Logical connectives (e.g., or, if, and not) are central to everyday conversation, and the inferences they generate are made with little effort in pragmatically sound situations. In contrast, the neural substrates of logical inference-making have been studied exclusively in abstract tasks where pragmatic concerns are minimal. Here, we used fMRI in an innovative design that employed narratives to investigate the interaction between logical reasoning and pragmatic processing in natural discourse. Each narrative contained three premises followed by a statement. In Fully-deductive stories, the statement confirmed a conclusion that followed from two steps of disjunction–elimination (e.g., Xavier considers Thursday, Friday, or Saturday for inviting his girlfriend out; he removes Thursday before he rejects Saturday and declares “I will invite her out for Friday”). In Implicated-premise stories, an otherwise identical narrative included three premises that twice removed a single option from consideration (i.e., Xavier rejects Thursday for two different reasons). The conclusion therefore necessarily prompts an implication (i.e., Xavier must have removed Saturday from consideration as well). We report two main findings. First, conclusions of Implicated-premise stories are associated with more activity than conclusions of Fully-deductive stories in a bilateral frontoparietal system, suggesting that these regions play a role in inferring an implicated premise. Second, brain connectivity between these regions increases with pragmatic abilities when reading conclusions in Implicated-premise stories. These findings suggest that pragmatic processing interacts with logical inference-making when understanding arguments in narrative discourse.