Commentary Aristotle Posterior Analytics
Lecture 17
Demonstration does not proceed from common principles, but from principles proper to the thing demonstrated b37. It is clear that if the conclusion is to show an attribute inhering as such, nothing can be demonstrated except from its ‘appropriate’ basic truths. Consequently a proof even from true, indemonstrable, and immediate premises does not constitute knowledge.
b40. Such proofs are like Bryson’s method of squaring the circle; for they operate by taking as their middle a common character — a character, therefore, which the subject may share with another — and consequently they apply equally to subjects different in kind. They therefore afford knowledge of an attribute only as inhering accidentally, not as belonging to its subject as such: otherwise they would not have been applicable to another genus. Our knowledge of any attribute’s connexion with a subject is accidental unless we know that connexion through the middle term in virtue of which it inheres, and as an inference from basic premises essential and ‘appropriate’ to the subject — unless we know, e.g., the property of possessing angles equal to two right angles as belonging to that subject in which it inheres essentially, and as inferred from basic premises essential and ‘appropriate’ to that subject: so that if that middle term also belongs essentially to the minor, the middle must belong to the same kind as the major and minor terms.
a8. The only exceptions to this rule are such cases as theorems in harmonics which are demonstrable by arithmetic. Such theorems are proved by the same middle terms as arithmetical properties, but with a qualification — the fact falls under a separate science (for the subject genus is separate), but the reasoned fact concerns the superior science, to which the attributes essentially belong. Thus, even these apparent exceptions show that no attribute is strictly demonstrable except from its ‘appropriate’ basic truths, which, however, in the case of these sciences have the requisite identity of character.
a17. It is no less evident that the peculiar basic truths of each inhering attribute are indemonstrable; for basic truths from which they might be deduced would be basic truths of all that is, and the science to which they belonged would possess universal sovereignty.
a19. This is so because he knows better whose knowledge is deduced from higher causes, for his knowledge is from prior premises when it derives from causes themselves uncaused: hence, if he knows better than others or best of all, his knowledge would be science in a higher or the highest degree.
a23. But, as things are, demonstration is not transferable to another genus, with such exceptions as we have mentioned of the application of geometrical demonstrations to theorems in mechanics or optics, or of arithmetical demonstrations to those of harmonics.
It is clear that if the conclusion. Above the Philosopher showed that demonstration does not proceed from extraneous principles; here he shows that it does not proceed from common principles.
Concerning this he does two things.
First, he states his proposition.
Second, he draws a conclusion from what he has said, at: it is no less evident (76a17; [146]).
Concerning the first he does two things.
First (75b37), he states his intention, saying that since it is clear that it is not just through anything at random that something is demonstrated, but it is required that demonstration be from one or another of a thing’s principles in such a way that what is demonstrated be in line with what the thing as such is, that is, it is required that the principles of the demonstration belong per se to that which is demonstrated; if, I say, this is so, then in order that something be scientifically known, it is not enough that it be demonstrated from true and immediate principles, but it is further required that it be demonstrated from proper principles.
Second, at: such proofs are like Bryson’s method (75b40), he proves his proposition, namely, that it is not enough to demonstrate something from true and immediate principles, because then something could be demonstrated in the way that Bryson proved squaring, that is, the squaring of a circle. For he showed that some square is equal to a circle, using common principles in the following way: In any matter in which it is possible to have something greater and something less than something else, one can find something equal to it. But in the genus of squares it is possible to find one which is less than a given circle, namely, one inscribed in the circle, and another which is greater, namely, one circumscribed about the circle. Therefore, one can be found which is equal to the circle.
But this proof is according to something common: for equal and greater and less transcend the genus of square and circle. Hence such characteristics demonstrate according to something common, because the middle is present in other things besides the one with which the demonstration is concerned. Therefore, such characteristics, since they belong to other things as well, do not belong to the things of which they are said as to proximate subjects.
