Advances in ProofTheoretic Semantics
On the Relation Between Heyting's and Gentzen's Approaches to MeaningIntroductionHeyting's Approach to MeaningGentzen's Approach to MeaningA First Comparison Between Heyting's and Gentzen's ApproachesFurther Development of Gentzen's IdeasArgument StructuresArgumentsValidity of ArgumentsWeak and Strong Validity and Their FeaturesMappings of Valid Arguments on BHKProofs and Vice VersaExtending the Mapping Proof to Arguments for AExtending the Mapping Arg to BHKProofs of AConcluding RemarksReferencesKreisel's Theory of Constructions, the KreiselGoodman Paradox, and the Second ClauseIntroductionPredicativity, Decidability, and the BHK InterpretationThe Theory of Constructions and the Second ClauseAn Overview of the Theory of ConstructionsThe Language of TThe Axiomatization of TFormalizing the BHK Interpretation in TSoundness, Completeness, and InternalizationThe KreiselGoodman ParadoxThe Reception of the Theory of Constructions and the Second ClauseShifting OpinionsGuilt by Association?Diagnosing the ParadoxSelfReference and TypingStratificationDecidabilityReflectionInternalizationConclusions and Further WorkReferencesOn the Paths of CategoriesFunctions of LanguageDeductions Not Necessarily Based on PropositionsDeductions in CategoriesDeductions in Multicategories and PolycategoriesRules for DeductionsReferencesSome Remarks on ProofTheoretic SemanticsBackground on General Elimination RulesIs Bullet a Logical Constant?The GErule for Implication and the TypeTheoretic Dependent Product TypeGERules in GeneralSeveral IRulesIRule Has Several PremissesPremiss of IRule Discharges Some AssumptionsGE Harmony: A CounterExampleAnother [Counter]ExampleIn Other WordsConclusionReferencesCategorical Harmony and Paradoxes in ProofTheoretic SemanticsIntroductionThe Principle of Categorical HarmonyCategorical Harmony in Comparison with Other PrinciplesDegrees of Paradoxicality of Logical ConstantsConcluding Remarks: From Semantic Dualism to DualityReferencesThe Paradox of Knowability from an Intuitionistic StandpointIntroductionAn Intuitionistic Solution(21) is intuitionistically validDefinition 1Truth NotionsInternal and Intuitive TruthUnknown StatementsNeoVerificationist ApproachesHow Is a Rational Discussion Possible?ConclusionReferencesExplicit Composition and Its Application in Proofs of NormalizationIntroductionNotation for Natural DerivationsStrong Normalization by Bar InductionConcluding Remarks and Further ApplicationsReferencesTowards a ProofTheoretic Semantics of EqualitiesFrege's QuestionEquality Versus IdentityThe Mode of PresentationMorning Star Versus Evening Star RevisitedEqualityEquality of SensesProofTheoretic SemanticsReferencesOn the ProofTheoretic Foundations of Set TheoryIntroductionDefining SetsFunctional Closure, Local Logic and the Notion of AbsolutenessThe Functional ClosureLocal LogicAbsolutenessA ProofTheoretic InterpretationSetsFoundational IssuesReferencesA Strongly Differing Opinion on ProofTheoretic Semantics?Tarski's Definition of Logical ConsequenceModel TheoryModelTheoretic SemanticsDefining Meanings in GeneralDefining Meanings: Specialise Then GeneraliseRepresenting the MeaningDefining Logical ConsequenceReferencesComments on an OpinionOn Dummett's “ProofTheoretic Justifications of Logical Laws”Analysis of the MethodBoundary RulesSchematic InferencesAssessmentAppendixAuthor's Postscript, January 2015ReferencesSelfcontradictory ReasoningIntroductionMeaning ConditionsThe Liar ParadoxSelfcontradictory Reasoning in N−∀∃=Selfcontradictory Reasoning in N−∃=Selfcontradictory Reasoning in N−=AppendixNaïve Set TheoryNormal Deductions in a Fragment of NReductions of Deductions in N−=Propositional LogicReferencesCompleteness in ProofTheoretic SemanticsIntroductionPrawitz's ConjectureFailure of Completeness for Intuitionistic LogicGoldfarb's Account of Dummett's ApproachDefinition 7ProofTheoretic Validity for Generalized Atomic SystemsGeneralized Atomic SystemsProofTheoretic ValidityLemma 3Failure of Strong CompletenessStrong Completeness ResultsFailure of CompletenessComparison with Kripke SemanticsA Completeness Result for Intuitionistic LogicDefinition 18Completeness Results for Classical LogicDefinition 19RemarksConclusionReferencesOpen Problems in ProofTheoretic SemanticsIntroductionThe Nature of Hypotheses and the Format of ProofsOpen Proofs and the Placeholder ViewThe NoAssumptions ViewBidirectionalityLocal and Global ProofTheoretic SemanticsThe Problem of HarmonyHarmony Based on Generalised RulesHarmony Based on EquivalenceThe Need for an Intensional Notion of HarmonyTowards a Definition of Strong HarmonyProofTheoretic Semantics Beyond LogicDefinitional ReflectionLogic, Paradoxes, Partial DefinitionsVariables and SubstitutionOutlook: Applications and Extensions of Definitional ReflectionReferences
