Proof Theory and Algebra in Logic PDF
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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 ©SpringerNatureSingaporePteLtd.2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore 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.
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