Proof Analysis: A Contribution to Hilbert’s Last Problem

280 Pages·2011·1.66 MB·english

by  Sara Negri

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ProofAnalysis
This book continues from where the authors’ previous book,
StructuralProofTheory,ended.Itpresentsanextensionofthemethods
of analysis of proofs in pure logic to elementary axiomatic systems
and to what is known as philosophical logic. A self-contained brief
introduction to the proof theory of pure logic that serves both the
mathematicallyandphilosophicallyorientedreaderisincluded.The
methodisbuiltupgradually,withexamplesdrawnfromtheoriesof
order,latticetheoryandelementarygeometry.Theaimis,ineachof
the examples, to help the reader grasp the combinatorial behaviour
ofanaxiomsystem,whichtypicallyleadstodecidabilityresults.The
lastpartpresents,asanapplicationandextensionofallthatprecedes
it,aproof-theoreticalapproachtotheKripkesemanticsofmodaland
relatedlogics,withagreatnumberofnewresults,providingessential
readingformathematicalandphilosophicallogicians.
sara negri isDocentofLogicattheUniversityofHelsinki.Sheis
theco-authorofStructuralProofTheory (Cambridge,2001,withJan
vonPlato)andshehaspublishedanumberofpapersonmathematical
andphilosophicallogic.
jan von plato is Professor of Philosophy at the Univer-
sity of Helsinki. He is the author of Creating Modern Probability
(Cambridge,1994)andtheco-author(withSaraNegri)ofStructural
ProofTheory (Cambridge,2001),andhehaspublishedanumberof
papersonlogicandepistemology.

Proof Analysis
A Contribution to Hilbert’s Last Problem
sara negri
jan von plato
UniversityofHelsinki
cambridge university press
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Informationonthistitle:www.cambridge.org/9781107008953
(cid:1)C CambridgeUniversityPress2011
Thispublicationisincopyright.Subjecttostatutoryexception
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permissionofCambridgeUniversityPress.
Firstpublished2011
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Negri,Sara,1967–
Proofanalysis:acontributiontoHilbert’slastproblem/SaraNegri,JanvonPlato.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-1-107-00895-3(hardback)
1.Prooftheory. I.VonPlato,Jan. II.Title.
QA9.54.N438 2011
511.3(cid:2)6–dc23 2011023026
ISBN978-1-107-00895-3Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
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Contents
Preface [pageix]
Prologue:Hilbert’slastproblem [1]
1 Introduction [3]
1.1 Theideaofaproof [3]
1.2 Proofanalysis:anintroductoryexample [4]
1.3 Outline [9]
part i proof systems based on natural
deduction
2 Rulesofproof:naturaldeduction [17]
2.1 Naturaldeductionwithgeneraleliminationrules [17]
2.2 Normalizationofderivations [23]
2.3 Fromaxiomstorulesofproof [29]
2.4 Thetheoryofequality [32]
2.5 Predicatelogicwithequalityanditswordproblem [35]
NotestoChapter2 [37]
3 Axiomaticsystems [39]
3.1 Organizationofanaxiomatization [39]
3.2 Relationaltheoriesandexistentialaxioms [46]
NotestoChapter3 [49]
4 Orderandlatticetheory [50]
4.1 Orderrelations [50]
4.2 Latticetheory [52]
4.3 Thewordproblemforgroupoids [57]
4.4 Rulesystemswitheigenvariables [62]
NotestoChapter4 [67]
5 Theorieswithexistenceaxioms [68]
5.1 Existenceinnaturaldeduction [68]
5.2 Theoriesofequalityandorderagain [71]
5.3 Relationallatticetheory [73]
NotestoChapter5 [82]
v
vi Contents
part ii proof systems based on sequent
calculus
6 Rulesofproof:sequentcalculus [85]
6.1 Fromnaturaldeductiontosequentcalculus [85]
6.2 Extensionsofsequentcalculus [97]
6.3 Predicatelogicwithequality [106]
6.4 Herbrand’stheoremforuniversaltheories [110]
NotestoChapter6 [111]
7 Linearorder [113]
7.1 PartialorderandSzpilrajn’stheorem [113]
7.2 Thewordproblemforlinearorder [119]
7.3 Linearlattices [123]
NotestoChapter7 [128]
part iii proof systems for geometric theories
8 Geometrictheories [133]
8.1 Systemsofgeometricrules [133]
8.2 Prooftheoryofgeometrictheories [138]
8.3 Barr’stheorem [144]
NotestoChapter8 [145]
9 Classicalandintuitionisticaxiomatics [147]
9.1 Thedualityofclassicalandconstructivenotionsandproofs [147]
9.2 Fromgeometrictoco-geometricaxiomsandrules [150]
9.3 Dualityofdependenttypesanddegeneratecases [155]
NotestoChapter9 [156]
10 Proofanalysisinelementarygeometry [157]
10.1 Projectivegeometry [157]
10.2 Affinegeometry [173]
10.3 Examplesofproofanalysisingeometry [180]
NotestoChapter10 [181]
part iv proof systems for non-classical
logics
11 Modallogic [185]
11.1 Thelanguageandaxiomsofmodallogic [185]
11.2 Kripkesemantics [187]
11.3 FormalKripkesemantics [189]
11.4 Structuralpropertiesofmodalcalculi [193]
11.5 Decidability [201]
Contents vii
11.6 Modalcalculiwithequality,undefinabilityresults [210]
11.7 Completeness [213]
NotestoChapter11 [219]
12 Quantifiedmodallogic,provabilitylogic,&other
non-classicallogics [222]
12.1 Addingthequantifiers [222]
12.2 Provabilitylogic [234]
12.3 Intermediatelogics [239]
12.4 Substructurallogics [249]
NotestoChapter12 [251]
Bibliography [254]
Indexofnames [262]
Indexofsubjects [264]