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Advances in Proof-Theoretic Semantics - Thomas Piecha


Year 2016

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Advances in Proof-Theoretic Semantics: IntroductionOn the Relation Between Heyting's and Gentzen's Approaches to Meaning1 Introduction2 Heyting's Approach to Meaning3 Gentzen's Approach to Meaning4 A First Comparison Between Heyting's and Gentzen's Approaches5 Further Development of Gentzen's Ideas5.1 Argument Structures5.2 Arguments5.3 Validity of Arguments6 Weak and Strong Validity and Their Features7 Mappings of Valid Arguments on BHK-Proofs and Vice Versa7.1 Extending the Mapping Proof to Arguments for A7.2 Extending the Mapping Arg to BHK-Proofs of A8 Concluding RemarksReferencesKreisel's Theory of Constructions, the Kreisel-Goodman Paradox, and the Second Clause1 Introduction2 Predicativity, Decidability, and the BHK Interpretation3 The Theory of Constructions and the Second Clause3.1 An Overview of the Theory of Constructions3.1.1 The Language of T3.1.2 The Axiomatization of T3.1.3 Formalizing the BHK Interpretation in T3.1.4 Soundness, Completeness, and Internalization3.2 The Kreisel-Goodman Paradox4 The Reception of the Theory of Constructions and the Second Clause4.1 Shifting Opinions4.2 Guilt by Association?5 Diagnosing the Paradox5.1 Self-Reference and Typing5.2 Stratification5.3 Decidability5.4 Reflection5.5 Internalization6 Conclusions and Further WorkReferencesOn the Paths of Categories1 Functions of Language2 Deductions Not Necessarily Based on Propositions3 Deductions in Categories4 Deductions in Multicategories and Polycategories5 Rules for DeductionsReferencesSome Remarks on Proof-Theoretic Semantics1 Background on General Elimination Rules2 Is Bullet a Logical Constant?3 The GE-rule for Implication and the Type-Theoretic Dependent Product Type4 GE-Rules in General4.1 Several I-Rules4.2 I-Rule Has Several Premisses4.3 Premiss of I-Rule Discharges Some Assumptions4.4 GE Harmony: A Counter-Example4.5 Another [Counter-]Example4.6 In Other Words5 ConclusionReferencesCategorical Harmony and Paradoxes in Proof-Theoretic Semantics1 Introduction2 The Principle of Categorical Harmony3 Categorical Harmony in Comparison with Other Principles4 Degrees of Paradoxicality of Logical Constants5 Concluding Remarks: From Semantic Dualism to DualityReferencesThe Paradox of Knowability from an Intuitionistic Standpoint1 Introduction2 An Intuitionistic Solution2.1 (21) is intuitionistically validDefinition 12.2 Truth Notions2.3 Internal and Intuitive Truth2.4 Unknown Statements3 Neo-Verificationist Approaches4 How Is a Rational Discussion Possible?5 ConclusionReferencesExplicit Composition and Its Application in Proofs of Normalization1 Introduction2 Notation for Natural Derivations4 Strong Normalization by Bar Induction5 Concluding Remarks and Further ApplicationsReferencesTowards a Proof-Theoretic Semantics of Equalities1 Frege's Question2 Equality Versus Identity3 The Mode of Presentation4 Morning Star Versus Evening Star Revisited5 Equality6 Equality of Senses7 Proof-Theoretic SemanticsReferencesOn the Proof-Theoretic Foundations of Set Theory1 Introduction2 Defining Sets3 Functional Closure, Local Logic and the Notion of Absoluteness3.1 The Functional Closure3.2 Local Logic3.3 Absoluteness4 A Proof-Theoretic Interpretation5 Sets6 Foundational IssuesReferencesA Strongly Differing Opinion on Proof-Theoretic Semantics?1.1 Tarski's Definition of Logical Consequence1.2 Model Theory1.3 Model-Theoretic Semantics2 Defining Meanings in General2.1 Defining Meanings: Specialise Then Generalise2.2 Representing the Meaning3 Defining Logical ConsequenceReferencesComments on an OpinionOn Dummett's “Proof-Theoretic Justifications of Logical Laws”1 Analysis of the Method2 Boundary Rules3 Schematic Inferences4 AssessmentAppendixAuthor's Postscript, January 2015ReferencesSelf-contradictory Reasoning1 Introduction2 Meaning Conditions3 The Liar Paradox4 Self-contradictory Reasoning in N−∀∃=5 Self-contradictory Reasoning in N−∃=6 Self-contradictory Reasoning in N−=AppendixNaïve Set TheoryNormal Deductions in a Fragment of NReductions of Deductions in N−=Propositional LogicReferencesCompleteness in Proof-Theoretic Semantics1 Introduction2 Prawitz's Conjecture3 Failure of Completeness for Intuitionistic Logic4 Goldfarb's Account of Dummett's ApproachDefinition 75 Proof-Theoretic Validity for Generalized Atomic Systems5.1 Generalized Atomic Systems5.2 Proof-Theoretic ValidityLemma 35.3 Failure of Strong Completeness5.4 Strong Completeness Results5.5 Failure of Completeness5.6 Comparison with Kripke Semantics5.7 A Completeness Result for Intuitionistic LogicDefinition 186 Completeness Results for Classical LogicDefinition 196.2 Remarks7 ConclusionReferencesOpen Problems in Proof-Theoretic Semantics1 Introduction2 The Nature of Hypotheses and the Format of Proofs2.1 Open Proofs and the Placeholder View2.2 The No-Assumptions View2.3 Bidirectionality2.4 Local and Global Proof-Theoretic Semantics3 The Problem of Harmony3.1 Harmony Based on Generalised Rules3.2 Harmony Based on Equivalence3.3 The Need for an Intensional Notion of Harmony3.4 Towards a Definition of Strong Harmony4 Proof-Theoretic Semantics Beyond Logic4.1 Definitional Reflection4.2 Logic, Paradoxes, Partial Definitions4.3 Variables and Substitution4.4 Outlook: Applications and Extensions of Definitional ReflectionReferences
 
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