 This leads to a further complication. The denial of “possibly \(P\)” for Aristotle is “either necessarily \(P\) or necessarily not \(P\)”. The denial of “necessarily \(P\)” is still more difficult to express in terms of a combination of modalities: “either possibly \(P\) (and thus possibly not \(P\)) or necessarily not \(P\)” This is important because of Aristotle’s proof procedures, which include proof through impossibility. If we give a proof through impossibility in which we assume a necessary premise, then the conclusion we ultimately establish is simply the denial of that necessary premise, not a “possible” conclusion in Aristotle’s sense. Such propositions do occur in his system, but only in exactly this way, i.e., as conclusions established by proof through impossiblity from necessary assumptions. Somewhat confusingly, Aristotle calls such propositions “possible” but immediately adds “ not in the sense defined”: in this sense, “possibly \(Oab\)” is simply the denial of “necessarily \(Aab\)”. Such propositions appear only as premises, never as conclusions.