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We study two logics of knowledge and belief stemming from the work of Stalnaker (2006), omitting positive introspection for knowledge. The two systems are equivalent with positive introspection, but not without. We show that while the logic of beliefs remains unaffected by omitting introspection for knowledge in one system, it brings significant changes to the other. The resulting logic of belief is non-normal, and its complete axiomatization uses an infinite hierarchy of coherence constraints. We conclude by returning to the philosophical interpretation underlying both models of belief, showing that neither is strong enough to support a probabilistic interpretation, nor an interpretation in terms of certainty or the "mental component" of knowledge.