Possibilistic Logic: From Certainty-Qualified Statements to Two-Tiered Logics – A Prospective Survey

Possibilistic logic (PL) is more than thirty years old. The paper proposes a survey of its main developments and applications in artificial intelligence, together with a short presentation of works in progress. PL amounts to a classical logic handling of certainty-qualified statements. Certainty is estimated in the setting of possibility theory as a lower bound of a necessity set-function. An elementary possibilistic formula is a pair made of a classical logic formula, and a certainty level belonging to a bounded scale. Basic PL handles only conjunctions of such formulas, and PL bases can be viewed as classical logic bases layered in terms of certainty. Semantics is in terms of epistemic states represented by fuzzy sets of interpretations. A PL base is associated with an inconsistency level above which formulas are safe from inconsistency. Applications include reasoning with default rules, belief revision, Bayesian possibilistic networks, information fusion, and preference modeling (in this latter case, certainty is turned into priority). Different extensions of basic PL are briefly reviewed, where levels take values in lattices, are replaced by vectors of levels, or are handled in a purely symbolic manner (without being instantiated). This latter extension may be of interest for explanation purposes. A paraconsistent treatment of inconsistency is also discussed. Still another extension allows for associating possibilistic formulas with sets of agents or sources that support them. In generalized possibilistic logic (GPL), negation and disjunction can be applied as well as conjunction, to possibilistic formulas. It may be viewed as a fragment of modal logic (such as KD45) where modalities cannot be nested. GPL can be still extended to a logic involving both objective and non-nested multimodal formulas. Applications of GPL to the modeling of ignorance, to the representation of answer set programs, to reasoning about other agents’ beliefs, and to a logic of argumentation are outlined. Generally speaking, the interest and the strength of PL relies on a sound alliance between classical logic and possibility theory which offers a rich representation setting allowing an accurate modeling of partial ignorance. The paper focuses more on ideas than on technicalities and provides references for details (Invited talk presented by the second author).