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    Functional Interpretation of Logical Deduction

    ISBN-10: 9814360953
    ISBN-13: 9789814360951
    Author(s): Ruy J. G. B. de Queiroz, Anjolina G. De Oliveira, Dov M. Gabbay
    Description: This comprehensive book provides an adequate framework to establish various calculi of logical inference. Being an 'enriched' system of natural deduction, it helps to formulate logical calculi in an operational manner. By uncovering a certain  More...
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    Publisher: World Scientific Publishing Company, Incorporated
    Binding: Hardcover
    Pages: 300
    Size: 6.50" wide x 9.50" long x 0.75" tall
    Weight: 1.210
    Language: English

    This comprehensive book provides an adequate framework to establish various calculi of logical inference. Being an 'enriched' system of natural deduction, it helps to formulate logical calculi in an operational manner. By uncovering a certain harmony between a functional calculus on the labels and a logical calculus on the formulas, it allows mathematical foundations for systems of logic presentation designed to handle meta-level features at the object-level via a labelling mechanism, such as the D Gabbay's Labelled Deductive Systems. The book truly demonstrates that introducing of 'labels' is useful to understand the proof-calculus itself, and also to clarify its connections with model-theoretic interpretations.

    Preface
    Overview
    Labelled Natural Deduction
    The r�le of the labels
    Dividing the tasks: A functional calculus on the labels, a logical calculus on the formula
    Reassessing Frege's two-dimensional calculus
    Canonical proofs and normalisation
    Canonical proofs
    Normalisation
    The Functional Interpretation of Implication
    Introduction
    Origins
    Types and propositions
    A-abstraction and implication
    Consistency proof
    Systems of implication and combinators
    Finale
    The Existential Quantifier
    Motivation
    The pairing interpretation
    Quantifiers and normalisation
    Introducing variables for the Skolem dependency functions
    The hiding principle
    Other approaches to existential quantification
    Systems of natural deduction based on direct existential instantiation
    Axiomatic systems based on the notion of 'such that'
    Model-theoretic semantics
    Constants versus variables revisited
    Eliminability and conservative extensions
    Finale
    Extensions to higher-order existentials
    Further connections to model-theoretic interpretations
    Examples of deduction
    Generic examples
    Specific examples
    Normalisation
    Introduction
    Proof transformations in labelled deduction
    Equivalences between proofs in LND
    The term rewriting system for LND
    Defining the LND-TRS
    The sort decreasing property
    Defining an order
    Proving the termination property
    Proving the confluence property
    Examples of transformations between proofs
    Final remarks
    Natural Deduction for Equality
    Introduction
    Labelled deduction
    Identifiers for (compositions of) equalities
    The proof rules
    Finale
    Normalisation for the Equality Fragment
    General rules
    The 'subterm substitution' rule
    The �- and �-rules
    Term rewriting systems
    Termination property
    Confluence property
    The completion procedure
    The transformations between proofs in the equational fragment of the LND
    Reductions on the rewrite sequence
    Transformations on the rewrite reasons
    The rewriting system for the LND equational logic
    Termination property for the LND<sub>EQ</sub>-TRS
    Confluence property for the LND<sub>EQ</sub>-TRS
    The normalization theorems
    The normalization procedure: some examples
    Final remarks
    Appendix: The �- and �-reductions for the LND system
    �-type reductions
    �-type reductions
    �-type transformations: The permutative transformations
    Termination property for LND<sub>EQ</sub>-TRS
    Modal Logics
    The functional interpretation
    Handling assumptions with 'world' variables
    Natural deduction with an extra parameter
    Connections with Kripke's truth definition
    Connections with Gentzen's sequent calculus
    Modal logics and the functional interpretation
    Chellas' RM rule for standard normative logics
    Modal logic K
    Modal logic D
    Modal logic T
    Modal logic B
    Modal logic S4 and its parallel with intuitionistic logic
    Grzegorczyk's extension of S4
    Modal logic S5 and its parallel with classical logic
    Non-normal modal logics
    Problematic cases: The scope of 'necessity'
    Finale
    Meaning and Proofs: A Reflection on Proof-Theoretic Semantics
    Proof-theoretic semantics
    Meaning, use and consequences
    Meaning and purpose
    Meaning and use
    Meaning and the explanation of consequences
    Use and the explanation of consequences
    Early signs of 'meaning-use/usefulness-consequences'
    Normalisation of proofs: the explanation of the consequences
    Concluding remarks
    Bibliography
    Index

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