Proof Theory and Algebra in Logic

164 Pages·2019·2.221 MB·English

by  Hiroakira Ono

Short Textbooks in Logic
Hiroakira Ono
Proof Theory and
Algebra in Logic
Short Textbooks in Logic
Series Editors
Fenrong Liu, Department of Philosophy, Tsinghua University, Beijing, China
HiroakiraOno,SchoolofInformationScience,JapanAdvancedInstituteofScience
and Technology, Nomi City, Ishikawa, Japan
EricPacuit,DepartmentofPhilosophy,UniversityofMaryland,CollegePark,MD,
USA
Jeremy Seligman, University of Auckland, Auckland, New Zealand
Thisisasystematicallydesignedbookseriesthatcomprisesoftextbooksonvarious
topicsinlogic.Thougheachbookcanbereadindependently,theseriesasawhole
gives readers a comprehensive view of logic of the present time. Each book in the
series is written clearly and concisely, and at the same time supplies plenty of
well-planned examples and exercises to the point. The series is also aimed at
providing readers with adequate explanations of scope and motivation of topics.
Thetopicsdiscussedintheseriesrangefrommathematicalandphilosophicallogic
to logical methods applied to computer science. Some will be introductory and
some other will be advanced but not too much specific. Its targeted readers are
advanced undergraduate as well as graduate students in philosophy, mathematics,
computer science and the related fields. The series is also suitable for self-taught
learning.
More information about this series at http://www.springer.com/series/15706
Hiroakira Ono
Proof Theory and Algebra
in Logic
123
Hiroakira Ono
JapanAdvancedInstitute ofScience
andTechnology
Nomi, Japan
ISSN 2522-5480 ISSN 2522-5499 (electronic)
Short Textbooks inLogic
ISBN978-981-13-7996-3 ISBN978-981-13-7997-0 (eBook)
https://doi.org/10.1007/978-981-13-7997-0
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Preface
This is an introductory textbook on proof theory and algebra in logic, focusing on
bothinterrelatingandcomplementaryfeaturesofthesetwotopics.Prooftheoryand
algebra are two major mathematical branches of study in nonclassical logics, each
of which is a core of syntactic and semantical approaches, respectively. Syntactic
study of logic is based on combinatorial argument and symbolic manipulation,
which will produce deep and concrete results. In contrast, standard mathematical
arguments will be fully used in the semantical study, from which general and
abstractresultscanbeoftenderived.Thoughthesetwoapproacheshavebeenoften
discussed separately, the recent development of study, in particular, in both modal
and substructural logics, has shown that intrinsic connections exist between them,
whicharemuchmorecloserthanwhatwethoughtbefore.Thus,thepresentbookis
intended to cover these two approaches together but concisely, and to foster
understandingbetweenthem.This,wehope,willmakeitpossibleforreaderstoget
things in proper perspective.
The book will be used as an introductory textbook on logic for a course at the
undergraduateorgraduatelevel.Wetriedtomakeitaselementaryaspossibleand
to minimize mathematical prerequisites. Many examples will be taken from topics
ofvariousnonclassicallogicstoshowhowthesetechniquesareappliedtothefield.
Hence,thebookcanbeusedalsoasaprimaryintroductiontononclassicallogicfor
a course at the undergraduate or graduate level, which covers modal logics,
many-valued logics, superintuitionistic, and substructural logics.
Though the present textbook is intended to explain both proof theory and
algebraicstudyofnonclassicallogics,weorganizeitsothatreaderscanreadPartI
and Part II independently without much effort. In particular, after a brief look at
preliminaries in Sect. 1.1, one may start to read the book from Part II in order to
study basics of algebraic logic and universal algebra with examples of their
applicationsinlogic.TolookupnotionsanddefinitionsgiveninPartIeasilywhen
necessary, we prepare a detailed index at the end of the book. As this is an
introductorytextbook,wehavenotgivenadetailedreferenceinordernottodisturb
smoothreading.Instead,wepreparealistofmajorbooksonthetopicsattheendof
v
vi Preface
each Part as a guide for further reading, and also a list of basic materials and
primary sources in the references at the end.
Thepresentbookoriginatedpartlyfrommysurveypapers[Ono98,Ono10b]on
proof theory in nonclassical logic and algebraic logic, respectively, and also from
notes of my talks for undergraduate and graduate students at various places in
recent years. They include Tbilisi Summer School in 2011, State University of
Campinas, University of Tehran, National Taiwan University, Southwest
University at Chongqing, Tsinghua University and Peking University at Beijing.
I am deeply grateful to people who offered me opportunities for giving such talks.
I am indebted to many friends and colleagues for their valuable discussions,
suggestions, and encouragement on various occasions. I would like to thank
Tomasz Kowalski for his support and encouragement for many years. I thank also
Majid Alizadeh, Katsuhiko Sano, and anonymous reviewers for their helpful sug-
gestionsandimportantcommentsonearlyversionofthemanuscript,andalsoRyo
Hatano for his technical help, in particular, in drawing diagrams. As the present
Short Textbooks series in Logic was initiated and promoted in cooperation with
FenrongLiu,takingthisopportunityIwouldliketothankher.Finally,Iwouldlike
to express gratitude to my wife Kazue for her constant support.
Kanazawa, Japan Hiroakira Ono
February 2019
Contents
Part I Proof Theory
1 Sequent Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Prologue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Sequent Systems LK for Classical Logic . . . . . . . . . . . . . . . . . 7
1.3 Completeness and Cut Elimination. . . . . . . . . . . . . . . . . . . . . . 17
1.4 Sequent System LJ for Intuitionistic Logic. . . . . . . . . . . . . . . . 19
