A min-max semantics for fuzzy likelihood

The problem of formalizing the notion of likelihood is considered, as expressed by adjectives such as likely, very likely, rather unlikely, certain, and so on. A formal logic is devised which operates at two distinct semantic levels: a lower level where propositions are assigned a degree of likelihood in the interval [0,1] and a second level where the likelihood adjectives are represented as modal operators. Additionally, at the lower level a formal distinction is made between those propositions which must be true or false purely because of their syntactic structure, and those which may be true or false depending on their interpretations, and min is applied for conjunctions and max for disjunctions only in the case of the latter. This leads to a coherent method of reasoning with fuzzy likelihood