We shall show that the “certain premises” they enumerate do not produce knowledge of things that actually exist; consequently, they fail to achieve the objective that purifies the soul. Nor do they yield knowledge of external realities as they truly are except in the same way that knowledge is gained through argument by analogy. Indeed, the comprehensive logical syllogism they call “demonstrative” can never furnish knowledge that is not equally attainable through the analogical inference they deem weak. That syllogism inevitably contains a universal proposition, and one cannot be sure that a proposition is truly universal unless one is first certain that all of its individual instances share the common attribute—a certainty arrived at only through analogy.

Both kinds of inference are beneficial when some of their universal premises are taken from the report of the infallible Messenger, especially when employed in theology, as the Qur’an does with priority. Without these two, however, they are of no use in the divine sciences, nor do they provide demonstrative scientific benefit in the natural sciences; at best they yield customary claims that may prove unreliable and thus belong merely to what generally, but not invariably, holds true.

As for pure mathematics—arithmetic and geometry—stripped of sensible matter, its propositions are valid in themselves, yet they have no directly known object outside the mind; they are nothing but mental posits of number and magnitude. True, they correspond to whatever is counted or measured in the external world, but demonstration deals only with what is in the mind, not with what is outside it. Most of their toil in geometry was spent in pursuit of celestial science: describing the heavenly spheres and the stars, their dimensions and their motions. Some of this is demonstratively established, but most is not, and their disputes over it are too numerous to recount. Thus, the knowledge they obtain through their demonstrations—mathematical or otherwise—neither purifies the soul nor discloses existing realities except in the same manner as analogical reasoning.

This becomes clearer when we examine the material of the syllogism. They unanimously maintain that every syllogism must include a universal proposition; no syllogism in any of the figures is valid if it rests on two negatives, on two particulars, or on a negative minor together with a particular major—though it may rest on a particular minor with a universal negative major. The point is that in every form of syllogism a universal proposition is indispensable, and in most