  If we take very seriously the common view that Aristotle claims that every immediate premise of a demonstration expresses something about an existing entity, then one may well wonder how the principles of demonstration, axioms and posits, can be premises of demonstrations. Existence claims and stipulations do not express something about an existing entity. Since Aristotle calls the axioms, ‘those from which (demonstration arises),’ some have suggested that the axioms alone form the premises for a science and that a proof in any science arises by placing genus terms and their definitions in the axioms and then substituting terms like ‘triangle’ for their definitions when they arise in proofs. However, besides pointing to the inadequacy of the axioms for this job, it may be objected that Aristotle also calls the principles of demonstration immediate statements protaseis), i.e., axioms and posits. Another possibility is that he regards even stipulations and existence claims as premises, as well as other hypotheses, but treats the axioms as somehow more fundamentally the source of proofs. In this case, he has a looser conception of what counts as a premise than many readers would expect. In any case, if his proof theory is to work at all, he must allow many more immediate premises than one would find in the introduction to a standard text of ancient Greek mathematics.