Among the rules they assign to a proposition are the contradictory (al-naqīḍ), the simple converse (al-ʿaks al-mustawī), and the contra-positive (ʿaks al-naqīḍ). Whenever the original statement is valid, its contradictory is necessarily invalid, while both its converse and its contra-positive are valid. Thus, if one says, “Every human being is an animal,” the contradictory—“No human being is an animal”—is false; the converse—“Some animals are human beings”—is true; and the contra-positive—“Whatever is not an animal is not a human being”—is likewise true. Contradiction, they explain, is the opposition of two propositions by affirmation and negation in such a way that the truth of one entails the falsehood of the other. Conversion is to exchange the subject and predicate while preserving truth, whereas the contra-positive (or the conversion of the negative) exchanges the subject and predicate and at the same time changes affirmation into negation.
They further divide propositions into universal and particular, affirmative and negative—that is, general versus specific, and assertive versus denying. Before tackling propositions, they discuss the simple elements that compose them, such as the notions of universal and particular, essential and accidental. Prior to that, they examine the words that indicate these meanings, distinguishing, for example, between indication by full correspondence, by implication, and by necessary entailment.
The whole purpose of these preliminaries is definition (ḥadd) and syllogism, the latter being the chief goal; and the highest form of syllogism, they say, is the demonstrative syllogism. Demonstration, in their view, is a syllogism whose matter consists of certainties that must be accepted, as mentioned earlier. The rhetorical syllogism draws upon commonly accepted premises suitable for addressing the general public, whether those premises yield certainty or only probability. The dialectical syllogism is built on whatever the disputant concedes, be it certain or probable, famous or obscure. This, they claim, is the best way to characterize the three kinds.
Many of them, however, redefine the classes as follows: the demonstrative syllogism is the one whose premises are necessarily to be accepted; the dialectical syllogism is the one whose premises are widely held, whether true or false, necessary, impossible, or merely possible; and the rhetorical syllogism is the one whose premises are probable, whatever their source. Hence the rhetorical syllogism yields only conjecture, whether its premises are conceded or simply widespread, whereas the dialectical syllogism rests on premises that are well known.