We consider a basic logic with two primitive uni-modal operators: one for certainty and the other for plausibility. The former is assumed to be a normal operator (corresponding - semantically - to a binary Kripke relation), while the latter is merely a classical operator (corresponding - semantically - to a neighborhood structure). We then define belief, interpreted as "maximally plausible possibility", in terms of these two notions: the agent believes φ if (1) she cannot rule out φ (that is, it is not the case that she is certain that ¬φ), (2) she judges φ to be plausible and (3) she does not judge ¬φ to be plausible. We consider several interaction properties between certainty and plausibility and study how these properties translate into properties of belief (positive and negative introspection, their converses, conjunction, etc.). We then prove that all the logics considered are minimal logics for the highlighted theorems. We also consider a number of possible interpretations of plausibility and identify the corresponding logics.
Certainty plausibility belief positive and negative introspection modal logic