Mathematical logic: Foundations for information science

274 Pages·2010·0.965 MB·English

by  Li W.

Progress in Computer Science and Applied Logic
Volume 25
Editor:
John C. Cherniavsky, National Science Foundation
Associate Editors
Robert Constable, Cornell University
Jean Gallier, University of Pennsylvania
Richard Platek, Cornell University
Richard Statman, Carnegie-Mellon University

Mathematical Logic
Foundations for
Information Science
Wei Li
Birkhäuser
Basel · Boston · Berlin
Author:
Wei Li
State Key Laboratory of Software Development Environment
Beihang University
37 Xueyuan Road, Haidian District
Beijing 100191
China
e-mail: [email protected]
2000 Mathematics Subject Classification: 83C05, 83C35, 58J35, 58J45, 58J05, 53C80
Library of Congress Control Number: 2009940118
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ISBN 978-3-7643-9976-4 Birkhäuser Verlag AG, Basel – Boston – Berlin
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English version based on, 数理逻辑:基本原理与形式演算 (Mathematical Logic – Basic
Principles and Formal Calculus), 978-7-03020096-9, Science Press, Beijing, China, 2007.
ISBN 978-3-7643-9976-4 e-ISBN 978-3-7643-9977-1
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Contents
Preface ix
Chapter1 SyntaxofFirst-OrderLanguages 1
1.1 Symbolsoffirst-orderlanguages . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Logicalformulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Freevariablesandsubstitutions . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Go¨deltermsofformulas . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Proofbystructuralinduction . . . . . . . . . . . . . . . . . . . . . . . . 15
Chapter2 ModelsofFirst-OrderLanguages 19
2.1 Domainsandinterpretations . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Assignmentsandmodels . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 Semanticsofterms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Semanticsoflogicalconnectivesymbols . . . . . . . . . . . . . . . . . . 25
2.5 Semanticsofformulas . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6 Satisfiabilityandvalidity . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.7 Validformulaswith↔ . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.8 Hintikkaset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.9 Herbrandmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.10 Herbrandmodelwithvariables . . . . . . . . . . . . . . . . . . . . . . . 38
2.11 Substitutionlemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.12 Theoremofisomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Chapter3 FormalInferenceSystems 45
3.1 Ginferencesystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 Inferencetrees,prooftreesandprovablesequents . . . . . . . . . . . . . 52
3.3 SoundnessoftheGinferencesystem . . . . . . . . . . . . . . . . . . . . 57
3.4 Compactnessandconsistency. . . . . . . . . . . . . . . . . . . . . . . . 61
3.5 CompletenessoftheGinferencesystem . . . . . . . . . . . . . . . . . . 63
3.6 Somecommonlyusedinferencerules . . . . . . . . . . . . . . . . . . . 66
3.7 Prooftheoryandmodeltheory . . . . . . . . . . . . . . . . . . . . . . . 68
Chapter4 Computability&Representability 71
4.1 Formaltheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2 Elementaryarithmetictheory . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 P-kernelonN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.4 Church-Turingthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5 Problemofrepresentability . . . . . . . . . . . . . . . . . . . . . . . . . 81
vi Contents
4.6 StatesofP-kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.7 OperationalcalculusofP-kernel . . . . . . . . . . . . . . . . . . . . . . 84
4.8 Representationsofstatements. . . . . . . . . . . . . . . . . . . . . . . . 86
4.9 Representabilitytheorem . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Chapter5 Go¨delTheorems 97
5.1 Self-referentialproposition . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2 Decidablesets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.3 FixedpointequationinΠ . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4 Go¨del’sincompletenesstheorem . . . . . . . . . . . . . . . . . . . . . . 107
5.5 Go¨del’sconsistencytheorem . . . . . . . . . . . . . . . . . . . . . . . . 109
5.6 Haltingproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Chapter6 SequencesofFormalTheories 117
6.1 Twoexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.2 Sequencesofformaltheories . . . . . . . . . . . . . . . . . . . . . . . . 122
6.3 Proschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.4 Resolventsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.5 Defaultexpansionsequences . . . . . . . . . . . . . . . . . . . . . . . . 130
6.6 Forcingsequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.7 Discussionsonproschemes . . . . . . . . . . . . . . . . . . . . . . . . . 136
Chapter7 RevisionCalculus 139
7.1 Necessaryantecedentsofformalconsequences . . . . . . . . . . . . . . 140
7.2 Newconjecturesandnewaxioms . . . . . . . . . . . . . . . . . . . . . . 143
7.3 Refutationbyfactsandmaximalcontraction . . . . . . . . . . . . . . . . 144
7.4 R-calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.5 Someexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.6 Specialtheoryofrelativity . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.7 Darwin’stheoryofevolution . . . . . . . . . . . . . . . . . . . . . . . . 156
7.8 ReachabilityofR-calculus . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.9 SoundnessandcompletenessofR-calculus . . . . . . . . . . . . . . . . 163
7.10 Basictheoremoftesting . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Chapter8 VersionSequences 169
8.1 Versionsandversionsequences . . . . . . . . . . . . . . . . . . . . . . . 171
8.2 TheProschemeOPEN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
8.3 Convergenceoftheproscheme . . . . . . . . . . . . . . . . . . . . . . . 176
8.4 Commutativityoftheproscheme . . . . . . . . . . . . . . . . . . . . . . 178
8.5 Independenceoftheproscheme. . . . . . . . . . . . . . . . . . . . . . . 180
8.6 Reliableproschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Contents vii
Chapter9 InductiveInference 187
9.1 Groundterms,basicsentences,andbasicinstances . . . . . . . . . . . . 190
9.2 InductiveinferencesystemA . . . . . . . . . . . . . . . . . . . . . . . . 192
9.3 Inductiveversionsandinductiveprocess . . . . . . . . . . . . . . . . . . 197
9.4 TheProschemeGUINA . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
9.5 ConvergenceoftheproschemeGUINA . . . . . . . . . . . . . . . . . . . 204
9.6 CommutativityoftheproschemeGUINA . . . . . . . . . . . . . . . . . . 206
9.7 IndependenceoftheproschemeGUINA . . . . . . . . . . . . . . . . . . . 207
Chapter10 WorkflowsforScientificDiscovery 209
10.1 Threelanguageenvironments . . . . . . . . . . . . . . . . . . . . . . . . 209
10.2 Basicprinciplesofthemeta-languageenvironment . . . . . . . . . . . . 213
10.3 Axiomatization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
10.4 Formalmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
10.5 Workflowofscientificresearch . . . . . . . . . . . . . . . . . . . . . . . 225
Appendix1 SetsandMaps 229
Appendix2 SubstitutionLemmaandItsProof 233
Appendix3 ProofoftheRepresentabilityTheorem 237
A3.1 RepresentationofthewhilestatementinΠ . . . . . . . . . . . . . . . . 237
A3.2 RepresentabilityoftheP-procedurebody . . . . . . . . . . . . . . . . . 244
Bibliography 253
Index 257

