On Lockean beliefs that are deductively closed and minimal change

TL;DR

This paper explores Lockean belief sets, providing characterizations for those closed under classical logic and proposing a minimal revision approach for belief updates within probabilistic frameworks.

Contribution

It offers new characterizations of Lockean belief sets that are deductively closed and introduces a minimal revision method for belief updates.

Findings

Characterizations of Lockean belief sets closed under classical logic

A minimal revision approach for belief updates

Deductive closure achieved with minimal changes

Abstract

Within the formal setting of the Lockean thesis, an agent belief set is defined in terms of degrees of confidence and these are described in probabilistic terms. This approach is of established interest, notwithstanding some limitations that make its use troublesome in some contexts, like, for instance, in belief change theory. Precisely, Lockean belief sets are not generally closed under (classical) logical deduction. The aim of the present paper is twofold: on one side we provide two characterizations of those belief sets that are closed under classical logic deduction, and on the other we propose an approach to probabilistic update that allows us for a minimal revision of those beliefs, i.e., a revision obtained by making the fewest possible changes to the existing belief set while still accommodating the new information. In particular, we show how we can deductively close a belief set via a minimal revision.