In this paper we design a new logical system to explicitly model the different deductive reasoning steps of a boundedly rational agent. We present an adequate system in line with experimental findings about an agent’s reasoning limitations and the cognitive effort that is involved. Inspired by Dynamic Epistemic Logic, we work with dynamic operators denoting explicit applications of inference rules in our logical language. Our models are supplemented by (a) impossible worlds (not closed under logical consequence), suitably structured according to (...) the effect of inference rules, and (b) quantitative components capturing the agent’s cognitive capacity and the cognitive costs of rules with respect to certain resources (e.g. memory, time). These ingredients allow us to avoid problematic logical closure principles, while at the same time deductive reasoning is reflected in our dynamic truth clauses. We further show that our models can be reduced to awareness-like plausibility structures that validate the same formulas and we give a sound and complete axiomatization with respect to them. This approach to the agent’s internal deductive reasoning is finally combined with actions of external information. (shrink)