So-called quantificational accounts explicate logical consequence or validity as truth-preservation in all cases, cases being construed as admissible substitutional variants or as admissible interpretations with respect to non-logical terms. In the present study, which is the first from three successive studies devoted to quantification accounts, I focus on the beginning of systematic theorizing of consequence in Aristotle‘s work, which contains the rudiments of both modal and formal accounts of consequence. I argue, inter alia, that there is no evidence for the claim that Aristotle propounded a quantificational account, and that for a full-fledged quantificational approach in a modern style we need to turn to Bolzano’s substitutional approach, whose motivation, structure and problems are explained in the second part of this study.