A unified view of some formalisms handling incomplete and inconsistent information (Talk @ UNILOG 2018)
Résumé
Sets of formulas in classical logic are often called knowledge or belief bases, as containing explicit information held by an agent. This framework does not allow for reasoning about ignorance. The issue of reasoning about incomplete information or ignorance has been addressed independently in three communities: - in uncertainty management, scholars have for a long time used additive set-functions to represent belief often using numerical measurement methods like in subjective probability theory, and more recently using non-additive monotonic set functions like possibility and necessity measures, Shafer’s belief and plausibility functions, Walley’s upper and lower previsions. - in logic there has been two main trends. Very early in the XXth century, some logicians have tried to handle the notion of ignorance by means of an additional truth-value, like Kleene and Lukasiewicz for instance. More recently, the full power of modal logic has been exploited to develop epistemic or doxastic logics, especially using extensions of systemKD45. This paper proposes a formal framework in the form of a two-tiered propositional logic, which can capture the three approaches in the setting of possibility theory. We recall a simplified version of epistemic logic that can be extended to graded beliefs and can capture three-valued logics of in-complete information. The graded version of this minimal epistemic logic is an expressive generalization of possibilistic logic. Then we propose a general framework where any set function representing uncertainty can be accommodated. It can account for multiple conflicting sources of information, and in particular, Belnap logic can be encoded in this formalism.