Philosophers, in fact, present demonstration as having a substantial existence, one in one way, another in another. Therefore, if one would treat aright of each question, he cannot carry back the discourse to another more generally admitted fundamental principle than what is admitted to be signified by the term by all of the same nation and language.

Then, starting from this point, it is necessary to inquire if the proposition has this signification or not. And next, if it is demonstrated to have, it is necessary to investigate its nature accurately, of what kind it is, and whether it ever passes over the class assigned. And if it suffices not to say, absolutely, only that which one thinks (for one’s opponent may equally allege, on the other side, what he likes); then what is stated must be confirmed. If the decision of it be carried back to what is likewise matter of dispute, and the decision of that likewise to another disputed point, it will go on ad infinitum, and will be incapable of demonstration. But if the belief of a point that is not admitted be carried back to one admitted by all, that is to be made the commencement of instruction. Every term, therefore, advanced for discussion is to be converted into an expression that is admitted by those that are parties in the discussion, to form the starting point for instruction, to lead the way to the discovery of the points under investigation. For example, let it be the term “sun” that is in question. Now the Stoics say that it is “an intellectual fire kindled from the waters of the sea.” Is not the definition, consequently, obscurer than the term, requiring another demonstration to prove if it be true? It is therefore better to say, in the common and distinct form of speech, “that the brightest of the heavenly bodies is named the sun.” For this expression is more credible and clearer, and is likewise admitted by all.

CHAPTER III.
DEMONSTRATION DEFINED.

Similarly, also, all men will admit that demonstration is discourse,[1319] agreeable to reason, producing belief in points disputed, from points admitted.

Now, not only demonstration and belief and knowledge, but foreknowledge also, are used in a twofold manner. There is that which is scientific and certain, and that which is merely based on hope.

In strict propriety, then, that is called demonstration which produces in the souls of learners scientific belief. The other kind is that which merely leads to opinion. As also, both he that is really a man, possessing common judgment, and he that is savage and brutal,—each is a man. Thus also the Comic poet said that “man is graceful, so long as he is man.” The same holds with ox, horse, and dog, according to the goodness or badness of the animal. For by looking to the perfection of the genus, we come to those meanings that are strictly proper. For instance, we conceive of a physician who is deficient in no element of the power of healing, and a Gnostic who is defective in no element of scientific knowledge.

Now demonstration differs from syllogism; inasmuch as the point demonstrated is indicative of one thing, being one and identical; as we say that to be with child is the proof of being no longer a virgin. But what is apprehended by syllogism, though one thing, follows from several; as, for example, not one but several proofs are adduced of Pytho having betrayed the Byzantines, if such was the fact. And to draw a conclusion from what is admitted is to syllogize; while to draw a conclusion from what is true is to demonstrate.

So that there is a compound advantage of demonstration: from its assuming, for the proof of points in question, true premisses, and from its drawing the conclusion that follows from them. If the first have no existence, but the second follow from the first, one has not demonstrated, but syllogized. For, to draw the proper conclusion from the premisses, is merely to syllogize. But to have also each of the premisses true, is not merely to have syllogized, but also to have demonstrated.

And to conclude, as is evident from the word, is to bring to the conclusion. And in every train of reasoning, the point sought to be determined is the end, which is also called the conclusion. But no simple and primary statement is termed a syllogism, although true; but it is compounded of three such, at the least,—of two as premises, and one as conclusion.

Now, either all things require demonstration, or some of them are self-evident. But if the first, by demanding the demonstration of each demonstration we shall go on ad infinitum; and so demonstration is subverted. But if the second, those things which are self-evident will become the starting points [and fundamental grounds] of demonstration.