The functional interpretation of logical deduction /

The functional interpretation of logical deduction /

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Bibliographic Details
Author / Creator:Queiroz, Ruy J. G. B. de
Imprint:Singapore ; Hackensack, N.J. : World Scientific, c2012.
Description:xxxii, 266 p. ; 24 cm.
Language:English
Series:Advances in logic ; v. 5
Advances in logic ; v. 5.
Subject:
Logic.
Modality (Logic)
Logic.
Modality (Logic)
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/8681109
Hidden Bibliographic Details
Other authors / contributors:Oliveira, Anjolina G. de.
Gabbay, Dov M., 1945-
ISBN:9789814360951
9814360953
Notes:Includes bibliographical references (p. 253-264) and index.
Table of Contents:
  • Preface
  • Overview
  • 1. Labelled Natural Deduction
  • 1.1. The róle of the labels
  • 1.1.1. Dividing the tasks: A functional calculus on the labels, a logical calculus on the formula
  • 1.1.2. Reassessing Frege's two-dimensional calculus
  • 1.2. Canonical proofs and normalisation
  • 1.2.1. Canonical proofs
  • 1.2.2. Normalisation
  • 2. The Functional Interpretation of Implication
  • 2.1. Introduction
  • 2.2. Origins
  • 2.3. Types and propositions
  • 2.4. A-abstraction and implication
  • 2.5. Consistency proof
  • 2.6. Systems of implication and combinators
  • 2.7. Finale
  • 3. The Existential Quantifier
  • 3.1. Motivation
  • 3.1.1. The pairing interpretation
  • 3.2. Quantifiers and normalisation
  • 3.2.1. Introducing variables for the Skolem dependency functions
  • 3.2.2. The hiding principle
  • 3.3. Other approaches to existential quantification
  • 3.3.1. Systems of natural deduction based on direct existential instantiation
  • 3.3.2. Axiomatic systems based on the notion of 'such that'
  • 3.4. Model-theoretic semantics
  • 3.4.1. Constants versus variables revisited
  • 3.4.2. Eliminability and conservative extensions
  • 3.5. Finale
  • 3.5.1. Extensions to higher-order existentials
  • 3.5.2. Further connections to model-theoretic interpretations
  • 3.6. Examples of deduction
  • 3.6.1. Generic examples
  • 3.6.2. Specific examples
  • 4. Normalisation
  • 4.1. Introduction
  • 4.2. Proof transformations in labelled deduction
  • 4.3. Equivalences between proofs in LND
  • 4.4. The term rewriting system for LND
  • 4.4.1. Defining the LND-TRS
  • 4.4.2. The sort decreasing property
  • 4.4.3. Defining an order
  • 4.4.4. Proving the termination property
  • 4.4.5. Proving the confluence property
  • 4.5. Examples of transformations between proofs
  • 4.6. Final remarks
  • 5. Natural Deduction for Equality
  • 5.1. Introduction
  • 5.2. Labelled deduction
  • 5.2.1. Identifiers for (compositions of) equalities
  • 5.2.2. The proof rules
  • 5.3. Finale
  • 6. Normalisation for the Equality Fragment
  • 6.1. General rules
  • 6.2. The 'subterm substitution' rule
  • 6.3. The ¿- and ¿-rules
  • 6.4. Term rewriting systems
  • 6.4.1. Termination property
  • 6.4.2. Confluence property
  • 6.4.3. The completion procedure
  • 6.5. The transformations between proofs in the equational fragment of the LND
  • 6.5.1. Reductions on the rewrite sequence
  • 6.5.2. Transformations on the rewrite reasons
  • 6.6. The rewriting system for the LND equational logic
  • 6.6.1. Termination property for the LND EQ -TRS
  • 6.6.2. Confluence property for the LND EQ -TRS
  • 6.6.3. The normalization theorems
  • 6.7. The normalization procedure: some examples
  • 6.8. Final remarks
  • 6.9. Appendix: The ß- and ¿-reductions for the LND system
  • 6.9.1. ß-type reductions
  • 6.9.2. ¿-type reductions
  • 6.9.3. ¿-type transformations: The permutative transformations
  • 6.10. Termination property for LND EQ -TRS
  • 7. Modal Logics
  • 7.1. The functional interpretation
  • 7.1.1. Handling assumptions with 'world' variables
  • 7.1.2. Natural deduction with an extra parameter
  • 7.1.3. Connections with Kripke's truth definition
  • 7.1.4. Connections with Gentzen's sequent calculus
  • 7.2. Modal logics and the functional interpretation
  • 7.2.1. Chellas' RM rule for standard normative logics
  • 7.2.2. Modal logic K
  • 7.2.3. Modal logic D
  • 7.2.4. Modal logic T
  • 7.2.5. Modal logic B
  • 7.2.6. Modal logic S4 and its parallel with intuitionistic logic
  • 7.2.7. Grzegorczyk's extension of S4
  • 7.2.8. Modal logic S5 and its parallel with classical logic
  • 7.2.9. Non-normal modal logics
  • 7.2.10. Problematic cases: The scope of 'necessity'
  • 7.3. Finale
  • 8. Meaning and Proofs: A Reflection on Proof-Theoretic Semantics
  • 8.1. Proof-theoretic semantics
  • 8.2. Meaning, use and consequences
  • 8.3. Meaning and purpose
  • 8.4. Meaning and use
  • 8.5. Meaning and the explanation of consequences
  • 8.6. Use and the explanation of consequences
  • 8.7. Early signs of 'meaning-use/usefulness-consequences'
  • 8.8. Normalisation of proofs: the explanation of the consequences
  • 8.9. Concluding remarks
  • Bibliography
  • Index

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