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Advances in Proof-Theoretic Semantics
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Advances in Proof-Theoretic Semantics: Introduction
On the Relation Between Heyting's and Gentzen's Approaches to Meaning
1 Introduction
2 Heyting's Approach to Meaning
3 Gentzen's Approach to Meaning
4 A First Comparison Between Heyting's and Gentzen's Approaches
5 Further Development of Gentzen's Ideas
5.1 Argument Structures
5.2 Arguments
5.3 Validity of Arguments
6 Weak and Strong Validity and Their Features
7 Mappings of Valid Arguments on BHK-Proofs and Vice Versa
7.1 Extending the Mapping Proof to Arguments for A
7.2 Extending the Mapping Arg to BHK-Proofs of A
8 Concluding Remarks
References
Kreisel's Theory of Constructions, the Kreisel-Goodman Paradox, and the Second Clause
1 Introduction
2 Predicativity, Decidability, and the BHK Interpretation
3 The Theory of Constructions and the Second Clause
3.1 An Overview of the Theory of Constructions
3.1.1 The Language of T
3.1.2 The Axiomatization of T
3.1.3 Formalizing the BHK Interpretation in T
3.1.4 Soundness, Completeness, and Internalization
3.2 The Kreisel-Goodman Paradox
4 The Reception of the Theory of Constructions and the Second Clause
4.1 Shifting Opinions
4.2 Guilt by Association?
5 Diagnosing the Paradox
5.1 Self-Reference and Typing
5.2 Stratification
5.3 Decidability
5.4 Reflection
5.5 Internalization
6 Conclusions and Further Work
References
On the Paths of Categories
1 Functions of Language
2 Deductions Not Necessarily Based on Propositions
3 Deductions in Categories
4 Deductions in Multicategories and Polycategories
5 Rules for Deductions
References
Some Remarks on Proof-Theoretic Semantics
1 Background on General Elimination Rules
2 Is Bullet a Logical Constant?
3 The GE-rule for Implication and the Type-Theoretic Dependent Product Type
4 GE-Rules in General
4.1 Several I-Rules
4.2 I-Rule Has Several Premisses
4.3 Premiss of I-Rule Discharges Some Assumptions
4.4 GE Harmony: A Counter-Example
4.5 Another [Counter-]Example
4.6 In Other Words
5 Conclusion
References
Categorical Harmony and Paradoxes in Proof-Theoretic Semantics
1 Introduction
2 The Principle of Categorical Harmony
3 Categorical Harmony in Comparison with Other Principles
4 Degrees of Paradoxicality of Logical Constants
5 Concluding Remarks: From Semantic Dualism to Duality
References
The Paradox of Knowability from an Intuitionistic Standpoint
1 Introduction
2 An Intuitionistic Solution
2.1 (21) is intuitionistically valid
Definition 1
2.2 Truth Notions
2.3 Internal and Intuitive Truth
2.4 Unknown Statements
3 Neo-Verificationist Approaches
4 How Is a Rational Discussion Possible?
5 Conclusion
References
Explicit Composition and Its Application in Proofs of Normalization
1 Introduction
2 Notation for Natural Derivations
4 Strong Normalization by Bar Induction
5 Concluding Remarks and Further Applications
References
Towards a Proof-Theoretic Semantics of Equalities
1 Frege's Question
2 Equality Versus Identity
3 The Mode of Presentation
4 Morning Star Versus Evening Star Revisited
5 Equality
6 Equality of Senses
7 Proof-Theoretic Semantics
References
On the Proof-Theoretic Foundations of Set Theory
1 Introduction
2 Defining Sets
3 Functional Closure, Local Logic and the Notion of Absoluteness
3.1 The Functional Closure
3.2 Local Logic
3.3 Absoluteness
4 A Proof-Theoretic Interpretation
5 Sets
6 Foundational Issues
References
A Strongly Differing Opinion on Proof-Theoretic Semantics?
1.1 Tarski's Definition of Logical Consequence
1.2 Model Theory
1.3 Model-Theoretic Semantics
2 Defining Meanings in General
2.1 Defining Meanings: Specialise Then Generalise
2.2 Representing the Meaning
3 Defining Logical Consequence
References
Comments on an Opinion
On Dummett's “Proof-Theoretic Justifications of Logical Laws”
1 Analysis of the Method
2 Boundary Rules
3 Schematic Inferences
4 Assessment
Appendix
Author's Postscript, January 2015
References
Self-contradictory Reasoning
1 Introduction
2 Meaning Conditions
3 The Liar Paradox
4 Self-contradictory Reasoning in N−∀∃=
5 Self-contradictory Reasoning in N−∃=
6 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
1 Introduction
2 Prawitz's Conjecture
3 Failure of Completeness for Intuitionistic Logic
4 Goldfarb's Account of Dummett's Approach
Definition 7
5 Proof-Theoretic Validity for Generalized Atomic Systems
5.1 Generalized Atomic Systems
5.2 Proof-Theoretic Validity
Lemma 3
5.3 Failure of Strong Completeness
5.4 Strong Completeness Results
5.5 Failure of Completeness
5.6 Comparison with Kripke Semantics
5.7 A Completeness Result for Intuitionistic Logic
Definition 18
6 Completeness Results for Classical Logic
Definition 19
6.2 Remarks
7 Conclusion
References
Open Problems in Proof-Theoretic Semantics
1 Introduction
2 The Nature of Hypotheses and the Format of Proofs
2.1 Open Proofs and the Placeholder View
2.2 The No-Assumptions View
2.3 Bidirectionality
2.4 Local and Global Proof-Theoretic Semantics
3 The Problem of Harmony
3.1 Harmony Based on Generalised Rules
3.2 Harmony Based on Equivalence
3.3 The Need for an Intensional Notion of Harmony
3.4 Towards a Definition of Strong Harmony
4 Proof-Theoretic Semantics Beyond Logic
4.1 Definitional Reflection
4.2 Logic, Paradoxes, Partial Definitions
4.3 Variables and Substitution
4.4 Outlook: Applications and Extensions of Definitional Reflection
References
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