2 Cut Elimination for Sequent Systems . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Cut Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Subformula Property. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Proof-Theoretic Analysis of Logical Properties. . . . . . . . . . . . . . . . 35
3.1 Decidability of Intuitionistic Logic. . . . . . . . . . . . . . . . . . . . . . 35
3.2 Disjunction Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Craig’s Interpolation Property . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.4 Glivenko’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Modal and Substructural Logics. . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1 Standard Sequent Systems for Normal Modal Logics . . . . . . . . 48
4.2 Roles of Structural Rules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3 Sequent Systems for Basic Substructural Logics. . . . . . . . . . . . 54
5 Deducibility and Axiomatic Extensions . . . . . . . . . . . . . . . . . . . . . . 61
5.1 Deducibility and Deduction Theorem. . . . . . . . . . . . . . . . . . . . 61
5.2 Local Deduction Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.3 Axiomatic Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.4 Framework for Substructural Logics and Modal Logics. . . . . . . 69
5.5 A View of Substructural Logics. . . . . . . . . . . . . . . . . . . . . . . . 71
vii
viii Contents
Part II Algebra in Logic
6 From Algebra to Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.1 Lattices and Boolean Algebras. . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 Subalgebras, Homomorphisms and Direct Products. . . . . . . . . . 83
6.3 Representations of Boolean Algebras . . . . . . . . . . . . . . . . . . . . 86
6.4 Algebraic Completeness of Classical Logic . . . . . . . . . . . . . . . 88
6.5 Many-Valued Chains and the Law of Residuation . . . . . . . . . . 89
7 Basics of Algebraic Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.1 Heyting Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.2 Lindenbaum-Tarski Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.3 Locally Finite Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.4 Finite Embeddability Property and Finite Model Property . . . . . 105
7.5 Canonical Extensions of Heyting Algebras. . . . . . . . . . . . . . . . 107
8 Logics and Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.1 Lattice Structure of Superintuitionistic Logics. . . . . . . . . . . . . . 113
8.2 The Variety HA of All Heyting Algebras. . . . . . . . . . . . . . . . . 116
8.3 Subvarieties of HA and Superintuitionistic Logics. . . . . . . . . . . 119
8.4 Subdirect Representation Theorem. . . . . . . . . . . . . . . . . . . . . . 123
8.5 Algebraic Aspects of Logical Properties. . . . . . . . . . . . . . . . . . 125
9 Residuated Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
9.1 Residuated Lattices and FL-Algebras. . . . . . . . . . . . . . . . . . . . 129
9.2 FL-Algebras and Substructural Logics . . . . . . . . . . . . . . . . . . . 133
9.3 Residuations Over the Unit Interval . . . . . . . . . . . . . . . . . . . . . 135
10 Modal Algebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
10.1 Modal Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
10.2 Canonical Extensions and Jónsson-Tarski Theorem. . . . . . . . . . 142
10.3 Kripke Semantics from Algebraic Viewpoint . . . . . . . . . . . . . . 143
10.4 Gödel Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
References.... .... .... .... ..... .... .... .... .... .... ..... .... 151
Further Reading for Part I. ..... .... .... .... .... .... ..... .... 155
Further Reading for Part II..... .... .... .... .... .... ..... .... 156
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 157
Part I
Proof Theory
Prooftheoryisasyntacticcoreofmathematicalstudyofproofsinformalsystems.
The main concern of proof theory is to study and analyze structures of proofs. A
typical question in proof theory is “what kind of proofs a given formula will have
as long as it is provable” and in particular “whether it has a normal (or standard)
proofornot.”InPartIofthepresenttextbook,wewillintroducesequentsystems
forvariouslogicsfromclassicalandintuitionisticlogicstomodalandsubstructural
logics.Thoughthereexistalternativeapproachestoformalizetheselogics,likenat-
uraldeduction and tableau calculi,wewillfocusour attention onsequentsystems
andwilldeveloptheprooftheorybasedonthem.Thereasonisthatsequentsystems
arecarefullydesignedespeciallyforanalysisofproofs,andhencelogicalproperties
specifictothelogicrepresentedbyagivensequentsystemareinsharplyreflectedby
proofs,inparticularwhencuteliminationholds.Thus,formalizationbysequentsys-
temswillberegardedasthemostilluminatingandinformativewaywhenwedevelop
proof-theoreticstudyofnonclassicallogics.PartIwillbedevotedtopresentsequent
systems for various nonclassical logics and to develop proof-theoretic analysis in
thesesystems.
AsacentralissueofthisPartI,wewilldiscusscuteliminationtheoremindetail,
which was proved by G. Gentzen. The theorem says that when classical logic is
formalizedinasequentsystemcalledLK,everyformulahasanormalproof(infact,
a proof without any application of cut rule)as long as it is provable. We present a
proofofcuteliminationinaslightlysimplifiedway.Animportantconsequenceofcut
eliminationisthatwecanextractinterestinglogicalpropertiesbyanalyzingstructures
ofnormalproofs,sincesuchproofsoftencontainimplicitlynecessaryinformationon
theselogicalproperties.CuteliminationholdsnotonlyforLKofclassicallogicbut
alsomanyothersequentsystemsforbasicmodalandsubstructurallogics,including
intuitionistic logic, and consequently, we can obtain many interesting and useful
logical properties of these logics as consequences of cut elimination. As this is an
introductorytextbook,wedonottreatfurtherextensionsofsequentsystemswhich
have been actively studied in recent years, such as display calculi, nested sequent
systems,hypersequentsystemsandlabeledsequentsystems.