Mögliche-Welten-Semantik für indikative und kontrafaktische Konditionale? Eine formal-philosophisch Untersuchung der Chellas-Segerberg Semantik [Elektronische Ressource] / Matthias Unterhuber. Gutachter: Gerhard Schurz ; Manuel Bremer

Mögliche-Welten-Semantik für indikative und kontrafaktische Konditionale? Eine formal-philosophisch Untersuchung der Chellas-Segerberg Semantik [Elektronische Ressource] / Matthias Unterhuber. Gutachter: Gerhard Schurz ; Manuel Bremer

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PossibleWorldsSemanticsforIndicativeandCounterfactualConditionals?AFormal-PhilosophicalInquiryintoChellas-SegerbergSemanticsInaugural-DissertationzurErlangungdesDoktorgradesderPhilosophie(Dr. phil.)durchdiePhilosophischeFakultätderHeinrich-Heine-UniversitätDüsseldorfvorgelegtvonMatthiasUnterhuberausSchnaitseeBetreuer:Prof. Dr. GerhardSchurzDüsseldorfDezember2010D61TagderPrüfung: 9.3.2011Gutachter: Prof. Dr. GerhardSchurz&Prof. Dr. ManuelBremer(apl. Professor)iiiAbstractConditional logic is a sub-discipline of philosophical logic. It aims to provideanalternativeaccountofconditionalsincontrasttothetraditionalmaterialimpli-cation analysis. The present thesis focuses on a specific possible worlds seman-tics for conditional logics, the Chellas-Segerberg (CS) semantics (Chellas, 1975;Segerberg, 1989), which has not been widely investigated, save by Nejdl (1992)andDelgrande(1987,1988).ThemainthesisofthisdissertationisthatCS-semanticsisanadequateframeworkfor both (i) indicative and (ii) counterfactual conditionals. To argue for (i) and(ii)we, first,discussthegeneralneedofaconditionallogicapproach,whichgoesbeyondamaterialimplicationanalysis. Weaddressthedifferencebetweenindica-tive and counterfactual conditionals. We focus, then, on two arguments broughtforward by Bennett (2003) against accounts of indicative conditionals in termsof truth and falsehood, which arguably include possible worlds semantics suchas CS-semantics: (a) D.

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PossibleWorldsSemanticsforIndicativeand
CounterfactualConditionals?
AFormal-PhilosophicalInquiryinto
Chellas-SegerbergSemantics
Inaugural-Dissertation
zurErlangungdesDoktorgradesderPhilosophie(Dr. phil.)
durchdiePhilosophischeFakultätder
Heinrich-Heine-UniversitätDüsseldorf
vorgelegtvonMatthiasUnterhuber
ausSchnaitsee
Betreuer:Prof. Dr. GerhardSchurz
DüsseldorfDezember2010
D61
TagderPrüfung: 9.3.2011
Gutachter: Prof. Dr. GerhardSchurz&
Prof. Dr. ManuelBremer(apl. Professor)iii
Abstract
Conditional logic is a sub-discipline of philosophical logic. It aims to provide
analternativeaccountofconditionalsincontrasttothetraditionalmaterialimpli-
cation analysis. The present thesis focuses on a specific possible worlds seman-
tics for conditional logics, the Chellas-Segerberg (CS) semantics (Chellas, 1975;
Segerberg, 1989), which has not been widely investigated, save by Nejdl (1992)
andDelgrande(1987,1988).
ThemainthesisofthisdissertationisthatCS-semanticsisanadequateframework
for both (i) indicative and (ii) counterfactual conditionals. To argue for (i) and
(ii)we, first,discussthegeneralneedofaconditionallogicapproach,whichgoes
beyondamaterialimplicationanalysis. Weaddressthedifferencebetweenindica-
tive and counterfactual conditionals. We focus, then, on two arguments brought
forward by Bennett (2003) against accounts of indicative conditionals in terms
of truth and falsehood, which arguably include possible worlds semantics such
as CS-semantics: (a) D. Lewis’ (1976) triviality result and (b) Bennett’s (2003)
Gibbardianstand-offargument,whichgoesbacktoGibbard(1980).
We,furthermore,investigatealatticeofconditionallogicsbasedonthebasicproof
theoreticsystemforCS-semanticsplus29furtheraxioms. Thisframeworkallows
us to describe – as we shall show – a range of conditional logic systems, such as
the indicative conditional logic system by Kraus, Lehmann, and Magidor (1990)
and Lehmann and Magidor (1992) and the counterfactual system of D. Lewis
(1973/2001). ForourformalinvestigationwedistinguishbetweenChellasframes
(Chellas, 1975) on the one hand and Segerberg frames (Segerberg, 1989) on the
otherhand: WhileChellasframesaregeneralizationsofKripkeframes,Segerberg
frames rather correspond to what is often called ‘general frames’. We give, then,
correspondence proofs for the lattice of systems on the basis of Chellas frames
and discuss the notion of trivial frame correspondence. We, then, provide a com-
pletenessresultforthelatticeofconditionallogicsforstandardSegerbergframes.
