A syllogism is composed of two premises, each of which is a proposition that is either affirmative or negative; and each of these, in turn, is either universal or particular. Hence it is necessary to discuss propositions, their various kinds, and their modalities.

One may also argue on the basis of a proposition’s contradictory, its converse, or the converse of its contradictory. Once the proposition itself is sound, its contradictory is necessarily false, while both its converse and the converse of its contradictory are valid. It is therefore incumbent to treat the contradiction of propositions, their simple conversion, and the conversion of their contradictory.

A proposition is either categorical, conditional conjunctive, or conditional disjunctive. Accordingly, from the standpoint of form the syllogism breaks down into three types: 1) the inclusionary syllogism (tadākhul), namely the categorical syllogism; 2) the concomitant syllogism (talāzum), namely the conjunctive conditional; and 3) the antagonistic syllogism (taʿānud), also expressed as division or alternative reasoning (taqsīm wa-tardīd), namely the disjunctive conditional. This pertains to form, whereas by matter the syllogism is divided into the five classes mentioned earlier.

It is therefore necessary to address the materials of the syllogism—that is, the propositions by which other propositions are proved. All of the foregoing concerns the syllogism of inclusion. As for analogical reasoning (qiyās al-tamthīl) and induction (istiqrāʾ), they follow a different rule. They state that inferring from the universal to the particular is the syllogism of inclusion, whereas inferring from the particular to the universal is induction—complete when one knows that the universal encompasses all individuals, otherwise incomplete—and that inferring from one particular to another is analogical reasoning.

Elsewhere we have elaborated that every inclusionary syllogism ultimately reduces to analogy, just as every analogical syllogism reduces to inclusion, and that their claim that the inclusionary syllogism yields certainty while analogy does not is mistaken.

We have likewise recorded the dispute over the very term “qiyās.” Does it encompass both forms, as the majority hold? Is it literal for analogy but figurative for inclusion, as maintained by Abū Ḥāmid al-Ghazālī and Abū Muḥammad al-Maqdisī? Or is the reverse the case, as Ibn Ḥazm and other logicians contend? A full treatment of this question is given elsewhere.