Introduction to Metalogic: With an Appendix on Type-Theoretical Extensional and Intensional Logic

189 Pages·1997·22.74 MB·English

by  Imre Ruzsa

INTRODUCTION
TO METALOGIC
APPENDIX
WITH AN ON
TYPE·THEORETICAL
EXTENSIONAL AND INTENSIONAL LOGIC
BY
IMRERUZSA
* *
ARON PUBLISHERS BUDAPEST 1997
ISBN9638550457
ARONPUBLISHERS, H-1447BUDAPEST,P.O.BOX4~7,-HUNGARY
©ImreRuzsa,1997.
PrintedinHungary
Copiesofthisbookareavailableat
AronPublishers
H-1447BUDAPEST,P.O.BOX487,HUNGARY
and
L.EotvosUniversity,DepartmentofSymbolicLogic
H-1364BUDAPEST,P.O.BOX107, HUNGARY
Nyomdaie16cillft~s: CPSTUDIO,Budapest
· l':.
), ,. .f~.r-?,:~/i' :,. T i.. ~.
In'menuiriammywife (1942-1996)
? ' "
.
;
ACKNOWLEDGEMENTS
The material ofthe present monograph originates from a series of lectures held bythe
author at the Department ofSymbolic Logic ofL. Eotvos University, Budapest. The
questions and the critical remarks of my students and colleagues gave me a very
valuable help in developing my investigations. My sincere thanks are due to all of
them.
Special thanks are due toPROFESSORISTVAN NEMETIand Ms AGNES KURUCZ
whoread thefirstversionofthemanuscriptandmadeveryimportant critical remarks.
In preparing and printing this monograph, I got substantial help from my son
DR.FERENCRUZSAaswellasfrom mydaughterAGNESRUZSA.
* * *
This work was partly supported by the Hungarian Scientific Research Foundation
(OTKA II3, 2258) and by the Hungarian Ministry ofCulture and Education (MKM
384/94).
lmre Ruzsa
Budapest,June 1996.
vi
TABLE OF CONTENTS
Chapter1 Introduction 1
1.1.Thesubjectmatterofmetalogic 1
1.2.Basicpostulatesonlanguages 2
1.3.Speakingonlanguages 4
1.4.Syntaxandsemantics 5
Chapter2, Instrumentsofmetalogic 7
2.1.Grammaticalmeans 7
2.2.Variablesandquantifiers 10
2.3.Logicalmeans 15
2.4.Definitions 19
2.5.Classnotation 20
Chapter3 Languageradices 25
3.1.Definitionandpostulates 25
3.2.Thesimplestalphabets 28
Chapter4 Inductiveclasses 31
4.1.Inductivedefinitions 31
4.2.Canonicalcalculi 36
4.3.Somelogicallanguages 38
4.4. Hypercalculi 42
4.5.Enumerabilityanddecidability 47
Chapter5 Normalalgorithms 51
5.1.Whatisanalgorithm? 51
5.2.Definitionofnormalalgorithms 54
5.3.Decidingalgorithms 58
5.4.Definiteclasses 61
Chapter6 Thefirst-ordercalculus(QC) 66
6.1.Whatisalogicalcalculus? 66
6.2.First-orderlanguages 67
6.3.ThecalculusQC 70
6.4.MetatheoremsonQC 72
6.5.Consistency.First-ordertheories 74
vii
Chapter7 Theformal theory ofcanonicalcalculi(CC*) 76
7.1.Approachingintuitively 76
7.2.Thecanonicalcalculus1:* 78
7.3.Truthassignment 81
7.4.Undecidability:Church'sTheorem 83
Chapter8 Completenesswithrespect tonegation 85
8.1.TheformaltheoryCC 85
8.2.Diagonalization 87
8.3.Extensionsanddiscussions 90
Chapter9 Consistencyunprovable 93
9.1.Preparatorywork 93
9.2.TheproofoftheunprovabilityofCons 94
Chapter10 Settheory 98
10.1.Setsandclasses 9R
10.2.Relationsandfunctions 103
10.3.Ordinal,natural,andcardinalnumbers 106
10.4.Applications 110
References 114
Index 116
List ofsymbols 122
APPENDIX (LectureNotes): Type-TheoeticalExtensional
andIntensionalLogic 123
Part1:ExtensionalLogic 127
Part2:Montague's IntensionalLogic 148
References 182
viii
Chapter 1 INTRODUCTION
1.1 The Subject Matter ofMetalogic
Modern logic is not a single theory. It consists of a considerable (and ever growing)
number oflogical systems (oftencalled - regrettably- logics). Metalogic is the sci
enceoflogicalsystems.Itstheoremsarestatementseitheronaparticularlogicalsystem