ThistypeofSegerbergframesissolelybasedonstructuralconditionsandis–un-
likethenotionof(simple)Segerbergframecompleteness–nottrivialinthesense
thatanyconditionallogiciscompletew.r.t. someclassofframes.
We finally, provide an objective and a subjective interpretation of CS-semantics
by drawing on the notion of alethic modality and the Ramsey-test, respectively.
We, then, argue that our objective and subjective account of CS-semantics can
serveasbasisforindicativeandcounterfactualconditionals,respectively.ivContents
Preface xiii
I FoundationalIssues 1
1 AnArgumentforaConditionalLogic 3
1.1 CounterexamplestoaMaterialImplicationAnalysisofConditionals 4
1.2 StrongerCounterexamplestoaMaterialImplicationAnalysis . . . 8
1.2.1 CounterfactualConditionals.................. 9
1.2.2 NormicConditionals ......................10
1.3 ConversationalImplicatures–APossibleWayOut?.........20
1.4 NoMaterialImplicationAnalysis...................24
2 InterdisciplinaryDimensions 25
2.1 TheConditionalLogicProjectandRelatedProjects.........26
2.1.1 Overview.............................26
2.1.2 TheConditionalLogicProject ................26
2.1.3 TheLinguisticsofConditionalsProject ...........26
2.1.4 ThePhilosophyofCProject28
2.1.5 ThePsychologyofReasoningProject ............32
2.1.6 TheNon-MonotonicReasoningProject ...........34
2.2 GeneralIntelligence,Defaults,Non-MonotonicRulesandCondi-
tionals...................................35
2.2.1 AMotivationfortheStudyofNon-MonotonicLogics . . . 35
vvi
2.2.2 ReiterDefaults .........................37
2.2.3 Reformulations39
2.2.4 Consistency,Non-DerivabilityandDefaultLogics .....39
2.2.5 Non-DerivabilityandAxiomatizationofDefaultTheories . 43
2.2.6 ProblemsofPresentDefaultLogicAccounts ........45
2.2.7 Non-Monotonic Logics, Conditional Logics and Default
Logics..............................48
3 ADefenseofPossibleWorldsSemanticsforIndicativeConditionals 53
3.1 OurDefenseofPossibleWorldsSemantics..............54
3.2 Ramsey-Test Interpretations and Possible Worlds Semantics for
Conditionals...............................57
3.2.1 Ramsey’sOriginalProposal ..................58
3.2.2 Stalnaker’sVersionoftheRamseyTest,StalnakerSeman-
tics, Set Selection Semantics and Chellas-Segerberg Se-
mantics..............................60
3.2.3 Ordering Semantics. D. Lewis (1973/2001), Kraus et al.
(1990)andRelatedSemantics ................65
3.2.4 Contrasting Ramsey Test Interpretations of Conditionals
andOrderingSemantics....................76
3.2.5 The Consistency Requirement and The Principle of Con-
ditionalExcludedMiddle...................79
3.2.6 AGeneralRamseyTestRequirement?............85
3.3 DistinguishingIndicativeandCounterfactual
Conditionals...............................87
3.3.1 IndicativeConditionals.....................87
3.3.2 CriteriaforCounterfactualConditionals ...........89
3.3.3 Subjective or Objective Interpretations of Indicative and
CounterfactualConditionals? .................90
3.4 Fundamental Issues of Probabilistic Approaches to Conditional
Logic....................................93vii
3.4.1 SubjectiveandObjectiveProbabilisticSemantics......93
3.4.2 ConditionalorUnconditionalProbabilitiesas
Primitive.............................96
3.4.3 The Status of Conditionals and the Language of a Proba-
bilisticConditionalLogic ...................97
3.4.4 A Motivation for the Restriction of Languages in Proba-
bilisticSemantics ........................104
3.5 Adams’P-Systems............................105
∗ +3.5.1 TheSystemsP,P andP ...................106
3.5.2 ThresholdSemantics ......................113
3.5.3 Adams’ (1975) System P and Schurz’s (1997b) Modifi-

cation...............................113
3.5.4 PossibleWorldsSemanticsandTruth-Assignmentsinthe
AdamsApproaches.......................123
3.6 Lewis’(1976)TrivialityResults ....................129
3.6.1 Lewis’Proofs ..........................130
3.6.2 TrivialityduetoNestingsandIterationsofConditionals? . 134
