We present a formal semantics for epistemic logic, capturing the notion of knowability relative to information (KRI). Like Dretske, we move from the platitude that what an agent can know depends on her (empirical) information. We treat operators of the form forumla (‘B is knowable on the basis of information A’) as variably strict quantifiers over worlds with a topic- or aboutness-preservation constraint. Variable strictness models the non-monotonicity of knowledge acquisition while allowing knowledge to be intrinsically stable. Aboutness-preservation models the topic-sensitivity of information, allowing us to invalidate controversial forms of epistemic closure while validating less controversial ones. Thus, unlike the standard modal framework for epistemic logic, KRI accommodates plausible approaches to the Kripke-Harman dogmatism paradox which bear on non-monotonicity or on topic-sensitivity. KRI also strikes a better balance between agent idealization and a non-trivial logic of knowledge ascriptions.