Rethinking Logic: Logic in Relation to Mathematics, Evolution, and Method

391 Pages·2013·4.409 MB·English

by  Carlo Cellucci (auth.)

Logic, Argumentation & Reasoning 1
Carlo Cellucci
Rethinking Logic:
Logic in Relation
to Mathematics,
Evolution,
and Method
Logic, Argumentation & Reasoning
Interdisciplinary Perspectives from the Humanities
and Social Sciences
Volume 1
SeriesEditor
ShahidRahman
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sciences. The series isopen towards research from the Analyticand theContinental traditions,
andhasfourmainfocusareas:Pragmaticmodelsandstudiesthatdevelopadynamicapproachto
reasoning inwhichargumentation isstructured asaninteractionorasagame,inwhichtwoor
moreparticipantsplaymovesdefinedbythetypeofargumentationinquestion,communication,
languageandtechniquesofargumentation:studiesbetweenthepractical andtheoretical dimen-
sionsof argumentation, aswellas therelationships betweenargumentation and othermodes of
communication,reception,persuasionandpower:studiesinwhichreasoningpracticeisconsidered
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understanding what makes an argument performative, Diachronic transformations of reasoning
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performanceanditseffectiveness.
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Carlo Cellucci
Rethinking Logic: Logic
in Relation to Mathematics,
Evolution, and Method
123
CarloCellucci
SapienzaUniversityofRome
Rome
Italy
ISBN978-94-007-6090-5 ISBN978-94-007-6091-2(eBook)
DOI10.1007/978-94-007-6091-2
SpringerDordrechtHeidelbergNewYorkLondon
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Preface
Despite strenuous efforts by its proponents, the contemporary form of logic,
mathematical logic, has generally failed to convince mathematicians, natural sci-
entistsandhumanscientistsofitsrelevancetotheirwork,increasinglysointhelast
fewdecades.Thiscontrastswiththereputationlogicenjoyedinantiquity,notonly
asoneofthemainpartsofphilosophy,butalsoasasupplierofinstrumentsforthe
sciences.
Thepurposeofthisbookistoexplainhowthepresentconditionoflogiccame
about and to propose an alternative to it. To this end, the book first gives an
overviewofhowlogicanditsrelationtothescientificmethodhavebeenconceived
in antiquity and in the modern age, because this provides indications for a new
approachtothesubject.Thenthebookproposesanewviewoflogicanditsrelation
to evolution, language, reason, method and knowledge, particularly mathematical
knowledge.Italsoproposesanewviewofphilosophyanditsrelationtoknowledge,
because seeing logic in a wider context helps to place it on a more satisfactory
basis. In terms of the proposed new view, logic is primarily a logic of discovery.
Accordingly,thebookdealswiththerulesofdiscovery.
I am grateful to several people for the help they gave me in many ways.
Somereadachapterandmaderemarks.Someraisedquestionsincorrespondence.
Some made comments on views expressed in this book, presented at seminars or
conferences. For their help in whatever form, I am especially indebted to Arthur
Bierman, Mirella Capozzi, Riccardo Chiaradonna, Cesare Cozzo, Philip J. Davis,
Michèle Friend, Donald Gillies, Norma Goethe, Emily Grosholz, Reuben Hersh,
JeffKochan,ColinMcGinn,DanielleMacbeth,JulienMurzi,DanNesher,Marwan
Rashed, Stephen Read, Andrea Reichenberger, Stephen P. Schwartz, Giovanna
Sillitti, Hourya Benis Sinaceur, Fabio Sterpetti, Philip Sullivan, Robert Thomas,
Mario Vassalle, Johan van Benthem, Jan von Plato. This does not mean that they
share the views expressed in this book or are in any way responsible for any
remaininginaccuracies.
Iamalsogratefultotwoanonymousrefereesfortheircomments,andtoArlette
Dupuisforreadingthemanuscriptandsuggestingseverallinguisticimprovements.
Moreover, I want to thank the Series Editor, Shahid Rahman, for supporting the
v
vi Preface
bookidea, forhelpingoutin manyways, andforselectingthe bookto inaugurate
theseriesLogic,Argumentation& Reasoning,andChristiLueatSpringerforher
assistanceinthepublicationprocess.
