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Possibilistic logic and modal logic are knowledge representation frameworks sharing some common features, such as the duality between possibility and necessity, and the decomposability of necessity for conjunctions, as well as some obvious differences since possibility theory is graded. At the semantic level, possibilistic logic relies on possibility distributions and modal logic on accessibility relations. In the last 30 years, there have been a series of attempts for bridging the two frameworks in one way or another. In this paper, we compare the relational semantics of epistemic logics (such as KD45 and S5) with simpler possibilistic semantics of a fragment of such logics that only uses modal formulas of depth 1. This minimal epistemic logic handles both all-or-nothing beliefs and explicitly ignored facts. We also contrast epistemic logic with the S5-based rough set logic. Finally, this paper presents extensions of generalized possibilistic logic with objective and non-nested multimodal formulas, in the style of modal logics KD45 and S5.

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

This paper proposes an extension of the MEL logic to a language containing modal formulae of depth 0 or 1 only. MEL is a logic of incomplete information where an agent can express both beliefs and explicitly ignored facts, that only uses modal formulae of depth 1, and no objective ones. The extended logic, called MEL+ has the same axioms as, and is in some sense equivalent to, S5 with a restricted language, but with the same expressive power. The semantics is not based on Kripke models with equivalence relations, but on pairs made of an interpretation (representing the real state of facts) and a non-empty set of possible interpretations (representing an epistemic state). Soundness and com- pleteness are established. We provide a rationale for using our approach when an agent reasons about what is known of the epistemic state of another agent and compares it with what is known about the real world. Our approach can be viewed as an alternative to the basic epistemic logic not concerned with introspection. We discuss the difference with S5 used as a logic for rough sets, and the similarity with some previous non-monotonic logics of knowledge is highlighted.

This paper considers a number of alternative rough consequence logics which come in a natural way from the logics L r and LR previously studied by Chakraborty and Banerjee. The systems have been compared to variants of S5, and the logic Triv of Hughes and Cresswell. A comparison has also been made with L r and LR , and therefore with Jaśkowski’s discussive logic J, as J is equivalent toL R .