Abstract: Knowledge and belief are closed under conjunction if knowing(/believing) p and knowing(/believing) q entail knowing(/believing) the conjunction p and q. This closure principle is a popular idealization, especially in the tradition of epistemic and doxastic logic, and including in recent work by myself and Bernhard Salow ("Epistemology Normalized", Phil Review, 2023), where we try to understand the structure of knowledge and rational belief in terms of the comparative normality of possibilities compatible with one's evidence.
But even as an idealization, closure faces a serious challenge to do with probability. Here it is in a nutshell. Plausibly, for some notion probability, (i) we know(/rationally believe) some propositions whose probability is less than 1, (ii) if we know(/rationally believe) any propositions whose probability is less than 1, then the conjunction of all such propositions does not have high probability, (iii) we can only know(/rationally believe) propositions that have high probability. These three plausible claims are inconsistent with closure.
In our earlier work Salow and I have defended (i) and developed two alternative strategies for maintaining closure: multi-dimensionalism, which rejects (iii), and probabilism, which rejects (ii). After explaining these strategies, I will show how their respective ideas can be combined to offer a new probability spheres account of knowledge and rational belief that rejects closure in order to uphold (i), (ii) and (iii). I will then argue that this new account is preferable to the popular Lockean account of rational belief, since Lockeanism makes closure fail more egregiously and makes rational belief come apart from potential knowledge. I will conclude by suggesting that the failures of closure that remain on the probability spheres approach are not obviously problematic, since the most prominent considerations in favor of closure can be explained away by appeal to general facts about vagueness.