A logic for reasoning about independent alternatives

Abstract In recent work, Richard Booth develops a semantics for deontic modals based on inquisitive semantics that validates ‘independence inferences’: from a necessity or possibility modal with an embedded disjunction, one can infer that each disjunct can obtain without any of the other disjuncts obtaining. This paper investigates an extension of propositional inquisitive logic with a modal operator interpreted using Booth’s key semantic construction. The modal operator is used to express that all alternatives for the formula within the scope the operator are independently possible, in the sense that for each alternative, there is an accessible world where that alternative is true, whereas all other alternatives are false. Two versions of the semantics are considered: a local version where the modal operator quantifies over alternatives in submodels generated from the state of evaluation, and a global version where the modal operator quantifies over the alternatives in the full model of evaluation. The main results are sound and strongly complete axiom systems, both with respect to the local version, and with respect to the global version.