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 state-
ments. 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.