Hypothesis Generation in Linear Temporal Logic for Clauses in a Restricted Syntactic Form

Possibility theory and modal logic are two knowledge representation frameworks that share some common features, such as the duality between possibility and necessity, as well as some obvious di↵erences since possibility theory is graded but is not primarily a logical setting. In the last thirty years there have been a series of attempts, reviewed in this paper, for bridging the two frameworks in one way or another. Possibility theory relies on possibility distributions and modal logic on accessibility relations, at the semantic level. Beyond the observation that many properties of possibility theory have qualitative counterparts in terms of axioms of well-known modal logic systems, the first works have looked for (graded) accessibility relations that can account for the behavior of possibility and necessity measures. More recently, another view has emerged from the study of logics of incomplete information, which is no longer based on Kripke-like models. On the one hand, possibilistic logic, closely related to possibility theory, mainly handles beliefs having various strength. On the other hand, in the so-called meta-epistemic logic (MEL) an agent can express both beliefs and explicitly ignored facts (both without strength), by only using modal formulas of depth 1, and no objective ones; its semantics is based on epistemic states. The system MEL is an extension of MEL having the syntax of S5. Generalized possibilistic logic (GPL) extends both possibilistic logic and MEL, and has a semantics in terms of sets of possibility distributions. After a survey of these di↵erent attempts, the paper presents GPL, a graded counterpart of MEL that extends MEL by allowing objective (sub)formulas. The axioms of GPL are graded counterparts of those of S5 modal system, the semantics being based on pairs made of an interpretation (representing the real state of facts) and a possibility distribution (representing an epistemic state). Soundness and completeness are established. The paper also discusses the di↵erence with S5 used as a logic for rough sets that accounts for indiscernibility rather than incomplete information, using also the square of opposition as a common structure underlying modal logic, possibility theory, and rough set theory.