Generalized Quantification as Substructural Logic
Natasha Alechina, Michiel van Lambalgen
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 se
quent calculus, is increased control over relations of dependence between
variables. In the case of generalized quantifiers, explicit attention to re
lations 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 firstorder conditions on the dependence relation.