  Statistics has made tremendous advances since the times of Fisher, Neyman,
Jeffreys, and others, but the fundamental and practically relevant questions
about probability and inference that puzzled our founding fathers remain
unanswered. To bridge this gap, I propose to look beyond the two dominating
schools of thought and ask the following three questions: what do scientists
need out of statistics, do the existing frameworks meet these needs, and, if
not, how to fill the void? To the first question, I contend that scientists
seek to convert their data, posited statistical model, etc., into calibrated
degrees of belief about quantities of interest. To the second question, I argue
that any framework that returns additive beliefs, i.e., probabilities,
necessarily suffers from {\em false confidence}---certain false hypotheses tend
to be assigned high probability---and, therefore, risks systematic bias. This
reveals the fundamental importance of {\em non-additive beliefs} in the context
of statistical inference. But non-additivity alone is not enough so, to the
third question, I offer a sufficient condition, called {\em validity}, for
avoiding false confidence, and present a framework, based on random sets and
belief functions, that provably meets this condition. Finally, I discuss
characterizations of p-values and confidence intervals in terms of valid
non-additive beliefs, which imply that users of these classical procedures are
already following the proposed framework without knowing it.
