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Principia Logico-Metaphysica
(Draft/Excerpt)
Edward N. Zalta
Philosophy Department
Stanford University
With critical theoretical contributions by
Daniel Kirchner
Institut für Mathematik
Freie Universität Berlin
and
Uri Nodelman
Philosophy Department
Stanford University
September 6, 2022
https://mally.stanford.edu/principia.pdf
Copyright©2022byEdwardN.Zalta
Chapter14Copyright©2022byEdwardN.ZaltaandUriNodelman
Allrightsreserved. Nopartofthisbookmaybereproducedinanyformbyany
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LibraryofCongressCataloging-in-PublicationData
Zalta,EdwardN.,1952–
PrincipiaLogico-Metaphysica
Bibliography: p.
Doesn’tyetincludeindex.
1. Logic,Symbolic 2. Metaphysics(Philosophy). I.Title
CatalogNo.xxxxxx 2022 xxx-xxxx xx-xxxxx
ISBNX-XXX-XXXXX-X
Tomywife,SusanneZ.Riehemann
iv PrincipiaLogico-Metaphysica
Draft / Excerpt
NOTE: This is an excerpt from an incomplete draft of the monograph Prin-
cipiaLogica-Metaphysica. Themonographdraftcurrentlyhasfourparts:
PartI:Prophilosophy
PartII:Philosophy
PartIII:Metaphilosophy
PartIV:TechnicalAppendices,Bibliography,Index
ThisexcerptwasgeneratedonSeptember6,2022andcontains:
• PartII:
Page# PDF#
Ch.7: TheLanguage 173 17
Ch.8: TheAxioms 235 79
Ch.9: TheDeductiveSystem 259 103
Ch.10: BasicLogicalObjects 412 256
Ch.11: PlatonicForms 486 330
Ch.12: Situations,Worlds,Times,Fictions 511 355
Ch.13: Concepts 609 453
Ch.14: NaturalNumbers(w/UriNodelman) 679 523
• PartIV:
Appendix: ProofsofTheoremsandMetarules 962 640
Bibliography 1,339 1,017
Consequently, this excerpt omits the Preface, Part I/Chapters 1–6 (which
need revision), Part II/Chapter 15 (which is being written), Part III (which
is mostly unwritten), and some Appendices in Part IV. The present excerpt
sometimes contains references to the omitted content and active links in the
TableofContentstoomittedcontentwon’twork.
The work is ongoing and so the monograph changes constantly. Any citations
tothismaterialshouldexplicitlyreferencethisversionofSeptember6,2022,
sincepagenumbers,chapternumbers,sectionnumbers,item(definition,the-
orem)numbers,etc.,mayallchangeinfutureversions.
Contents
Preface xii
Acknowledgments xv
I Prophilosophy 1
1 Introduction 3
1.1 HistoricalContext . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 LogicandMetaphysics . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 TheCentralDistinction . . . . . . . . . . . . . . . . . . . . . . . . 15
2 DevelopinganExactScience 21
2.1 TheBasicParadoxesofEncoding . . . . . . . . . . . . . . . . . . 22
2.2 ASolutiontotheParadoxes . . . . . . . . . . . . . . . . . . . . . 28
2.3 ConsistencyoftheTheory . . . . . . . . . . . . . . . . . . . . . . 31
2.4 ReintroducingtheParadoxesviaDescriptions . . . . . . . . . . . 33
3 AczelModelsinDetail 36
3.1 AczelModelsFormallyDefined . . . . . . . . . . . . . . . . . . . 36
3.2 ConsistencyoftheComprehensionSchemata . . . . . . . . . . . 40
3.3 TheSmallestAczelModel . . . . . . . . . . . . . . . . . . . . . . 41
3.4 IntensionalAczelModels . . . . . . . . . . . . . . . . . . . . . . . 42
3.5 TheSmallestIntensionalAczelModel . . . . . . . . . . . . . . . 44
3.6 SomeObservationsAboutAczelModels . . . . . . . . . . . . . . 46
4 PhilosophicalQuestionsAbouttheTheory 48
5 AModalLanguageandItsInterpretation 52
5.1 TheSyntaxoftheLanguage . . . . . . . . . . . . . . . . . . . . . 52