or about some interrelations between certain logical systems. In fact, every system of
logichas its own metalogic that, among others, describes the construction of the sys
tem, investigates the structure of proofs in the system, and so on. Many theorems
usuallyknownas"lawsoflogic" are,infact, metalogicalstatements. Forexample, the
statement sayingthatmodus ponens is avalidruleofinference- say,in classical first
order logic - is a metalogical theorem about a system of logic. A more deeper meta
logical theorem about theclassical frrst-orderlogictellsus that acertainlogical calcu
lusissound andcomplete withrespecttotheset-theoreticalsemantics ofthissystemof
logic.
Remark. Itisassumedherethatthisessayisnotthefirstencounterofthereaderwithlogic(neither
withmathematics),sothattheexamplesaboveandsomesimilaroneslateron,areintelligible.However,
thisdoesnotmeanthattheunderstandingofthisbookisdependentonsomepreviousknowledgeoflogicor
mathematics.
Another veryimportant task of metalogicis toanswer the problem: How is logic
possible? Togivesomeinsight intotheseriousnessofthis question, letmerefertothe
well-knownfactthatmodernlogicusesmathematical meansintensively,whereas mod
em mathematical theoriesarebased onsomesystem(s)oflogic.Is therea wayoutfrom
this - seemingly - vicious circle? This is thefoundational problem of logic, and its
solutionisthetaskoftheintroductorypart ofmetalogic.The greaterpart ofthis essay
isdevotedtothisfoundationalproblem.
The device we shall apply in the course consists in dealing alternatively with
mathematical and logical knowledge, without drawing a sharp borderline between
mathematical and logical means. No knowledge ofa logical ora mathematical theory
will be presupposed. The onlypresupposition we shall exploit will be that the reader
knows the elements of the grammar of some natural language (e.g., hislher mother
tongue), can read, write and count, and is able tofollowso-calledformal expositions.
(Ofcourse,theabilitylastmentionedassumes- tacitly- someskills whichcanbebest
mastered inmathematics.)
The introductory part of metalogic is similar to the discipline created by David
Hilbert, called metamathematics. (See, e.g., HILBERT 1926.) The aim of metamathe
matics was tofindasolidfoundationofsomemathematical systems (e.g.,number the
ory,settheory)byusingonlyso-calledfinite means.InHilbert'sview,fmitemathemat-
1
ics is sufficient for proving the consistency of transfinite (non-fmite, infinite) mathe
matical theories. In asense, thefoundationofalogicalcalculus (which is sufficientfor
mathematicalproofsinmostcases) wasincludedinHilbert'sprogramme. (perhaps this
isthereason thatscientists whothinkthatmodem logicisjustabranch ofmathematics
- calledmathematicallogic- oftensaymetamathematics insteadofmetalogic.)
Metalogicis notparticularly interested inthefoundation ofmathematical theories.
Itisinterested inthefoundation oflogicalsystems.Initsbeginning, metalogic willuse
verysimple, elementary means which could be qualified asfinite ones. However, the
author does not dare to draw a borderline between thefinite and the transfinite. We
shall proceed starting with simple means and using them to construct more complex
systems.
Every system ofmodem logic is based on aformal language. As a consequence,
our investigation will start with the problem: How is it possible to construct a lan
guage? Someofourresults maytum outtobeapplicable notonlyforformallanguages
butfornatural languagesaswell.
