A traditional theory of uncertainty says that beliefs come in degrees. Degrees of belief (“credences”) have real number values between  and , where  conventionally represents certain belief,  represents certain disbelief, and values in between represent degrees of uncertainty. We have elegant, well-understood normative theories for credences: norms for how credences should hang together at ...

We use the martingale-theoretic approach of game-theoretic probability to incorporate imprecision into study randomness. In particular, we define several notions randomness associated with interval, rather than precise, forecasting systems, and their properties. The richer mathematical structure that thus arises lets us, amongst other things, better understand place existing results for precise...