5.1.1 SimpleTermsandFormulas . . . . . . . . . . . . . . . . . 57
5.2 ASemanticInterpretation . . . . . . . . . . . . . . . . . . . . . . 61
v
vi CONTENTS
5.2.1 Interpretations. . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2.2 AssignmentstoVariables . . . . . . . . . . . . . . . . . . . 63
5.2.3 Denotation,andTruth,withrespecttoI andf . . . . . . 64
5.2.4 Truth,Validity,andLogicalConsequence . . . . . . . . . 67
5.3 QuantifiedFormulas . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3.1 BasicIdeas . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.3.2 TheExistentialQuantifier . . . . . . . . . . . . . . . . . . 71
5.3.3 QuantifiersandNecessityOperators . . . . . . . . . . . . 74
5.4 FormulasWithanActualityOperator . . . . . . . . . . . . . . . . 75
5.5 DefiniteDescriptions . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.5.1 InterpretingDescriptions . . . . . . . . . . . . . . . . . . 78
5.5.2 DescriptionsandNecessity. . . . . . . . . . . . . . . . . . 82
5.5.3 Non-DenotingDescriptionsandRelationExistence. . . . 85
5.5.4 CanonicalDescriptions . . . . . . . . . . . . . . . . . . . . 90
5.6 Complexn-aryRelationTerms(n≥0). . . . . . . . . . . . . . . . 91
5.6.1 FromExistenceClaimston-aryRelationTerms . . . . . . 92
5.6.2 InterpretingComplexn-aryRelationTerms . . . . . . . . 98
5.6.3 GlobalSemanticConstraints . . . . . . . . . . . . . . . . . 102
6 Preview:NoteworthyPrinciples 112
6.1 Logicalvs. Non-logicalAxioms . . . . . . . . . . . . . . . . . . . 112
6.2 Negations,ConditionalsandTautologies . . . . . . . . . . . . . . 116
6.2.1 BasicNotions . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.2.2 PrimeandNon-PrimeFormulas . . . . . . . . . . . . . . . 120
6.2.3 Truth-FunctionalValuationsandTheirExtensions . . . . 123
6.2.4 SatisfactionandTautologies . . . . . . . . . . . . . . . . . 124
6.2.5 TautologicalImplication . . . . . . . . . . . . . . . . . . . 126
6.2.6 AnEffectiveProcedure . . . . . . . . . . . . . . . . . . . . 128
6.3 PrinciplesforIdentity . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.3.1 ObjectIdentity. . . . . . . . . . . . . . . . . . . . . . . . . 137
6.3.2 RelationIdentity . . . . . . . . . . . . . . . . . . . . . . . 144
6.4 NoteworthyQuantificationalPrinciples . . . . . . . . . . . . . . 152
6.4.1 AccomodatingDescriptions . . . . . . . . . . . . . . . . . 153
6.4.2 Show(39.5.a)isValid . . . . . . . . . . . . . . . . . . . . . 155
6.5 AxiomsforActuality . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.6 AxiomsforNecessity . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.7 AxiomsforNecessityandActuality . . . . . . . . . . . . . . . . . 159
6.8 AxiomsGoverningDescriptions . . . . . . . . . . . . . . . . . . . 159
6.8.1 OldStuffandNotes . . . . . . . . . . . . . . . . . . . . . . 159
6.8.2 DescriptionsandSubstitutionofIdenticals . . . . . . . . 162
6.9 AxiomsforComplexRelationTerms . . . . . . . . . . . . . . . . 164
6.9.1 α-Conversion . . . . . . . . . . . . . . . . . . . . . . . . . 164
CONTENTS vii
6.9.2 β-Conversion . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.9.3 η-Conversion . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.9.4 ı-Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . 170
II Philosophy 171
7 TheLanguage 173
7.1 MetatheoreticalDefinitions . . . . . . . . . . . . . . . . . . . . . 173