3.6.3 ProbabilisticSemanticsandRestrictionoftheLanguage . 137
3.6.4 Lewis’TrivialityResultandTruth-ValueAccounts.....141
3.6.5 Conclusion............................143
3.7 Bennett’sArgumentagainstTruth-ValueSemanticsandObjective
ProbabilisticSemantics .........................143
3.7.1 Bennett’sGibbardianStand-OffsArgument.........144
3.7.2 Truth-ValueAccounts .....................147
3.7.3 ObjectiveProbabilisticApproaches..............148
3.7.4 SubjectiveP .............149
3.7.5 Summary.............................151
3.8 Conclusion................................151viii
II FormalResultsforChellas-SegerbergSemantics 153
4 FormalFramework 155
4.1 WhyChelas-SegerbergSemantics?..................155
4.2 Proof-TheoreticNotions.........................156
− ∗4.2.1 LanguagesL ,L ,L ,L andL .........156KL KL rKL rKL rrKL
4.2.2 Logics ..............................163
4.2.3 Non-Monotonicity .......................164
4.2.4 ConsistencyandMaximality..................165
4.2.5 APropositionalBasisforConditionalLogics ........165
4.2.6 SystemCK ...........................166
4.2.7 AlternativeAxiomatizationsofSystemCK .........169
4.3 Model-TheoreticNotions ........................171
4.3.1 ChellasFramesandChellasModels .............172
4.3.2 ADiscussionofChellasModelsandFrames ........173
4.3.3 SegerbergFramesandSegerbergModels ..........176
4.3.4 Validity,LogicalConsequenceandSatisfiability ......179
4.3.5 NotionsofFrameCorrespondence ..............181
4.3.6 Standard and Non-Standard Chellas Models and Seger-
bergFrames...........................184
4.3.7 NotionsofSoundnessandCompleteness ..........186
5 FrameCorrespondence 193
5.1 Non-TrivialFrameConditionsforaLatticeofConditionalLogics. 195
5.2 TheNotionsofTrivialandNon-TrivialFrameConditions.....201
5.2.1 ATranslationProcedurefromAxiomSchematatoTrivial
FrameConditions........................201
5.2.2 ANon-TrivialityCriterion . ..................205
5.3 ChellasFrameCorrespondenceProofs ................206
5.3.1 SystemP.............................206
5.3.2 ExtensionsofSystemP ....................210ix
5.3.3 AxiomsfromWeakProbabilityLogic(ThresholdLogic) . 212
5.3.4 MonotonicPrinciples......................213
5.3.5 BridgePrinciples ........................215
5.3.6 CollapseConditionsMaterialImplication ..........217
5.3.7 TraditionalExtensions .....................218
5.3.8 IterationPrinciples .......................221
6 SoundnessandCompletenessProofsforaLatticeofConditionalLog-
ics 223
6.1 GeneralOverview............................223
6.1.1 FocusofOurCompletenessProofs..............223
6.1.2 Discussion of Segerberg Frame Completeness and Chel-
lasFramesCompletenesProofs...............225
6.2 SingletonFramesforCS-Semantics..................228
6.3 Soundnessw.r.t. ClassesofChellasFrames..............230
6.4 StandardSegerbergFrameCompleteness231
6.4.1 GeneralPrinciples .......................231
6.4.2 CanonicalModels........................233
6.4.3 CanonicityProofsforIndividualPrinciples .........235
7 CSSemanticsforIndicativeandCounterfactualConditionals 247
7.1 TheBasicCSSystems(SystemsCKandCKR)...........250
7.1.1 ObjectiveandSubjectiveInterpretationsofCS-Semantics
forIndicativeandCounterfactualConditionals .......251
7.1.2 AlternativeAxiomatizationsofSystemCKR ........258
7.2 ConditionalLogicswithoutBridgePrinciples ............260
7.2.1 SystemC.............................260
7.2.2 SystemCL............................267
7.2.3 SystemP268
7.2.4 SystemR275
7.2.5 Lewis’(1973/2001)SystemV.................278x
7.2.6 Monotonic Systems without Bridge Principles (Systems
CMandM)...........................286
7.3 ConditionalLogicswithBridgePrinciples..............293
∗7.3.1 Adams’(1965,1966,1975)OriginalSystemP ......293
7.3.2 Lewis’(1973)SystemVC ...................300
7.3.3 StalnakerandThomason’sSystemS .............300
7.3.4 TheMaterialCollapseSystemMC..............302
8 ConcludingRemarks 307
References 311