Logics For Qualitative Reasoning

Abstract

Assertions and arguments involving vague notions occur often both in ordinary language and in many branches of science. The vagueness may be plainly expressed by “modifiers”, such as ‘generally’, ‘rarely’, ‘most’, ‘many’, etc., or, less obviously, conveyed by objects termed ‘representative’, ‘typical’ or ‘generic’. A precise treatment of such ideas has been a basic motivation for logics of qualitative reasoning. Here, we present some logical systems with generalized quantifiers for these modifiers, also handling ‘generic’ reasoning. Other possible applications for these and related logics for qualitative reasoning are indicated. These (monotonic) generalized logics, with simple sound and complete deductive calculi, are proper conservative extensions of classical first-order logic, with which they share various properties. For generic reasoning, special individuals can be introduced by means of ‘generally’, and internalized as representative constants, thereby producing conservative extensions where one can reason about generic objects as intended. Some interesting situations, however, require such assertions to be relative to various universes, which cannot be captured by relativization. Thus we extend our generalized logics to sorted versions, with qualitative notions relative to the universes, which can also be compared.