7.2 DefinitionsExtendingtheObjectLanguage . . . . . . . . . . . . 190
7.2.1 TheClassical,Sentence-FormingOperators . . . . . . . . 196
7.2.2 ExistenceintheLogicalSense . . . . . . . . . . . . . . . . 197
7.2.3 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
7.3 ExplanatoryRemarks: Digression . . . . . . . . . . . . . . . . . . 205
8 Axioms 235
8.1 AxiomsforNegationsandConditionals . . . . . . . . . . . . . . 236
8.2 AxiomsforQuantificationandExistence . . . . . . . . . . . . . . 236
8.3 AxiomsfortheSubstitutionofIdenticals . . . . . . . . . . . . . . 239
8.4 AxiomsforActuality . . . . . . . . . . . . . . . . . . . . . . . . . 240
8.5 AxiomsforNecessity . . . . . . . . . . . . . . . . . . . . . . . . . 241
8.6 AxiomsforNecessityandActuality . . . . . . . . . . . . . . . . . 242
8.7 AxiomsforDefiniteDescriptions . . . . . . . . . . . . . . . . . . 242
8.8 AxiomsforRelations . . . . . . . . . . . . . . . . . . . . . . . . . 243
8.9 AxiomsforEncoding . . . . . . . . . . . . . . . . . . . . . . . . . 245
8.10 SummaryoftheAxioms . . . . . . . . . . . . . . . . . . . . . . . 247
8.11 ExplanatoryRemarks: Digression . . . . . . . . . . . . . . . . . . 249
9 DeductiveSystemsofPLM 259
9.1 PrimitiveRuleofPLM:ModusPonens . . . . . . . . . . . . . . . 259
9.2 (ModallyStrict)ProofsandDerivations . . . . . . . . . . . . . . . 260
9.3 TwoFundamentalMetarules: GENandRN . . . . . . . . . . . . 264
9.4 TheInferentialRoleofDefinitions . . . . . . . . . . . . . . . . . 273
9.5 TheTheoryofNegationsandConditionals . . . . . . . . . . . . . 276
9.6 TheTheoryofQuantification . . . . . . . . . . . . . . . . . . . . 287
9.7 LogicalExistence,Identity,andTruth . . . . . . . . . . . . . . . . 295
9.8 TheTheoryofActualityandDescriptions . . . . . . . . . . . . . 308
9.8.1 TheTheoryofActuality . . . . . . . . . . . . . . . . . . . 308
9.8.2 TheTheoryofDescriptions . . . . . . . . . . . . . . . . . 315
9.9 TheTheoryofNecessity . . . . . . . . . . . . . . . . . . . . . . . 324
9.9.1 PropositionalModalLogic . . . . . . . . . . . . . . . . . . 324
9.9.2 QuantifiedModalLogic . . . . . . . . . . . . . . . . . . . 335
viii CONTENTS
9.9.3 Conditionsfor,andConsequencesof,ModalCollapse . . 335
9.10 TheTheoryofRelations. . . . . . . . . . . . . . . . . . . . . . . . 343
9.10.1 PrinciplesGoverningComplexRelationTerms . . . . . . 343
9.10.2 FactsAboutRelations . . . . . . . . . . . . . . . . . . . . . 351
9.11 TheTheoryofObjects . . . . . . . . . . . . . . . . . . . . . . . . . 369
9.11.1 OrdinaryObjects . . . . . . . . . . . . . . . . . . . . . . . 369
9.11.2 AbstractObjects . . . . . . . . . . . . . . . . . . . . . . . . 374
9.11.3 DiscernibleObjects . . . . . . . . . . . . . . . . . . . . . . 387
9.12 PropositionalProperties . . . . . . . . . . . . . . . . . . . . . . . 395
9.13 ExplanatoryRemarksonDefinitions . . . . . . . . . . . . . . . . 397
10 BasicLogicalObjects 412
10.1 Truth-Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
10.2 ExtensionsofPropositions . . . . . . . . . . . . . . . . . . . . . . 421
10.3 ExtensionsofProperties: NaturalClasses. . . . . . . . . . . . . . 422
10.3.1 BasicDefinitionsandTheorems . . . . . . . . . . . . . . . 426
10.3.2 NaturalClasses,LogicalSets,andModality . . . . . . . . 431
10.4 ReconstructingtheFregeanExtensionofF . . . . . . . . . . . . . 433
10.5 Interlude: RestrictedVariables . . . . . . . . . . . . . . . . . . . . 439
