UID:
almahu_9949602159902882
Format:
1 online resource (281 pages)
Edition:
1st ed.
ISBN:
9783319226866
Series Statement:
Trends in Logic Series ; v.43
Note:
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.
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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.
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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.
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4.4 Outlook: Applications and Extensions of Definitional Reflection -- References.
Additional Edition:
Print version: Piecha, Thomas Advances in Proof-Theoretic Semantics Cham : Springer International Publishing AG,c2015 ISBN 9783319226859
Language:
English
Keywords:
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URL:
http://dx.doi.org/10.1007/978-3-319-22686-6
