Table Of ContentINFINITEREGRESSARGUMENTS
Argumentation Library
VOLUME 17
SeriesEditors
FransH.vanEemeren,UniversityofAmsterdam,TheNetherlands
ScottJacobs,UniversityofIllinoisatUrbana-Champaign,USA
ErikC.W.Krabbe,UniversityofGroningen,TheNetherlands
JohnWoods,UniversityofBritishColumbia,Vancouver,Canada
Forfurthervolumes:
http://www.springer.com/series/5642
INFINITE REGRESS
ARGUMENTS
Claude Gratton
AntelopeValleyCollege
123
Dr.ClaudeGratton
AntelopeValleyCollege
3041WestAvenueK
LancasterCA93536-5426
USA
cgratton@avc.edu
ISSN1566-7650
ISBN978-90-481-3340-6 e-ISBN978-90-481-3341-3
DOI10.1007/978-90-481-3341-3
SpringerDordrechtHeidelbergLondonNewYork
LibraryofCongressControlNumber:2009940669
©SpringerScience+BusinessMediaB.V.2010
Nopartofthisworkmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformorby
anymeans,electronic,mechanical,photocopying,microfilming,recordingorotherwise,withoutwritten
permissionfromthePublisher,withtheexceptionofanymaterialsuppliedspecificallyforthepurpose
ofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Acknowledgements
Themuchshorterandmuchearlierversionofthiswork,whenitwasmyPh.D.dis-
sertation, improved greatly from the constructive comments of Derek Allen and
Robert Tully. I would like to thank some of my colleagues at Antelope Valley
College: Santi Tafarella (English) for his comments on the first chapter; Ron
Halcrow (Economics) for his encouragement after reading the first two chapters;
andDebraAnderson(Mathematics)forhermeticulousreadingsofpartsofthefirst
chapter.Iamgladtoreportthattheyarestillmyfriends!Iamalsoverygratefulfor
the constructive criticism of three anonymous reviewers of this work. Of course, I
aloneamresponsiblefortheremainingweaknessesinthiswork.
v
Contents
1 WhatisanInfiniteRegressArgument? . . . . . . . . . . . . . . . . 1
1.1 TheGeneralStructureofInfiniteRegressArguments . . . . . . . 1
1.2 BoundariesofanInfiniteRegressArgument . . . . . . . . . . . . 5
1.2.1 BoundarieswhenanInfiniteRegressisVicious . . . . . . 6
1.2.2 BoundarieswhenanInfiniteRegressisBenign . . . . . . 9
1.3 AHypothesisAbouttheNatureofInfiniteRegresses . . . . . . . 12
1.4 TestingHypothesisH . . . . . . . . . . . . . . . . . . . . . . . . 18
1.5 TestingHypothesisHwithNonconcatenatingRegresses . . . . . . 21
1.6 PotentiallyInfiniteandActuallyInfiniteRegresses. . . . . . . . . 25
1.7 TheNecessaryQuantityofTermsandRelations . . . . . . . . . . 28
1.8 ApplicationsofHypothesisHtoVariousExamples . . . . . . . . 31
1.8.1 Plato’sCouch . . . . . . . . . . . . . . . . . . . . . . . . 31
1.8.2 TeachersTaughtbyTeachers . . . . . . . . . . . . . . . . 32
1.8.3 GodsGivingMeaningtoGods . . . . . . . . . . . . . . . 33
1.8.4 MapsofMaps . . . . . . . . . . . . . . . . . . . . . . . . 35
1.8.5 LewisCarroll’s“WhattheTortoiseSaidtoAchilles” . . . 38
1.9 LogicalFunctionsofInfiniteRegresses. . . . . . . . . . . . . . . 44
1.9.1 BenignRegresses . . . . . . . . . . . . . . . . . . . . . . 45
1.9.2 SuperfluousRegresses . . . . . . . . . . . . . . . . . . . 49
1.10 CogencyandBenignRegresses . . . . . . . . . . . . . . . . . . . 52
2 TheFormalandNonformalLogicofInfiniteConcatenatingRegresses 57
2.1 RecurringTerms,Loops,andRegressFormulas . . . . . . . . . . 57
2.2 TheRelationofTermsandObjectsofanInfiniteRegress . . . . . 63
2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4 RecurringTerms,Loops,andInfiniteConcatenatingRegresses . . 68
2.5 RelationsandLoops. . . . . . . . . . . . . . . . . . . . . . . . . 72
2.6 BlockingAllPossibleLoops . . . . . . . . . . . . . . . . . . . . 75
2.7 Are Irreflexivity, or Asymmetry or Transitivity
NecessarytoBlockLoops? . . . . . . . . . . . . . . . . . . . . . 78
2.8 ConcatenatingRelationsinRegressFormulas . . . . . . . . . . . 81
2.9 DirectionsofInfiniteConcatenatingRegresses . . . . . . . . . . . 82
2.9.1 TheImportanceoftheDirectionofanInfiniteRegress . . 83
vii
viii Contents
2.9.2 TheFormalDirectionofanInfiniteRegress . . . . . . . . 84
2.9.3 TheSemanticDirectionofanInfiniteRegress . . . . . . . 86
2.10 Non-formalConsiderationsinRegressFormulas . . . . . . . . . . 87
2.10.1 RelationsandTheirImplications . . . . . . . . . . . . . . 88
2.10.2 UnstatedPropertiesofRelationsandTerms . . . . . . . . 89
2.10.3 StatedPropertiesofObjectsorConditionsina
RegressFormula . . . . . . . . . . . . . . . . . . . . . . 90
2.10.4 UnstatedPropertiesofObjectsDesignatedbyTerms. . . . 91
2.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.12 EvaluativeQuestions . . . . . . . . . . . . . . . . . . . . . . . . 99
3 Viciousness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.1 AreThereInherentlyViciousRegresses? . . . . . . . . . . . . . . 102
3.2 ClarkonViciousness . . . . . . . . . . . . . . . . . . . . . . . . 105
3.3 JohnstoneandViciousness . . . . . . . . . . . . . . . . . . . . . 107
3.4 UncompletabilityandViciousness . . . . . . . . . . . . . . . . . 111
