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The inferential model (IM) framework produces data-dependent, non-additive degrees of belief about the unknown parameter that are provably valid. The validity property guarantees, among other things, that inference procedures derived from the IM control frequentist error rates at the nominal level. A technical complication is that IMs are built on a relatively unfamiliar theory of random sets. Here we develop an alternative -- and practically equivalent -- formulation, based on a theory of possibility measures, which is simpler in many respects. This new perspective also sheds light on the relationship between IMs and Fisher's fiducial inference, as well as on the construction of optimal IMs.