New semantics for numerical values given to possibility measures are provided. For epistemic possibilities, the new approach is based on the semantics of the transferable belief model, itself based on betting odds. It is shown that the least informative among the belief structures that are compatible with prescribed betting rates is a possibility measure. It is also proved that the idempotent conjunctive combination of two possibility measures corresponds to the hyper-cautious conjunctive combination of the belief functions induced by the possibility measures. For objective possibility degrees, the semantics is based on the most informative possibilistic approximation of a probability measure. We show how the idempotent disjunctive combination of possibility functions is related to the convex mixture of probability distributions.
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