Belief Update Through Semiorders

Belief change is a core component of intelligent reasoning, enabling agents to adapt their beliefs in response to new information. A prominent form of belief change is belief revision, which involves altering an agent’s beliefs about a static (unchanging) world in light of new evidence. A foundational framework for modeling rational belief revision was introduced by Alchourrón, Gärdenfors, and Makinson (AGM), who formalized revision functions based on total preorders over possible worlds—that is, orderings that encode the relative plausibility of alternative states of affairs. Building on this, Peppas and Williams later characterized AGM-style revision functions using weaker preference structures known as semiorders, which, unlike total preorders, permit intransitive indifference between alternatives. In this article, we extend the framework of Peppas and Williams to the context of belief update. In contrast to belief revision, belief update concerns maintaining coherent beliefs in response to actual changes in a dynamic, evolving environment. We provide both axiomatic and semantic characterizations of update functions derived from semiorders, establishing corresponding representation theorems. These results essentially generalize the classical belief-update framework of Katsuno and Mendelzon, which relies on total preorders, thereby offering a broader and more flexible perspective. The intransitivity of indifference inherent in semiorders plays a central role in our framework, enabling the representation of nuanced plausibility distinctions between possible states of affairs—an essential feature for realistically modeling belief dynamics.