Advances in Proof-Theoretic Semantics.
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| Other Authors: | |
| Format: | eBook |
| Language: | English |
| Published: |
Cham :
Springer International Publishing AG,
2015.
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| Edition: | 1st ed. |
| Series: | Trends in Logic Series
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| Subjects: | |
| Online Access: | Click to View |
Table of Contents:
- Intro
- Contents
- Advances in Proof-Theoretic Semantics: Introduction
- References
- 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
- 3.2 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 (2) is intuitionistically valid
- 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
- 3 Normalization by Hilfssatz
- 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 Straw Model Theory
- 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
- References
- On Dummett's ``Proof-Theoretic Justifications of Logical Laws''
- 1 Analysis of the Method
- 2 Boundary Rules
- 3 Schematic Inferences
- 4 Assessment
- References
- Self-contradictory Reasoning
- 1 Introduction
- 2 Meaning Conditions
- 3 The Liar Paradox
- 4 Self-contradictory Reasoning in N-forall=
- 5 Self-contradictory Reasoning in N-=
- 6 Self-contradictory Reasoning in N-=
- 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
- 5 Proof-Theoretic Validity for Generalized Atomic Systems
- 5.1 Generalized Atomic Systems
- 5.2 Proof-Theoretic Validity
- 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
- 6 Completeness Results for Classical Logic
- 6.1 Other Logical Constants
- 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.