Incomplete induction is illustrated by the judgment that whenever an animal eats, its lower jaw moves while chewing—because this is observed in most animals—although in the cases not yet examined the opposite may occur, as with the crocodile. The complete form is useful for establishing certainties, whereas the incomplete form, though serviceable in dialectical debates, does not yield certainty.
Analogical inference (qiyās al-tamthīl), on the other hand, passes a ruling on one thing on the basis of the ruling given to another, by virtue of a common feature between them. Thus they say, “The universe exists, therefore it must be eternal, like the Creator,” or, “The universe is a body, therefore it must be originated, like a human being.” This type of argument contains four elements: the branch (farʿ), the original case (aṣl), the cause (ʿillah), and the ruling (ḥukm). In the foregoing examples the branch is the universe, the original case is the Creator (in the first) or the human being (in the second), the cause is existence or corporeality, and the ruling is eternality or origination.
They add that analogical inference differs from induction in that the subject of judgment in analogy may be a particular, whereas in induction it is always a universal.
Objections raised by Muslim dialecticians to analogical inference:
They maintain that it does not yield certainty. The mere fact that two things share a common attribute does not necessarily entail that they share the ruling assigned to one of them—unless it can be shown that the shared attribute is the very cause of that ruling. Any evidence offered to establish such causality remains conjectural, for those who defend analogical inference in rational matters rely on nothing more than positive and negative concomitance (ṭard and ʿaks) and the method of elimination and division (sabr wa-taqsīm).
Positive and negative concomitance amount to no more than claiming that the ruling and its cause invariably accompany each other, both in presence and in absence—something that can be known only through induction. Yet such induction cannot be carried out with respect to the branch, for that is the point at issue; the induction is therefore incomplete. Moreover, the cause of the ruling in the original case may be composite, consisting of the shared attribute together with other qualities found in the original. When the shared attribute is present in the original, the ruling holds because its complete cause is present; when it is absent, the ruling is absent because the cause is deficient. Consequently, the presence of the shared attribute in the branch does not necessarily entail the ruling, since the remaining qualities, or some of them, may be missing.