On the relation between possibilistic logic and modal logics of belief and knowledge

Abstract

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 differences 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 general Kripke-like models. On the one hand, possibilistic logic, closely related to possibility theory, mainly handles beliefs having various strengths. On the other hand, in the so-called meta-epistemic logic (MEL) an agent can express both all-or-nothing beliefs and explicitly ignored facts, by only using modal formulas of depth 1, and no objective ones; its semantics is based on subsets of interpretations viewed as epistemic states. The system MEL + is a KD45-like extension of MEL with objective formulas. 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 different attempts, the paper presents GPL + , a graded counterpart of MEL + that extends both MEL and GPL by allowing objective (sub)formulas. The axioms of GPL + are graded counterparts of those of MEL + , the semantics being based on pairs made of an interpretation (representing the real state of facts) and a possibility distribution (representing an epistemic state). S5-like extensions of MEL + and GPL + , called MEL ++ and GPL ++ respectively, are also considered. Soundness and completeness of GPL + and and GPL ++ are established. The paper also discusses the difference between MEL and S5 used as a standard epistemic logic, or used as a logic for rough sets that accounts for indiscernibility rather than incomplete information. We highlight the square of opposition as a common structure underlying modal logic, possibility theory, and rough set theory.

Possibility Theory and Possibilistic Logic: Tools for Reasoning Under and About Incomplete Information

“…GPL is in fact just a two-tiered standard propositional logic, in which propositional formulas are encapsulated by weighted modal operators, forming higher order propositional formulas, interpreted in terms of uncertainty measures from possibility theory. GPL can be still extended to a logic involving both objective and non-nested multimodal formulas [4].…”

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

“…Note that when α is fixed we get a fragment of the modal logic KD. See [5] for a survey of studies of the links between modal logics and possibility theory, and extensions of GPL with objective and non-nested multimodal formulas, as in KD45 and S5.…”

A Crash Course on Generalized Possibilistic Logic

“…The close connection between MEL and modal epistemic logic S5 and doxastic logic KD45 is very clear, as MEL uses a fragment of their language. Moreover, all inferences that can be made in MEL can equally be made in KD45 as proved in [11]. However there are noticeable differences, reflecting the fact that MEL is in some sense a minimal uncertainty logic:…”

“…So MEL can be viewed as the poor man's epistemic or doxastic logic; we cannot see the difference here because axiom T: p → p cannot be written in MEL (but we can include it by extending MEL to allow for objective formulas, see [11]). While KD45 is supposed to model an agent reflecting on its own beliefs, MEL is rather dedicated to represent what an agent knows about the epistemic state of another agent.…”

“…Note in particular that when α is fixed we get a fragment of the modal logic KD. See [11] for a survey of previous studies on the links between modal logics and possibility theory. When k = 1, GPL with a value scale Λ 1 coincides with MEL.…”