and division,” and which dialecticians may also call “division and the posing of alternatives,” it amounts to inferring the negation of one of two contradictories from the affirmation of the other, and the affirmation of one from the negation of the other. “It has four types.” Thus, in a disjunction that precludes both joint truth and joint falsity, all four forms of exceptive inference apply: if this is affirmed, its contradictory is negated, and likewise for the other; if this is negated, its contradictory is affirmed, and likewise for the other.

A disjunction that precludes joint truth involves inferring the negation of one of two contraries from the affirmation of the other, since the two are mutually incompatible. A disjunction that precludes joint falsity involves both contradiction and necessary entailment. Two contradictories cannot both be false, so it precludes their both being absent. However, it concerns the existence of one thing and the nonexistence of another, not the existence and nonexistence of the same thing. The existence of one thing and the nonexistence of another may stand in a relation whereby one necessarily follows from the other, even though they cannot both be absent: their simultaneous absence would entail the simultaneous absence of both the existence and the nonexistence of a thing.

In short, everything has something it necessarily entails, without which it cannot exist, and something incompatible with and opposed to its existence. Its existence can therefore be inferred from the existence of something that necessarily entails it, and its nonexistence from the absence of something it necessarily entails. Its nonexistence can also be inferred from the existence of something incompatible with it, and its existence from the absence of that incompatible thing, provided the possibilities are confined to these two, so that they can neither both be absent nor both exist.

Such inference yields knowledge of a thing’s states, what necessarily entails them, and what they necessarily entail. Once the natural disposition conceives of this, it expresses it in various ways and casts it into various forms of argument. None of this is confined to the form they describe for syllogism, let alone what they call demonstration. For demonstration, they stipulate particular material content: the propositions they have specified. They exclude from self-evident first principles what they call “estimative propositions” and what they call “commonly accepted propositions,” although the natural disposition assents to both—especially what they call estimative propositions—more strongly than it assents to many of the certainties they have made the material content of demonstration.