Counting Objects

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Abstract
We survey results on axiomatic completeness and complexity of formal languages for reasoning about finite quantities. These languages are interpreted on domains with and without structure. Expressions dealt with include ‘at least n Xs are Ys,’ in the case of unstructured domains, and ‘at least n ’related‘ Xs are Ys,’ in the structured case. The results contained in this paper are brought together from generalized quantifier theory, modal logic, and knowledge representation, while a few new results are also included. A unifying approach is offered to the wide variety of formalisms available in the literature.

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