QUALITATIVE MODALITIES

Abstract

We add a binary operator ≥ to the logical language, with intended meaning of φ<ψ: ‘φ is at least as likely, probable, or trustworthy, as ψ’. The operator ≥ is interpreted on Kripke structures, making it possible to define the standard necessity operator □ in terms of ≥. The operator ≥ provides us with an intermediate for the K-axiom, in the sense that we have both □(p→q)→(q≥p) and (q≥p)→(□p→□q). We discuss two semantics for this binary modal operator. It turns out that, as shown by Gärdenfors and Segerberg, ≥ is not only too weak to distinguish finite models from infinite ones or to distinguish countable additivity from finite additivity, ≥ also cannot distinguish sophisticated ways of assigning exact probabilities to events (‘measuring’) from the conceptually simpler task of just counting them.