TheviewsexpressedinthebookareadevelopmentofthosepresentedinCellucci
1998,2003,2008a,2008b,2012.Chapter18isarevisedversionofCellucci2011.
IthankCambridgeScholarsPublishingfortheirkindpermissiontousethismaterial.
Awordaboutpronounsandgender.Constantuseof‘heorshe’maybeclumsy,
while constantuse of ‘she’ may give rise to misunderstandings.Therefore,I have
preferred to use the generic ‘he’ while stipulating here that I mean it to refer to
personsofbothgenders.
Contents
1 Introduction................................................................. 1
1.1 TheIntendedPurposeofMathematicalLogic..................... 1
1.2 TheBasicAssumptionsofMathematicalLogic................... 1
1.3 InadequacyoftheBasicAssumptionsofMathematicalLogic... 3
1.4 TheReceptionoftheLimitativeResults........................... 5
1.5 MathematicsandAxiomaticFormalTheories..................... 6
1.6 MathematicsandtheLossofCertainty............................ 7
1.7 TheTop-DownandBottom-UpApproachestoMathematics .... 10
1.8 TheTop-DownandBottom-UpApproachestoScience.......... 11
1.9 LimitationsoftheTop-DownApproach........................... 12
1.10 SeekingaNewRoleforMathematicalLogic ..................... 13
1.11 TheCriticismofScholasticLogic ................................. 14
1.12 ScholasticLogicandMathematicalLogic......................... 15
1.13 MathematicalLogicandDiscovery................................ 16
1.14 TheNeedforanAlternativeLogicParadigm ..................... 17
1.15 TowardsanAlternativeLogicParadigm........................... 18
1.16 CharactersoftheAlternativeLogicParadigm..................... 18
1.17 TheAlternativeLogicParadigmandPhilosophy ................. 19
1.18 TheReconstructionofLogic....................................... 19
1.19 OrganizationoftheBook........................................... 21
1.20 Notations,Quotations,Transliterations............................ 22
PartI AncientPerspectives
2 TheOriginofLogic ........................................................ 25
2.1 SixClaimsbyGreekPhilosophers................................. 25
2.2 Universe,MindandDivinity....................................... 26
2.3 HumanMindandDivineMind..................................... 26
2.4 MethodandUniverse............................................... 27
2.5 MethodandLogic................................................... 27
2.6 LogicandDiscovery................................................ 28
vii
viii Contents
2.7 Logic,IntuitiveThinkingandDiscursiveThinking............... 29
2.8 AnArticulatedViewoftheOriginofLogic....................... 29
2.9 OriginoftheConnectionBetweenUniverse,GodandMind..... 30
2.10 FromChaostoOrder ............................................... 31
2.11 UniverseandEunomia.............................................. 31
2.12 IntuitiveThinking,DiscursiveThinkingandGreekMythology.. 33
2.13 OriginoftheName‘Logic’ ........................................ 33
2.14 Aristotle’sNamesforLogic........................................ 34
2.15 OriginoftheName‘Method’...................................... 35
3 AncientLogicandScience................................................. 37
3.1 ConceptionsofScience............................................. 37
3.2 Parmenides’ConceptionofScience ............................... 37
3.3 ParmenidesontheRoleofLogicandIntuitioninScience ....... 38
3.4 Plato’sConceptionofScience...................................... 39
3.5 PlatoandKnowledgeasJustifiedTrueBelief..................... 41
3.6 PlatoontheRoleofLogicandIntuitioninScience............... 42
3.7 PlatoontheImpedimentsoftheBody............................. 42
3.8 Plato’sCriticismoftheAxiomaticMethod ....................... 43
3.9 HippocratesofCos’CriticismoftheAxiomaticMethod......... 44
3.10 Aristotle’sConceptionofScience.................................. 45
3.11 Aristotle’sConceptionofDemonstration.......................... 46
3.12 AristotleonRoleofLogicandIntuitioninScience............... 47
3.13 AristotleonProperandCommonPrinciples ...................... 49
3.14 Aristotle’sConceptionofDefinition............................... 49
3.15 Aristotle’sSeparationofKindsandPrinciples.................... 50
