Concerning Measures in a First-order Logic with Actions and Meta-beliefs

The unification of logic and probability has been seen as a long-standing concern in philosophy and mathematical logic. In this paper, we propose a new general probabilistic modal logic of belief and only-believing in the situation calculus. Our logic can express both continuous and discrete degrees of belief. More importantly, expressing degrees of belief for arbitrary first-order formulas in a dynamic setting is possible for the first time, going well beyond previous proposals where fluents are assumed to be nullary or discrete. We show that our notion of belief retains many of the properties known from the previous related work.