Advances in Proof-Theoretic Semantics -

On the Relation Between Heyting's and Gentzen's Approaches to Meaning Introduction Heyting's Approach to Meaning Gentzen's Approach to Meaning A First Comparison Between Heyting's and Gentzen's Approaches Further Development of Gentzen's Ideas Argument Structures Arguments Validity of Arguments Weak and Strong Validity and Their Features Mappings of Valid Arguments on BHK-Proofs and Vice Versa Extending the Mapping Proof to Arguments for A Extending the Mapping Arg to BHK-Proofs of A Concluding Remarks References Kreisel's Theory of Constructions, the Kreisel-Goodman Paradox, and the Second Clause Introduction Predicativity, Decidability, and the BHK Interpretation The Theory of Constructions and the Second Clause An Overview of the Theory of Constructions The Language of T The Axiomatization of T Formalizing the BHK Interpretation in T Soundness, Completeness, and Internalization The Kreisel-Goodman Paradox The Reception of the Theory of Constructions and the Second Clause Shifting Opinions Guilt by Association?Diagnosing the Paradox Self-Reference and Typing Stratification Decidability Reflection Internalization Conclusions and Further Work References On the Paths of Categories Functions of Language Deductions Not Necessarily Based on Propositions Deductions in Categories Deductions in Multicategories and Polycategories Rules for Deductions References Some Remarks on Proof-Theoretic Semantics Background on General Elimination Rules Is Bullet a Logical Constant?The GE-rule for Implication and the Type-Theoretic Dependent Product Type GE-Rules in General Several I-Rules I-Rule Has Several Premisses Premiss of I-Rule Discharges Some Assumptions GE Harmony: A Counter-Example Another [Counter-]Example In Other Words Conclusion References Categorical Harmony and Paradoxes in Proof-Theoretic Semantics Introduction The Principle of Categorical Harmony Categorical Harmony in Comparison with Other Principles Degrees of Paradoxicality of Logical Constants Concluding Remarks: From Semantic Dualism to Duality References The Paradox of Knowability from an Intuitionistic Standpoint Introduction An Intuitionistic Solution (21) is intuitionistically valid Definition 1 Truth Notions Internal and Intuitive Truth Unknown Statements Neo-Verificationist Approaches How Is a Rational Discussion Possible?Conclusion References Explicit Composition and Its Application in Proofs of Normalization Introduction Notation for Natural Derivations Strong Normalization by Bar Induction Concluding Remarks and Further Applications References Towards a Proof-Theoretic Semantics of Equalities Frege's Question Equality Versus Identity The Mode of Presentation Morning Star Versus Evening Star Revisited Equality Equality of Senses Proof-Theoretic Semantics References On the Proof-Theoretic Foundations of Set Theory Introduction Defining Sets Functional Closure, Local Logic and the Notion of Absoluteness The Functional Closure Local Logic Absoluteness A Proof-Theoretic Interpretation Sets Foundational Issues References A Strongly Differing Opinion on Proof-Theoretic Semantics?Tarski's Definition of Logical Consequence Model Theory Model-Theoretic Semantics Defining Meanings in General Defining Meanings: Specialise Then Generalise Representing the Meaning Defining Logical Consequence References Comments on an Opinion On Dummett's “Proof-Theoretic Justifications of Logical Laws”Analysis of the Method Boundary Rules Schematic Inferences Assessment Appendix Author's Postscript, January 2015 References Self-contradictory Reasoning Introduction Meaning Conditions The Liar Paradox Self-contradictory Reasoning in N−∀∃=Self-contradictory Reasoning in N−∃=Self-contradictory Reasoning in N−=Appendix Naïve Set Theory Normal Deductions in a Fragment of N Reductions of Deductions in N−=Propositional Logic References Completeness in Proof-Theoretic Semantics Introduction Prawitz's Conjecture Failure of Completeness for Intuitionistic Logic Goldfarb's Account of Dummett's Approach Definition 7 Proof-Theoretic Validity for Generalized Atomic Systems Generalized Atomic Systems Proof-Theoretic Validity Lemma 3 Failure of Strong Completeness Strong Completeness Results Failure of Completeness Comparison with Kripke Semantics A Completeness Result for Intuitionistic Logic Definition 18 Completeness Results for Classical Logic Definition 19 Remarks Conclusion References Open Problems in Proof-Theoretic Semantics Introduction The Nature of Hypotheses and the Format of Proofs Open Proofs and the Placeholder View The No-Assumptions View Bidirectionality Local and Global Proof-Theoretic Semantics The Problem of Harmony Harmony Based on Generalised Rules Harmony Based on Equivalence The Need for an Intensional Notion of Harmony Towards a Definition of Strong Harmony Proof-Theoretic Semantics Beyond Logic Definitional Reflection Logic, Paradoxes, Partial Definitions Variables and Substitution Outlook: Applications and Extensions of Definitional Reflection References