10.6 TheLawsofNaturalClassesandLogicalSets . . . . . . . . . . . 463
10.7 AbstractionviaEquivalenceConditions . . . . . . . . . . . . . . 475
10.8 AbstractionviaEquivalenceRelations . . . . . . . . . . . . . . . 478
10.8.1 DirectionsandShapes . . . . . . . . . . . . . . . . . . . . 478
10.8.2 GeneralAbstractionviaEquivalenceRelations . . . . . . 483
11 PlatonicForms 486
11.1 TheThinConceptionofForms . . . . . . . . . . . . . . . . . . . . 491
11.2 TheThickConceptionofForms . . . . . . . . . . . . . . . . . . . 497
12 Situations,Worlds,Times,andStories 511
12.1 Situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
12.1.1 BasicDefinitionsandTheorems . . . . . . . . . . . . . . . 514
12.1.2 TruthinaSituation . . . . . . . . . . . . . . . . . . . . . . 516
12.1.3 SituationIdentityandPartsofSituations. . . . . . . . . . 518
12.1.4 ComprehensionConditionsforSituations . . . . . . . . . 519
12.1.5 NullandTrivialSituations . . . . . . . . . . . . . . . . . . 523
12.1.6 ActualSituations . . . . . . . . . . . . . . . . . . . . . . . 524
12.1.7 Consistent,PossibleandIncompatibleSituations . . . . . 526
12.1.8 TheRoutleyStarOperationonSituations . . . . . . . . . 531
12.2 PossibleWorlds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536
12.3 World-IndexedLogicalObjectsandRelations . . . . . . . . . . . 563
12.3.1 World-IndexedTruth-Values . . . . . . . . . . . . . . . . . 563
CONTENTS ix
12.3.2 World-IndexedExtensions . . . . . . . . . . . . . . . . . . 566
12.3.3 World-IndexedRelations . . . . . . . . . . . . . . . . . . . 568
12.4 ImpossibleWorlds . . . . . . . . . . . . . . . . . . . . . . . . . . . 574
12.5 MomentsofTimeandWorld-States . . . . . . . . . . . . . . . . . 581
12.6 StoriesandFictionalIndividuals. . . . . . . . . . . . . . . . . . . 590
12.6.1 DataandMethodology . . . . . . . . . . . . . . . . . . . . 590
12.6.2 PrinciplesForAnalyzingFiction . . . . . . . . . . . . . . 596
12.6.3 AnalysisoftheData. . . . . . . . . . . . . . . . . . . . . . 602
12.6.4 ValidatingJudgmentsofLogicalConsequence. . . . . . . 604
12.6.5 FinalIssuesConcerningFictionalIndividuals . . . . . . . 607
13 Concepts 609
13.1 TheCalculusofConcepts . . . . . . . . . . . . . . . . . . . . . . . 613
13.1.1 ConceptAddition . . . . . . . . . . . . . . . . . . . . . . . 613
13.1.2 ConceptInclusionandContainment . . . . . . . . . . . . 618
13.1.3 ConceptInclusion,Addition,andIdentity . . . . . . . . . 620
13.1.4 TheAlgebraofConcepts . . . . . . . . . . . . . . . . . . . 621
13.1.5 TheMereologyofConcepts . . . . . . . . . . . . . . . . . 628
13.2 ConceptsofPropertiesandIndividuals . . . . . . . . . . . . . . . 641
13.2.1 ConceptsofProperties . . . . . . . . . . . . . . . . . . . . 642
13.2.2 ConceptsofOrdinaryIndividuals . . . . . . . . . . . . . . 648
13.2.3 ConceptsofGeneralizations . . . . . . . . . . . . . . . . . 653
13.3 TheContainmentTheoryofTruth . . . . . . . . . . . . . . . . . . 654
13.4 TheModalMetaphysicsofConcepts . . . . . . . . . . . . . . . . 660
13.4.1 Realization,Appearance,Mirroring . . . . . . . . . . . . . 663
13.4.2 Possible-IndividualConcepts . . . . . . . . . . . . . . . . 666
13.4.3 Compossibility . . . . . . . . . . . . . . . . . . . . . . . . 669
13.4.4 Counterparts. . . . . . . . . . . . . . . . . . . . . . . . . . 670
13.4.5 World-RelativeConceptsofIndividuals . . . . . . . . . . 670
