  Bayesian inference is limited in scope because it cannot be applied in
idealized contexts where none of the hypotheses under consideration is true and
because it is committed to always using the likelihood as a measure of
evidential favoring, even when that is inappropriate. The purpose of this paper
is to study inductive inference in a very general setting where finding the
truth is not necessarily the goal and where the measure of evidential favoring
is not necessarily the likelihood. I use an accuracy argument to argue for
probabilism and I develop a new kind of argument to argue for two general
updating rules, both of which are reasonable in different contexts. One of the
updating rules has standard Bayesian updating, Bissiri et al's (2016) general
Bayesian updating, Douven's (2016) IBE-based updating, and Vassend's (2019a)
quasi-Bayesian updating as special cases. The other updating rule is novel.

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