3.16 AristotleonTruthasCorrespondence ............................. 51
3.17 AristotleonTruthasIntuitionoftheEssence..................... 52
3.18 AristotleonNousandIntuition .................................... 52
4 TheAnalyticMethod....................................................... 55
4.1 StatementoftheAnalyticMethod ................................. 55
4.2 InferenceandContainment......................................... 55
4.3 ThePlausibilityTestProcedure.................................... 56
4.4 PlausibilityandProbability......................................... 56
4.5 Non-deductiveRules,PlausibilityandExperience................ 57
4.6 DeductiveRules,PlausibilityandExperience..................... 57
4.7 TheDoubleMovementoftheAnalyticMethod................... 58
4.8 TheAnalyticNotionofDemonstration............................ 58
4.9 OriginoftheAnalyticMethod..................................... 59
4.10 BasicFeaturesoftheAnalyticMethod............................ 62
4.11 AnalyticMethodandInfiniteRegress ............................. 63
4.12 Non-finalityofSolutionstoProblems ............................. 64
4.13 OriginalFormulationoftheAnalyticMethod..................... 65
4.14 OriginalFormulationoftheAnalyticMethodandIntuition...... 67
Contents ix
4.15 TheAxiomaticMethod............................................. 68
4.16 TheAxiomaticNotionofDemonstration.......................... 68
4.17 AnalyticMethodvs.AxiomaticMethod........................... 69
4.18 TheMethodofAncientMedicine.................................. 70
4.19 LimitationsoftheOriginalFormulation
oftheAnalyticMethod............................................. 71
4.20 FortuneoftheAnalyticMethod.................................... 72
5 TheAnalytic-SyntheticMethod........................................... 75
5.1 Aristotle’sChangestotheAnalyticMethod....................... 75
5.2 Aristotle’sAnalytic-SyntheticMethod ............................ 76
5.3 Original Formulation of Aristotle’s
Analytic-SyntheticMethod......................................... 76
5.4 AnExampleofAristotle’sAnalytic-SyntheticMethod........... 78
5.5 The Direction of Analysis in Aristotle’s
Analytic-SyntheticMethod......................................... 80
5.6 Aristotle’sAnalytic-SyntheticMethodandIntuition ............. 81
5.7 PlausiblePremisesandEndoxa .................................... 81
5.8 The ControversyBetween Plato and Aristotle
ConcerningMethod................................................. 82
5.9 Pappus’Analytic-SyntheticMethod............................... 83
5.10 OriginalFormulationofPappus’Analytic-SyntheticMethod.... 84
5.11 AnExampleofPappus’Analytic-SyntheticMethod.............. 85
5.12 The Direction of Analysis in Pappus’
Analytic-SyntheticMethod......................................... 87
5.13 FortuneoftheAnalytic-SyntheticMethod ........................ 88
5.14 Analytic-SyntheticMethodandAxiomaticMethod .............. 89
5.15 RelationswithReductiontotheImpossible....................... 90
5.16 TheReasonforUseofReductiontotheImpossible.............. 91
5.17 AnalyticMethodvs.Analytic-SyntheticMethod ................. 93
6 Aristotle’sLogic:TheDeductivistView ................................. 95
6.1 TheDeductivistViewofAristotle’sLogic ........................ 95
6.2 Assertions ........................................................... 97
6.3 LogicalRelationsBetweenAssertions............................. 99
6.4 Syllogisms........................................................... 100
6.5 ConcerningtheName‘Syllogism’................................. 101
6.6 FiguresandMoods.................................................. 102
6.7 SingularandIndeterminateAssertionsinSyllogisms............. 104
6.8 CompleteandIncompleteSyllogisms ............................. 104
6.9 ConversionRulesandStrongReductiontotheImpossible....... 105
6.10 TheCompletionofSyllogisms..................................... 107
6.11 TheReductionofSyllogisms....................................... 108
6.12 Syllogistic........................................................... 110
6.13 LimitationsoftheDeductivistView ............................... 112