Consequently, one who knows through such characteristics does not know according to what the thing is as such, that is, per se, but only per accidens. For if it were based on what the thing is according to itself, the demonstration would not apply to something of another genus. For we know a thing according to what it accidentally is, when we do not know it according to what it is in virtue of its own principles, that is, according to what comes per se from its principles, as “having three angles equal to two right angles” belongs per se to triangle, that is, according to what it is in virtue of its principles. Consequently, if the middle which one used belonged per se to the conclusion, it would necessarily have to be in the same proximity, that is, proximate according to genus, to the conclusion.
Third, at: the only exceptions to this rule (76a8), he excludes a doubt.
For it sometimes happens that the middle of a demonstration is not in the same genus as the conclusion. How this can happen he shows when he suggests that if the middle is not in the same proximate genus as the conclusion, but is present in the way that theorems in harmonics, (that is, in music), are demonstrable by arithmetic, it is nevertheless true that such things are also demonstrated. (For in a lower science there is demonstration through the principles of a higher science, as we have established, just as in the higher science there is demonstration through the principles of that higher science.) But there is this difference: on the part of the other science, namely, the lower one, there is only knowledge quia [that is, knowledge of the fact], (for the generic subject of the lower science is different from the generic subject of the higher science, from which the principles are borrowed.) But knowledge propter quid [that is, of the cause why] is proper to the higher science to which those proper attributes belong per se.
For since it is in virtue of the middle that a proper attribute is in a subject, that science will consider the 'propter quid' to which pertains the middle in which the proper attribute being demonstrated inheres per se. But if the subject belongs to another science, it will not pertain to that science to know 'propter quid' but only 'quia'. Nor will the proper attribute demonstrated of such a subject belong per se to it, but it will be through an alien middle. However, if the middle and the subject pertain to the same science, then it will be characteristic of that science to know 'quia' and 'propter quid'.
Having settled this doubt, he draws the conclusion chiefly intended, saying that in the light of the foregoing it is clear that one should not demonstrate haphazardly, that is, in any random way, but in such a way that a demonstration is made from the principles proper to each thing. But even the principles proper to the particular sciences have something common prior to them.
Then, at: it is no less evident (76a17) he draws a conclusion that follows from the aforesaid.
And in regard to this he does three things.
First, he draws the conclusion that if demonstrations in the particular sciences do not proceed from common principles, and if the principles of those sciences have something prior to them which is common, then it is clear that it is not the business of each science to prove its proper principles. For those prior principles through which the proper principles of the particular sciences can be proved are principles common to all; and the science which considers such common principles is proper to all, that is, is related to things which are common to all, as those other particular sciences are related to things respectively proper to each. For example, since the subject of arithmetic is number, arithmetic considers things proper to number. In like manner, first philosophy, which considers all principles, has for its subject “being,” which is common to all. Therefore, it considers the things proper to being (which are common to all) as proper to itself.
Second, at: this is so because he knows (76a19), he shows the pre-eminence of this science which considers common principles, namely, first philosophy, over the others. For that through which something is proved must always be scientifically more known or at least more known. For one whose science of something proceeds from higher causes must understand those causes even better, because his science proceeded from the absolutely prior, since it is not through something caused that he knows these causes. For when one’s knowledge of causes proceeds from caused things, he does not derive his knowledge from the absolutely prior and better known, but from what is prior and better known in reference to us. But when the principles of a lower science are proved by the principles of a higher science, the process is not from caused things to causes, but conversely. Therefore, such a process must go from the absolutely prior and from the absolutely more known. Thus, those items of a higher science that are used in proving matters of a lower science must be more known. Furthermore, that by which all other things are proved and which is itself not proved by anything prior must be the most known. Consequently, a higher science will be science in a fuller sense than a lower; and the highest science, namely, first philosophy, will be science in the fullest sense.
Third, at: but, as things are (76a23), he returns to the principal conclusion.
He says that demonstration does not cross over into another genus except, as already mentioned, when a demonstration from geometry is applied to certain subordinate sciences, as the mechanical arts, which employ measurements, or the perspective arts, such as the sciences which deal with vision, as optics which deals with the visual. The same applies to arithmetic in relation to harmonics, that is, music.