A Logic for Uncertain Probabilities

In standard logic, propositions are considered to be either true or false. However, a fundamental aspect of the human condition is that nobody can ever determine with absolute certainty whether a proposition about the world is true or false. In addition, whenever the truth of a proposition is assessed, it is always done by an individual, and it can never be considered to represent a general and objective belief. This indicates that important aspects are missing in the way standard logic captures our perception of reality, and that it is more designed for an idealised world than for the subjective world in which we are all living. Several alternative calculi and logics which take uncertainty and ignorance into consideration have been proposed and quite successfully applied to practical problems where conclusions have to be made based on insufficient evidence (see for example Hunter 1996 or Motro & Smets 1997 for an analysis of some uncertainty logics and calculi). Although including uncertainty in the belief model is a significant step forward, it only goes half the way in realising the real nature of human beliefs. It is also necessary to take into account that beliefs always are held by individuals and that beliefs for this reason are fundamentally subjective. In this paper we describe subjective logic (see Jøsang 1997 for an earlier version) as a logic which operates on subjective beliefs about the world, and use the term opinion to denote the representation of a subjective belief. Subjective logic operates on opinions and contains standard logical operators in addition to some non-standard operators which specifically depend on belief ownership. An opinion can be interpreted as a probability measure containing secondary uncertainty, and as such subjective logic can be seen as an extension of both probability calculus and binary logic.