Journal of Symbolic Logic

Generalized Quantification as Substructural Logic

Natasha Alechina and Michiel van Lambalgen

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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.

Article information

Source
J. Symbolic Logic, Volume 61, Issue 3 (1996), 1006-1044.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183745089

Mathematical Reviews number (MathSciNet)
MR1412522

Zentralblatt MATH identifier
0858.03051

JSTOR
links.jstor.org

Citation

Alechina, Natasha; van Lambalgen, Michiel. Generalized Quantification as Substructural Logic. J. Symbolic Logic 61 (1996), no. 3, 1006--1044. https://projecteuclid.org/euclid.jsl/1183745089


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