3.5 Occam’sRazor:OntologicalExtravagance . . . . . . . . . . . . . 116
3.6 BlockingViciousInfiniteRegresses . . . . . . . . . . . . . . . . 119
3.6.1 Hume . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.6.2 Miller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.6.3 LaurenceandMargolis . . . . . . . . . . . . . . . . . . . 125
3.6.4 TheGeneralformoftheArgumentforBlockingRegresses 127
4 Circular Definitions, Circular Explanations,
andInfiniteRegresses . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.1 AFormalDerivationofInfiniteRegressesfromCircularDefinitions 132
4.2 InfinitelyManyInfiniteRegresses . . . . . . . . . . . . . . . . . 134
4.3 SemanticConsiderations . . . . . . . . . . . . . . . . . . . . . . 135
4.4 RegressesIndependentofCircularity . . . . . . . . . . . . . . . . 138
4.5 TheViciousnessofInfiniteRegressesEntailedbyCircular
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.6 TheDerivationofInfiniteRegressesfromCircularExplanations . 142
5 InfiniteRegressesandRecurringQuestions . . . . . . . . . . . . . . 147
5.1 Recurring Questions and the Derivation
ofInfiniteRegresses. . . . . . . . . . . . . . . . . . . . . . . . . 149
5.2 RecurringQuestionsandViciousRegresses . . . . . . . . . . . . 153
6 InfiniteRegressesofRecurringProblemsandResponses . . . . . . 159
6.1 Plato’sAviaryintheTheatetus . . . . . . . . . . . . . . . . . . . 161
6.2 McTaggart’sDiscontinualRegress . . . . . . . . . . . . . . . . . 163
6.3 Mackie’sDiscontinualRegress . . . . . . . . . . . . . . . . . . . 167
6.4 Armstrong’sContinualRegress . . . . . . . . . . . . . . . . . . . 172
6.5 AContinualRegressinDefenseofCantor’sDiagonalMethod . . 178
6.6 Lehrer’s Regress of Recurring Possible Problems
andPossibleResponses . . . . . . . . . . . . . . . . . . . . . . . 182
6.7 EvaluativeQuestions . . . . . . . . . . . . . . . . . . . . . . . . 188
Contents ix
AppendixA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
AppendixB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
Introduction
Infinite regresses (e.g., event caused event , event caused event , event caused
2 1 3 2 4
event ,adinfinitum;statement justifiesstatement ,statement justifiesstatement ,
3 2 1 3 2
statement justifies statement , ad infinitum) have been used as premises in argu-
4 3
ments on a great variety of topics in both Eastern and Western philosophy since
ancienttimes.Infiniteregressargumentsarepartofaphilosopher’stoolkitofargu-
mentation. But how sharp or strong is this tool? How effectively is it used? The
typical presentation of infinite regress arguments throughout history is so succinct
and has so many gaps that it is often unclear how an infinite regress is derived, or
whyaninfiniteregressislogicallyproblematic,andasaresult,itisoftendifficultto
evaluateinfiniteregressarguments.Ourcustomarywayofusingthistoolindicates
that there definitely is a need for a theory to re-orient our practice. Consequently,
afterwellovertwothousandyearsofusinginfiniteregressesaspremises,onewould
have expected that at least some theory of infinite regress arguments would have
emerged.Noneexists.Therehavebeenonlyafewarticlesoninfiniteregressargu-
ments,buttheyarebasedontheexaminationofonlyasmallnumberofexamples,
discussonlyafewlogicalorrhetoricalaspectsofinfiniteregressarguments,andso
theyhelptomeettheneedforatheoryinonlyalimitedway.
Given the situation, I examined many infinite regress arguments to clarify the
variousaspectsofthederivationofinfiniteregresses,andexplainthedifferentways
inwhichcertaininfiniteregressesareunacceptable.Mygeneralapproachconsisted
of collecting and evaluating as many infinite regress arguments as possible, com-
paring and contrasting many of the formal and non-formal properties, looking for
recurringpatterns,andidentifyingthepropertiesthatappearedessentialtothosepat-
terns.Thesixchaptersofthisbookgraduallyemergedfromthisapproach.Twovery
generalquestionsguidedthiswork:(1)Howareinfiniteregressesgeneratedininfi-
niteregressarguments?(2)Howdoinfiniteregresseslogicallyfunctionaspremises
inanargument?InansweringthesequestionsIavoidedasmuchaspossibleaddress-
ingthephilosophicalcontentandhistoricalbackgroundoftheargumentsexamined.
Due to the already extensive work done on causal regresses and regresses of jus-
tification, only a few references are made to them. However, the focus is on other
issuesthathavebeenneglected,andthatdocontributetoageneraltheoryofinfinite
regressarguments:Iclarifythenotionofaninfiniteregress;identifydifferentlogi-
calformsofinfiniteregresses;describedifferentkindsofinfiniteregressarguments;
xi
Description:Infinite regress arguments are part of a philosopher's tool kit of argumentation. But how sharp or strong is this tool? How effectively is it used? The typical presentation of infinite regress arguments throughout history is so succinct and has so many gaps that it is often unclear how an infinite r