We investigate the articulation between domain-general reasoning and interpretive processes in failures of deductive reasoning. We focus on illusory inferences from disjunction-like elements, a broad class of deductive fallacies studied in some detail over the past 15 years. These fallacies have received accounts grounded in reasoning processes, holding that human reasoning diverges from normative standards. A subset of these fallacies however can be analyzed differently: human reasoning is not to blame, instead the premises were interpreted in a non-obvious, yet perfectly predictable and reasonable way. Once we consider these interpretations, the apparent fallacious conclusion is no mistake at all. We give a two-factor account of these fallacies that incorporates both reasoning-based elements and interpretive elements, showing that they are not in real competition. We present novel experimental evidence in favor of our theory. Cognitive load such as induced by a dual-task design is known to hinder the interpretive mechanisms at the core of interpretation-based accounts of the fallacies of interest. In the first experiment of its kind using this paradigm with an inferential task instead of a simpler truth-value-judgment task, we found that the manipulation affected more strongly those illusions where our theory predicts that interpretive processes are at play. We conclude that the best way forward for the field to investigate the elusive line between reasoning and interpretation requires combining theories and methodologies from linguistic semantics and the psychology of reasoning.