.
In thefollowingwe shall make references thus:
1112 denotessection 2ofchapter III;
T2 denotestheorem 2 of thepresentsection;
T5.2.3 denotestheorem3of section 2ofchapter V;
[27]4 denotes number 4by author 27 in Church's Bibliog-
raphy, J.SymbolicLogic, vol. 1,no. 4;
[11]35 denotesthearticlebeginning orreviewed onpage 35 of
vol. 11, J.SymbolicLogic.