Preface
Classicalmathematicallogicisconsideredtobeanimportantcomponentofthefounda-
tion of mathematics. It is the study of mathematical methods, especially the properties
of axiom systems and the structure of proofs. The core of mathematical logic consists
ofdefiningthesyntaxoffirst-orderlanguages,studyingtheirmodels,formalizinglogical
inferenceandprovingitssoundnessandcompleteness.Italsocoversthetheoryofcom-
putabilityandGo¨del’sincompletenesstheorems.Thisprocessofabstractionstartedinthe
late19thCenturyandwasessentiallycompletedby1950.
In 1990, I began to give courses on mathematical logic. This teaching experience
mademerealizethat,althoughdeductivelogicwaswellanalyzed,theprocessofaxioma-
tizationhadnotbeenstudiedindepth.Severalyearslater,Iorganizedaseriesofseminars
asanensuingeffort.Thefirstfiveseminarscoveredclassicalmathematicallogicandthe
restwereapreliminaryoutlineoftheformaltheoryofaxiomatization.
As my understanding of mathematical logic became deeper, my desire to analyze
andformalizetheprocessofaxiomatizationbecamemoreintense.Ialsosawtheinfluence
ofmathematicallogicininformationtechnologyandscientificresearch.Thisinspiredme
towriteabookforstudentslivingintheinformationsociety.
The computer was invented in the 1940’s and high-level programming languages
weredefinedandimplementedsoonafterwards.Computersciencehasdevelopedrapidly
sincethen.Thisexertedaprofoundinfluenceonmathematicallogic,becauseitsconcepts
and theories were extensively applied. However, the development of computer science
has,inturn,madenewdemandsonmathematicallogic,whichhavebeenthefocusofmy
research and the motivation for this book. This motivation is guided by two considera-
tions.
Firstly,mathematicallogicwasoriginallyageneraltheoryaboutaxiomsystemsand
proofsinmathematics,butnow,itsconceptsandtheorieshavebeenadoptedbycomputer
scienceandhaveplayedaprincipalguidingroleinthedesignandimplementationofboth
softwareandhardware.
For example, the method of structural induction was invented to define the gram-
mar of first-order languages, but it is now used to define programming languages. This
suggeststhatthestudyofmathematicallogiccanbeappliedtomanyareasofcomputer
science.
AnotherexampleisgivenbyPeano’stheoryofarithmetic.Thisisaformaltheory
inafirst-orderlanguage,whilethenaturalnumbersystemisamodelofthattheory.The
distinction is essential in mathematical logic, because it is necessary in order to prove
importanttheoremssuchasthoseofGo¨del.However,manypeopleoutsidethisfieldfind
ithardtoseetheutilityofmakingthisdistinction.
But in computer science, it is vital to differentiate between a high-level program-
ming language and compiled executable codes. The difference between programs and
their compiled executables is precisely the same as that made between first-order lan-