On Lockean beliefs that are deductively closed and minimal change
This work offers incremental improvements to belief change theory by enabling deductive closure and minimal updates in probabilistic belief systems.
The paper addresses the issue that Lockean belief sets, defined by probabilistic confidence thresholds, are not generally closed under classical logical deduction, and provides characterizations for such closed sets along with a probabilistic update approach for minimal belief revision.
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.