13.4.6 FundamentalTheorems . . . . . . . . . . . . . . . . . . . 675
14 NaturalNumbers (withUriNodelman) 679
14.1 PhilosophicalContext . . . . . . . . . . . . . . . . . . . . . . . . . 679
14.2 EquinumerosityandDiscernibleObjects . . . . . . . . . . . . . . 692
14.3 NumberingPropertiesandNaturalCardinals . . . . . . . . . . . 702
14.4 AncestralsandRelationsonDiscernibles . . . . . . . . . . . . . . 715
14.5 Predecessor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723
14.6 DerivingtheNumber-TheoreticPostulates . . . . . . . . . . . . . 728
14.7 NumberTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733
14.8 FunctionsandRecursiveDefinitions . . . . . . . . . . . . . . . . 744
14.8.1 TotalFunctions . . . . . . . . . . . . . . . . . . . . . . . . 744
14.8.2 FunctionsFromDomainstoCodomains . . . . . . . . . . 752
x CONTENTS
14.8.3 RestrictedFunctionsandFunctionsGenerally . . . . . . . 765
14.8.4 NumericalOperations . . . . . . . . . . . . . . . . . . . . 768
14.8.5 Recursively-DefinedRelationsandFunctions . . . . . . . 772
14.9 Deriving2nd-OrderPeanoArithmetic . . . . . . . . . . . . . . . 784
14.10 Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787
15 TypedObjectTheoryanditsApplications 792
15.1 TheLanguageandItsInterpretation . . . . . . . . . . . . . . . . 793
15.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817
15.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 820
15.4 TheDeductiveSystemandBasicTheorems . . . . . . . . . . . . 826
15.4.1 TheSystem,Negations,Conditionals,Quantification . . . 826
15.4.2 LogicalExistence,Identity,andUniqueness . . . . . . . . 827
15.4.3 TheTheoryofActualityandDescriptions . . . . . . . . . 835
15.4.4 TheTheoryofNecessity . . . . . . . . . . . . . . . . . . . 837
15.4.5 TheTypedTheoryofRelations . . . . . . . . . . . . . . . 838
15.4.6 TheTypedTheoryofObjects . . . . . . . . . . . . . . . . 840
15.4.7 DiscernibleObjectsOfEveryType . . . . . . . . . . . . . 846
15.4.8 PropositionalProperties . . . . . . . . . . . . . . . . . . . 847
15.5 LogicalObjectsandWorlds. . . . . . . . . . . . . . . . . . . . . . 848
15.6 MathematicsandMathematicalLanguage . . . . . . . . . . . . . 848
15.6.1 MathematicalTheoriesandTruthinaTheory . . . . . . . 848
15.6.2 MathematicalIndividualsandRelations . . . . . . . . . . 851
15.6.3 AnalyzingtheDatafromMathematics . . . . . . . . . . . 852
15.7 Higher-OrderFictionsandLogicalObjects . . . . . . . . . . . . . 852
15.7.1 Higher-OrderFictions . . . . . . . . . . . . . . . . . . . . 852
15.7.2 Higher-OrderLogicalObjects . . . . . . . . . . . . . . . . 852
15.8 PropositionalAttitudesandReportsThereof . . . . . . . . . . . . 854
15.9 Type-Lowering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854
15.9.1 ObjectifiedProperties. . . . . . . . . . . . . . . . . . . . . 854
15.9.2 ObjectifiedStatesofAffairs . . . . . . . . . . . . . . . . . 854
III Metaphilosophy 855
16 PhilosophicalIssuesandObservations 857
16.1 Puzzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 857
16.2 CarnapandaMetaontologicalFramework . . . . . . . . . . . . . 857
16.3 NaturalismandDependentAbstracta . . . . . . . . . . . . . . . . 858
16.4 ExplanatoryPower . . . . . . . . . . . . . . . . . . . . . . . . . . 858
16.5 PhilosophicalInterpretationsoftheFormalism . . . . . . . . . . 858
16.6 MeinongianIdealObjects. . . . . . . . . . . . . . . . . . . . . . . 860