Probabilistic Deontic Logics for Reasoning about Uncertain Norms

In this article, we present a proof-theoretical and model-theoretical approach to probabilistic logic for reasoning about uncertainty about normative statements. We introduce two logics with languages that extend both the language of monadic deontic logic and the language of probabilistic logic. The first logic allows statements like "the probability that one is obliged to be quiet is at least 0.9". The second logic allows iteration of probabilities in the language. We axiomatize both logics, provide the corresponding semantics and prove that the axiomatizations are sound and complete. We also prove that both logics are decidable. In addition, we show that the problem of deciding satisfiability for the simpler of our two logics is in PSPACE, no worse than that of deontic logic.