Advances in Proof-Theoretic Semantics.

• Superior document: Trends in Logic Series ; v.43
• Author: Piecha, Thomas.
• TeilnehmendeR: Schroeder-Heister, Peter.
• Place / Publishing House: Cham : : Springer International Publishing AG,, 2015.; Ã2016.
• Year of Publication: 2015
• Edition: 1st ed.
• Language: English
• Series: Trends in Logic Series
• Physical Description: 1 online resource (281 pages)

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.