This essay is almost self-contained. Two exceptions where most proofs will be
omitted arethefirst-order calculus oflogic(calledhereQC,Chapter 6)and settheory
(Chapter 10).The author assumes thatthedetailed studyofthesedisciplines isthetask
ofothercoursesinlogic.
Technicalremarks. Detailed informationabout thestructureofthis bookistobe
foundin the Table ofContents. Attheend ofthebook, theIndex and the List ofSym
bolshelpsthereadertofindthedefinitionsofreferrednotionsandsymbols.Intheinner
reference, the abbreviations 'Ch.', 'Sect.', 'Def.', and 'Th.' are used for 'Chapter',
'Section', 'Definition',and 'Theorem',respectively.References forliterature aregiven,
as usually, bythe (last) name of the author(s) and the year of publication (e.g.,
'HILBERT 1926'); the full data are to be found in the References (at the end of the
book).- No technical term and symbol will be used without definition in this book.
At theendofalongerproofordefinition,abullet '.' indicates theend. - A consider
able part ofthe material in this bookisborrowed froma workbythe author written in
Hungarian (RUZSA1988).
1.2 Basic postulates on languages
On thebasis ofexperiences gained fromnatural languages, wecan formulate our first
postulateconcerning languages:
(L1)Each language isbasedonafinitesupply ofprimitiveobjects. Thissupply is
called thealphabetofthelanguage.
Inthecaseofformallanguages, thispostulatewillserveasanormative rule.
2
In spoken languages, the objects ofthe alphabet are called phonemes, in written
languages letters (orcharacters).Thephonemesare(atleasttheoretically)perceivable
as sound events, and the lettersas visible (writtenor printed) paint marks on the sur
faceofsomematerial (e.g.,paper) - this is thedifferencebetween spoken and written
languages.Weshallbeinterestedonlyinwrittenlanguages.
Inusing a language, we formfmite strings fromthe members ofthealphabet, al
lowing therepetition (repeatedoccurrence)ofthemembers ofthealphabet.In thecase
ofspokenlanguages,themembersofsuch a string areordered bytheirtemporal con
secution.Inthecaseofwrittenlanguages, theorderisregulated bycertainconventions
of writing. (The author assumes that no further details are necessary on this point.)
Finite strings formed from the members,of the alphabet are called expressions - or,
briefly,words - ofthat language. The reader maycomment herethat not all possible
strings ofletters (orofphonemes) are used in a (natural) language.Onlya part ofthe
totalityofpossibleexpressionsisuseful;thispartiscalledthetotalityofwell-formedor
meaningfulexpressions. (Butacounterexamplemayoccurin aformallanguage!)Beit
as itis, todefinethewell-formedexpressions, wecertainlymust refertothe totalityof
all expressions.Thus,ournotionofexpressions(words)isnotsuperfluous.
Note that one-member strings are not excluded from the totality of expressions.
Hence, thealphabet ofa language is always a part ofthe totalityofwords ofthat lan
guage. Moreover, by technical reasons, it is useful - although not indispensable - to
include the empty string.called empty word amongst the words'of a language. (We
shallrecurtothisproblemlateron.)
Oursecondpostulateisagainbasedone~perienceswithnaturallanguages:
(L2) Ifwe know the alphabet ofa language, we know the totality ofits words
(expressions). In other words: The alphabet ofa language uniquely determines the
totalityofitswords.
Toavoidphilosophicalandlogicaldifficultiesweneedthethirdpostulate:
(L3)The expressionsofany languageare idealobjects whichare sensiblyrealiz
able or representable (in any copies) by physical objects (paint marks) or events
(soundevents orothers).
This assumption is not a graver one than the view that natural numbers are ideal ob
jects. Anditmakesintelligibletheuseofalanguageincommunication.,
Thus, in speaking of an expression (of a language) we speak of an ideal object
ratherthan ofaperceivableobject,i.e.,aconcreterepresentationofthatidealobject.In
other words: Statements on an expression refer to all of its possible realizations, not
onlytosomeparticular representation oftheexpression. (Referencetoa concretecopy
ofanexpressionmustbeindicatedexplicitly.)