Qualitative Logic for 'Generally'

We examine logical systems with generalized quantifiers, for expressing and reasoning about assertions with 'generally'. The primary motivation is a qualitative, rather than quantitative, approach to some vague notions. Assertions and arguments involving notions, such as 'generally', 'most', 'many', etc., occur often in ordinary language and in some branches of science. Filters can be used for capturing an intuitive idea of 'generally'. This motivates the introduction of an operator to express 'generalized' assertions whose meaning is intended to be "belonging to a given filter (of 'important' sets)". These ideas are incorporated in a basic (unsorted) logic. Some interesting situations may require assertions relative to several universes, involving "most birds" and "most penguins" for instance. This leads to our sorted framework for reasoning about versions of 'generally' relative to various universes.