Commentary Aristotle Posterior Analytics
Lecture 19
How common principles differ from one another
b23. That which expresses necessary self-grounded fact, and which we must necessarily believe, is distinct both from the hypotheses of a science and from illegitimate postulate — I say ‘must believe’, because all syllogism, and therefore a fortiori demonstration, is addressed not to the spoken word, but to the discourse within the soul, and though we can always raise objections to the spoken word, to the inward discourse we cannot always object.
b27. That which is capable of proof but assumed by the teacher without proof is, if the pupil believes and accepts it, hypothesis, though only in a limited sense hypothesis — that is, relatively to the pupil; if the pupil has no opinion or a contrary opinion on the matter, the same assumption is an illegitimate postulate. Therein lies the distinction between hypothesis and illegitimate postulate: the latter is the contrary of the pupil’s opinion, demonstrable, but assumed and used without demonstration.
Definitions, namely, those which are not expressed as statements that anything is or is not, are not hypotheses: but it is in the premises of a science that its hypotheses are contained. Definitions require only to be understood, and this is not hypothesis — unless it be contended that the pupil’s hearing is also an hypothesis required by the teacher. Hypotheses, on the contrary, postulate facts on the being of which depends the being of the fact inferred.
b39. Nor are the geometer’s hypotheses false, as some have held, urging that one must not employ falsehood and that the geometer is uttering falsehood in stating that the line which he draws is a foot long or straight, when it is actually neither. The truth is that the geometer does not draw any conclusion from the being of the particular line of which he speaks, but from what his diagrams symbolize.
a3. A further distinction is that all hypotheses and illegitimate postulates are either universal or particular, whereas a definition is neither.
a5. So demonstration does not necessarily imply the being of Forms nor a One beside a Many, but it does necessarily imply the possibility of truly predicating one of many; since without this possibility we cannot save the universal, and if the universal goes, the middle term goes with it, and so demonstration becomes impossible. We conclude, then, that there must be a single identical term unequivocally predicable of a number of individuals.
That which expresses necessary self-grounded fact. After dividing common principles from proper principles, Aristotle now distinguishes among the common principles.
His treatment is divided into three parts.
In the first he lays down a distinction among the common principles.
In the second he shows the difference between a definition and a certain genus of common principles, at: the definition, namely, those (76b35; [163]).
In the third he excludes an error, at: so demonstration does not (77a5; [166]).
Concerning the first he does two things.
First, he distinguishes the common conceptions in the mind from postulates or suppositions.
Second, he distinguishes among the latter, at: that which is capable of proof (76b27; [162]).
He says the following about the first.
It should be noted that the common conceptions in the mind have something in common with the other principles of demonstration and something proper: something common, because both they and the others must be true in virtue of themselves. But what is proper to the former is that it is necessary not only that they be true of themselves but that they be seen to be such. For no one can think their contraries.
He says, therefore, that that principle of which it is not only required that it be in virtue of itself but further required that it be seen (76b23), namely, a common conception in the mind or a dignity, is neither a postulate nor a supposition. He proves this in the following way: A postulate and a supposition can be confirmed by a reason from without, that is, by some argumentation; but a common conception in the mind does not bear on a reason from without (because it cannot be proved by any argument), but bears on that reason which is in the soul, because it is made known at once by the natural light of reason. That it does not bear on any reason from without is shown by the fact that a syllogism is not formed to prove such common conceptions of the mind. Furthermore, that these are not made known by an outward reason but by the inward he proves by the fact that it is possible to contest an outward reason, either truly or apparently, but it is not always possible to do so with the inward reason.
This is so because nothing is so true that it cannot be denied orally. (For even this most evident principle that the same thing cannot be and not be has been orally denied by some). On the other hand, some things are so true that their opposites cannot be conceived by the intellect. Therefore, they cannot be challenged in the inward reason but only by an outward reason which is by the voice. Such are the common conceptions in the mind.
Then, at: that which is capable of proof (76b27) he distinguishes suppositions and postulates from one another.
Here, too, it should be noted that they have something in common and something in which they differ. What is common to them is that although they can be demonstrated, the demonstrator assumes them without demonstrating (chiefly because they are not demonstrable by his own science but by another, as explained above. Hence they are reckoned among the immediate principles because the demonstrator uses them without a middle, since they do not have a middle in that science).
Yet they do differ, because if such a proposition is accepted as reasonable by the pupil to whom a demonstration is being made, it is called a “supposition.” In that case it is not called a supposition absolutely, but relative to him. But if the pupil has no reason for or against the proposition, the demonstrator must request him to admit it as reasonable, and then it is called a “postulate.” However, if the pupil has a contrary opinion, then it will be a “question,” which must be settled between them. At any rate, what is common to all of them is that the demonstrator uses each of them without demonstrating, although they are demonstrable.
Then, at: the definition, namely (76b35) he distinguishes definitions from suppositions with two reasons, the second of which begins at: a further distinction (77a3; [165]).
First, he presents this reason: Every postulate or supposition declares something to be or not to be; but terms, that is, definitions, do not declare that something is or is not. Terms, therefore, taken by themselves, are neither postulates nor suppositions. But when they are employed in propositions, they are suppositions, as in the statement, “Man is a rational mortal animal.” But terms taken by themselves are only understood; and understanding is not supposing, any more than hearing is. Rather supposition bears on things that exist such that a conclusion is made from them in regard to what they are, that is, in virtue of the premises.
Second, at: nor are the geometer’s hypotheses (76b39), he excludes a doubt.
For there were some who said that a geometer uses a false supposition when he says that a line not one foot long is one foot long, or that a line traced in the sand is straight when it really is not. But he says that a geometer does not on this account suppose something false. For since the geometer demonstrates nothing of singulars but of universals, as explained above, and these lines are singulars, it is obvious that he is not demonstrating anything about these lines or from these lines; rather he is using them as examples of the universals (which are understood by examples) from which and about which he demonstrates.
Then, at: a further distinction (77a3) he presents the second reason which is this: every supposition or postulate is in whole or in part, that is, is either a universal proposition or a particular. But definitions are neither of these, because nothing is placed or predicated in them universally or particularly. Therefore...
Then, at: so demonstration does not (77a5) he shows from the foregoing that it is not necessary to posit “Ideas,” as Plato did. For it was shown above that demonstrations are concerned with universals and in that sense with eternal things. Therefore, it is not necessary for the validity of demonstration that there be forms, that is, “Ideas,” or a one outside the many, as Plato posited separated mathematical beings along with Ideas in order that thereby demonstrations might bear on eternal things. What is required is that there be one in many and about many, if there is to be demonstration, because it will not be universal unless it is one about many. And if it is not universal, there will not be a middle of demonstration and, consequently, no demonstration. That the middle of a demonstration must be universal is plain from the fact that the middle of a demonstration must be some one same thing predicated of many not equivocally but according to the same aspect, which is a universal aspect. But if it should happen to be equivocal, a defect in reasoning would occur.