Orthodox Bayesianism endorses revising by conditionalization. This paper investigates the zero-raising problem, or equivalently the certainty-dropping problem of orthodox Bayesianism: previously neglected possibilities remain neglected, although the new evidence might suggest otherwise. Yet, one may want to model open-minded agents, that is, agents capable of raising previously neglected possibilities. Different reasons can be given for open-mindedness, one of which is fallibilism. The paper proposes a family of open-minded propositional revisions depending on a parameter ϵ. The basic idea is this: first (...) extend the prior to the newly suggested possibilities by mixing the prior with the uniform probability on these possibilities, then conditionalize. This may put the agent back on the right track when her beliefs or evidence happen to be false. The paper justifies this family of equivocal epsilon-conditionalizations as minimal non-biased open-minded modifications of conditionalization. Several variations are discussed, such as mixing with an ad hoc or silent prior instead of the uniform prior, and a generalization to probabilistic information is given. The approach is compared to other accounts, such as Jeffrey’s Bayesianism, Gärdenfors’s probabilistic revision, maximizing entropy, and minimal revision. _1_ Introduction _2_ Downsides of Certainty Conservation _3_ Why Raise Zeros? _3.1_ Accuracy defects _3.2_ Fallibilism and dissolutions _3.3_ Weak fallibilism and partial solutions _4_ Epsilon-Conditionalization _4.1_ Definitions _4.2_ Revision types and information assumptions _4.3_ Epsilon interpretation and flavours _5_ Justifications _5.1_ Open-minded orthodox Bayesianism _5.2_ Minimal revision _5.3_ Transitivity _6_ Conclusion Appendix. (shrink)