A Reconstruction of Default Conditionals within Epistemic Logic

Abstract

Default conditionals are statements that express a condition of normality, in the form ‘if φ then normally ψ’ and are of primary importance in Knowledge Representation. There exist modal approaches to the construction of conditional logics of normality. Most of them are built on notions of preference among possible worlds, corresponding to the semantic intuition that φ ⇒ ψ is true in a situation if in the most preferred (most ‘normal’) situations in which φ is true, ψ is also true. It has been noticed that there exist natural epistemic readings of a default conditional, but this direction has not been thoroughly explored. A statement of the form ‘something known to be a bird, that can be consistently believed to fly, does fly’ involves well-known epistemic attitudes and allows the possibility of defining defaults within the rich framework of Epistemic Logic. We pursue this direction here and proceed to define conditionals within KBE, a recently introduced S4.2-based modal logic of knowledge, belief and estimation. In this logic, knowledge is a normal S4 operator, belief is a normal KD45 operator and estimation is a non-normal operator interpreted as a ‘majority’ quantifier over the set of epistemically alternative situations. We define and explore various conditionals using the epistemic operators of KBE, capturing φ ⇒ ψ in various ways, including ‘according to the agent’s knowledge, an estimation that φ is true implies the estimation that (φ∧ψ) is true’ or ‘if φ is known and there is no reason to believe ¬ψ then ψ can be plausibly inferred’. Overall, we define here three nonmonotonic default conditionals, one conditional satisfying monotonicity (strengthening the antecedent) and two nonmonotonic conditionals that do not satisfy the ubiquitous axiom ID (reflexivity). Our project provides concrete evidence that the machinery of epistemic logic can be exploited for the study of default conditionals.