Where Proof Begins

The oldest of the philosophers never let themselves be carried off into disputing and doubting. Still less should we, who hold to the philosophy that is really true, and whom the Scriptures charge to examine and investigate. It is the later Greek philosophers who, out of an empty and pointless love of reputation, are dragged into useless chatter — refuting, and wrangling. The barbarian philosophy, on the other hand, drives all contentiousness out, and says, “Seek, and you will find; knock, and it will be opened to you; ask, and it will be given to you.” [a]

So the question we put forward for inquiry and answer knocks, by that very inquiry, at the door of truth — as it appears to me. And when the obstacle gives way in the course of the search and an opening is made, what follows is a contemplation that really knows. To those who knock this way, I say, the matter under investigation is opened.

And to those who ask their questions this way in the Scriptures, God gives what they are after: the gift of the knowledge that comes from him, taken hold of through the real light that logical inquiry sheds. For it is impossible to find without having sought; or to have sought without having examined; or to have examined without having unfolded and opened up the question by asking about it, until it stands out clearly; or, again, to have gone through the whole search without receiving at the end, as the prize, the knowledge of the point in question.

Finding belongs to the one who has sought, and seeking belongs to the one who first grants that he does not know. Drawn by longing to the discovery of what is good, such a man searches with thought, and without any love of quarreling or of glory — asking, answering, and weighing besides what has been said. For we are bound to carry on this kind of investigation not only with the divine Scriptures but with the notions everyone shares, and to let the search come to rest at some point where it is of use.

Rowdy people, and the clever tricks of the courtroom, have another place waiting for them and another crowd. But a man who both loves the truth and learns from it should be peaceable even while he investigates, moving forward by proofs that yield real knowledge — not in love with himself, but in love with the truth — until he has a full grasp of the matter.

For a course of instruction like this one, what better or clearer place is there to begin than a discussion of the word put forward — carried through so plainly that everyone who speaks the same language can follow it? Is “demonstration” a word of the same kind as blityri, a mere noise that signifies nothing? Yet the philosopher does not bring forward demonstration as a term that means nothing, and neither does the public speaker, and no more does the judge; nor is anyone arguing a case in any doubt that the word does mean something. [e]

Philosophers do in fact present demonstration as something that really exists, though one describes it one way and another another. So if a man wants to handle each question rightly, there is no more widely granted first principle he can trace his argument back to than the meaning that everyone of the same people and the same language agrees the word carries.

Starting from that point, one has to ask whether the statement in question carries that meaning or not. Then, if it is shown to carry it, one has to examine its nature closely — what sort of thing it is, and whether it ever strays outside the class assigned to it. And since it is not enough simply to state, flatly, what you think — for your opponent may just as freely allege whatever he likes on his side — what is stated has to be established. If settling it is carried back to something that is itself in dispute, and the settling of that to another disputed point, it will run on with no end and can never be demonstrated. But if belief in a point not yet granted can be carried back to one that everybody grants, that granted point is to be made the beginning of the teaching.

Every term put forward for discussion, then, is to be turned into a form of words that the parties to the discussion accept, so as to form the starting point of instruction and lead the way to the discovery of the points under investigation. Take the word “sun” as the point at issue. The Stoics say it is “an intelligent fire kindled out of the waters of the sea.” Is that definition not darker than the word it defines, needing a further demonstration to show whether it is even true? Better to say, in the common and distinct way of speaking, that the brightest of the heavenly bodies is called the sun. That statement is easier to believe, and clearer, and everyone accepts it as well.

In the same way, everyone will grant that a demonstration is an argument, in accordance with reason, that produces belief about disputed points out of points already granted. [h]

Now demonstration and belief and knowledge — and foreknowledge too — are each used in two senses. There is the kind that yields real knowledge and is certain, and the kind that rests on nothing more than hope.

