A fundamental problem in knowledge representation is that reasoning, if based on classical logic, is inherently intractable or even undecidable. A principled approach to specifying more efficient inference mechanisms is to use weaker logics with non-standard model theories.In this paper we extend earlier work on a logic of belief for decidable deductive first-order reasoning by adding the ability to introspect. The new logic allows us to formally specify the beliefs of deductively limited yet fully introspective first-order knowledge bases. The complexity of reasoning in this framework reduces to the complexity of a special validity problem of the logic and we obtain various tractability/decidability results. To demonstrate the usefulness of the logic for knowledge representation purposes we show how it can be applied to the specification of routines to query and update a knowledge base. Other interesting aspects of the logic include distinctions between knowing that and knowing who by way of quantifying-in and a very tight coupling between the limited beliefs of a knowledge base and those that follow under the assumption of perfect logical reasoning.