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A representation theorem for a decision theory with conditionals. (English) Zbl 0963.03047

Summary: We investigate the role of conditionals in hypothetical reasoning and rational decision making. Our main result is a proof of a representation theorem for preferences defined on sets of sentences (and, in particular, conditional sentences), where an agent’s preference for one sentence over another is understood to be a preference for receiving the news conveyed by the former. The theorem shows that a rational preference ordering of conditional sentences determines probability and desirability representations of the agent’s degrees of belief and desire that satisfy, in the case of nonconditional sentences, the axioms of Jeffrey’s decision theory and, in the case of conditional sentences, Adams’ expression for the probabilities of conditionals. Furthermore, the probability representation is shown to be unique and the desirability representation is unique up to positive linear transformations.

MSC:

03B60 Other nonclassical logic
68T27 Logic in artificial intelligence
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