Strictly speaking, then, what is called a demonstration is what produces in the souls of learners a belief that amounts to knowledge; the other kind leads only to opinion. It is the same with the word “man”: the man who really is one, with ordinary good sense, and the man who is savage and brutish are each called a man. This is what the comic poet meant when he said, “What a graceful thing a man is, so long as he is a man.” The same holds with ox and horse and dog, according to whether the animal is a good one or a bad one. For it is by looking to the perfection of the kind that we come to the meanings that are strictly proper. So we form the idea of a doctor who lacks nothing of the power to heal, and of a man of knowledge who lacks nothing of real knowledge.

Demonstration is not the same as syllogism. What is demonstrated points to one thing, itself one and the same — as we say that being pregnant is the proof of being no longer a virgin. What is grasped by syllogism is one thing too, but it follows from several: not one but several proofs were brought forward that Pytho had betrayed the Byzantines, if in fact he did. To draw a conclusion from what is granted is to syllogize; to draw a conclusion from what is true is to demonstrate.

So demonstration carries a double advantage: it takes true premises for the proof of the points at issue, and it draws the conclusion that follows from them. If the premises are not there, but the conclusion does follow from them, a man has not demonstrated; he has only syllogized. Drawing the proper conclusion from the premises is merely to have syllogized. To have each of the premises true as well is not merely to have syllogized but to have demonstrated.

And to conclude, as the word itself shows, is to bring a thing to its conclusion. In every train of reasoning the point being settled is the end, and this is also called the conclusion. No simple and primary statement is called a syllogism, however true it is; a syllogism is put together out of at least three such statements — two as premises and one as conclusion.

Now either everything requires demonstration, or some things are plain in themselves. If everything does, then by demanding a demonstration of every demonstration we shall go on with no end, and demonstration is overthrown. But if some things are plain in themselves, those things will be the starting points of demonstration and the ground it stands on.

And in fact the philosophers admit that the first principles of all things cannot be demonstrated. So if there is any demonstration at all, there is an absolute necessity that there be something plain in itself, which is called primary and undemonstrable.

It follows that every demonstration is traced back to a faith that cannot be demonstrated.

It will also turn out that demonstrations have other starting points besides the source that rises out of faith: the things that show themselves clearly to the senses and to the understanding. What the senses report is simple and cannot be broken down further; what the understanding grasps is simple, rational, and primary. What is produced out of these is composite, but no less clear and reliable, and it has more to do with our power of reasoning than the first things do. For the native power of reason, which we all have by nature, works on agreement and disagreement. So if any argument turns out to be of the kind that can produce belief in what is not yet believed out of points already believed, we shall say that this is the very thing that demonstration is.

Demonstration, it is said, is twofold in nature, as belief is: one kind produces in the souls of the hearers persuasion and nothing more, the other produces knowledge.

If, then, a man begins from the things that are evident to the senses and to the understanding, and then draws the proper conclusion, he truly demonstrates. But if he begins from things that are only probable and not primary — evident neither to sense nor to understanding — and draws the right conclusion, he will indeed have syllogized, but he will not have produced a demonstration that yields knowledge. And if he does not draw the right conclusion, he will not have syllogized at all.

Demonstration also differs from analysis. Each of the points demonstrated is demonstrated by means of points that have themselves been demonstrated, and those by others before them, until we get back to what is plain in itself, or to what is evident to sense and to understanding; and that is called analysis. Demonstration is when the point in question reaches us through all the steps in between. The man who practices demonstration, then, ought to give great attention to the truth, while paying no mind to what the premises are called — whether you call them axioms, or premises, or assumptions. In the same way he must pay close attention to the suppositions a conclusion rests on, while remaining quite unconcerned whether anyone chooses to call it a conclusive or a syllogistic proposition.

For I say that these two things must be attended to by the man who means to demonstrate: to take true premises, and to draw from them the conclusion that legitimately follows — which some also call the inference, since it is what is brought in from the premises.

In each statement made about a question the premises must differ from one another, and yet be related to the statement laid down; and what is put forward must be reduced to a definition, and that definition must be one everyone accepts. But when premises are assumed that have nothing to do with the statement to be established, it is impossible to arrive at any right result, because the whole statement — which is also called the question of what the thing is — has been passed over.

In all questions, then, there is something known beforehand: the thing which, being plain in itself, is believed without demonstration. That must be made the starting point of the investigation and the test of the results it appears to reach.