or whether they belong to the class of things established by repeated experiment, by recurrent mass transmission, or by intuitive insight—according to those who regard some intuitions as among the certitudes that must be accepted.

An example is a person’s knowledge that the moon’s light is borrowed from the sun: he observes how the moon’s appearance changes with its varying alignment to the sun—one pattern after it separates from conjunction, as on the first crescent night, and another when it stands opposite the sun at full moon.

They disagree, however, as to whether intuition (ḥads) can in fact yield certainty.

Likewise, there are purely rational axioms, such as our statements: “One is half of two,” “The whole is greater than the part,” “Things equal to the same thing are equal to one another,” “Opposites cannot co-exist,” and “Contradictories can neither both prevail nor both vanish.”

Now, for every one of these universal propositions that they offer as a premise in demonstration, the conclusion can be known without that mediating demonstration; in practice this happens frequently.

When one knows that any single unit is half of any pair, and that every pair has one as its half, he immediately knows that this particular unit is half of these two—and so on in all individual cases—without first appealing to the universal statement.

Likewise, whenever a whole and a part are before him, he can recognize that the given whole is greater than its part without passing through the universal premise.

So too with contradictories: once he conceives the two as contradictories he realizes they cannot both obtain nor both be absent. Everyone knows that this specific thing cannot be both existent and non-existent, nor can it be devoid of both existence and non-existence, just as he knows the same of any other particular, without having to argue from the maxim that “nothing can be both existent and non-existent simultaneously.”

The same applies to opposites: a person knows that this object cannot be both black and white, or both in motion and at rest, and he knows the same of any other object without needing that universal statement to ground his knowledge.