This paper argues that a combined treatment of probabilities, time and actions is essential for an appropriate logical account of the notion of probability; and, based on this intuition, describes an expressive probabilistic temporal logic for reasoning about actions with uncertain outcomes. The logic is modal and higher-order: modalities annotated by actions are used to express possibility and necessity of propositions in the next states resulting from the actions, and a higher-order function is needed to express the probability operator. The proposed logic is shown to be an adequate extension of classical mathematical probability theory, and its expressiveness is illustrated through the formalization of the Monty Hall problem.
Many probabilistic logics are surveyed in @cite_2 . Among those logics, most depart from classical logic by adopting a probabilistic notion of validity and entailment. , on the other hand, remains strictly classical in this respect. The probabilistic modal logics described in Sections 4.1 and 4.2 of @cite_2 are probably the most similar to . However, they are propositional, lack the probabilistic diamond operator, and are atemporal.
@cite_2
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Logic and probability theory are two of the main tools in the formal study of reasoning, and have been fruitfully applied in areas as diverse as philosophy, artificial intelligence, cognitive science and mathematics. This entry discusses the major proposals to combine logic and probability theory, and attempts to provide a classification of the various approaches in this rapidly developing field.