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Generalized quantification as substructural logic

Published online by Cambridge University Press:  12 March 2014

Natasha Alechina  and
Michiel Van Lambalgen
Natasha Alechina
Affiliation:
Department of Mathematics and Computer Science, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands, E-mail: N.Alechina@cs.bham.ac.uk
Michiel Van Lambalgen
Affiliation:
Department of Mathematics and Computer Science, University of Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands, E-mail: michiell@fwi.uva.nl
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Abstract

We show how sequent calculi for some generalized quantifiers can be obtained by generalizing the Herbrand approach to ordinary first order proof theory. Typical of the Herbrand approach, as compared to plain sequent calculus, is increased control over relations of dependence between variables. In the case of generalized quantifiers, explicit attention to relations of dependence becomes indispensible for setting up proof systems. It is shown that this can be done by turning variables into structured objects, governed by various types of structural rules. These structured variables are interpreted semantically by means of a dependence relation. This relation is an analogue of the accessibility relation in modal logic. We then isolate a class of axioms for generalized quantifiers which correspond to first-order conditions on the dependence relation.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 3 , September 1996 , pp. 1006 - 1044
Copyright
Copyright © Association for Symbolic Logic 1996

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