The formalization of action and obligation using logic languages is a topic of increasing relevance in the field of ethics for AI. Having an expressive syntactic and semantic framework to reason about agents decisions in moral situations allows for unequivocal representations of components of behavior that are relevant when assigning blame (or praise) of outcomes to said agents. Two very important components of behavior in this respect are belief and belief-based action. In this work we present a logic of doxastic oughts by extending epistemic deontic stit theory with beliefs. On one hand, the semantics for formulas involving belief operators is based on probability measures. On the other, the semantics for doxastic oughts relies on a notion of optimality, and the underlying choice rule is maximization of expected utility. We introduce an axiom system for the resulting logic, and we address its soundness, completeness, and decidability results. These results are significant in the line of research that intends to use proof systems of epistemic, doxastic, and deontic logics to help in the testing of ethical behavior of AI through theorem-proving and model-checking.
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