In addition to these postulates, however, other premises are presupposed by the various alternative geometries—and indeed by any kind of systematic thought. For instance, all such systems require that a single proposition cannot be simultaneously true and false. This principle of non-contradiction (articulated by Aristotle) is not an arbitrary postulate, but a necessary axiom. Axioms, like postulates, are starting points for systematic thought. Unlike postulates, however, axioms (as the term is understood here) are necessarily accepted, at least implicitly.

An axiom is not an "inescapable conclusion" (since it is not a conclusion), but it is, in a sense, an "inescapable beginning."