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Chapter IX: Demonstration is founded not on general, but on special and indemonstrable principles; nor is it easy to know whether one really possesses knowledge drawn from these principlesEdit

All demonstration is derived from special principles, themselves indemonstrable, the knowledge of which, in each genus, is the supreme knowledge on which the whole deduction depends.

Since it is clear that nothing can be demonstrated except from its own elementary principles, that is to say when the thing demonstrated is an essential attribute of the subject, it does not suffice for the possession of knowledge that a thing shall have been demonstrated from true, indemonstrable and ultimate premises. Otherwise demonstrations would be admissible resembling that of Bryson demonstrating the squaring of the circle. Now such arguments demonstrate by means of a common principle which will apply to another science as well, so that the same arguments are of service in other sciences distinct in kind. Thus we have no essential but only an accidental knowledge of the thing, for otherwise the demonstration would not also be applicable to another kind of subjects.

We have more than an accidental knowledge of anything when we see it in the light of its essential nature, after starting from the elementary principles of the things as such. Thus we know the law that a triangle has two right angles when we know of what figure this is an essential attribute and know it after starting from the principles peculiar to Triangle. Hence if the attribute is essentially an attribute of the subject, the middle term of the demonstration must necessarily be included in the same genus, or, if not, one of the genera must be subordinate to the other, as when proportions in harmonics are proved by means of arithmetical premises. Such relations are proved in the same way as in arithmetic, but there is a difference between the two cases, for the question of the Fact falls under the one science (since the subjects of the two sciences differ generically) but the Cause is established by the superior science, to which the properties in question are essential. It is plain even from the case of the subordinate sciences that no absolute demonstration of a thing can be attained save by starting from its own elementary principles. In this case, however, the elementary principles of the sciences in question are not mutually exclusive.

If this be admitted it is also clear that it is impossible to demonstrate the special elementary principles of each science, for the principles of such a demonstration would be the elementary principles of everything, and the science formed by them would be the universal master science; seeing that one who learns a thing through the recognition of higher causes has a better knowledge of it, and the principles through which he learns the thing are anterior when they are causes not themselves produced by any higher cause. If then his knowledge be of this higher kind it must have attained to the highest possible degree, and if this subjective knowledge of his constitute a science, that science must be higher than any other, and in fact the highest science.

The demonstration of one thing is not applicable to another genus except in the case already mentioned, as illustrated by the application of geometrical demonstrations to mechanical or optical, or of arithmetical demonstrations to harmonic theorems.

Now it is hard to decide if we really know a thing or not, for it is hard to decide whether our knowledge is derived from the elementary principles of the subject or not, and it is in this that knowledge consists. We imagine that, if we possess a syllogism drawn from true and primary premises, we really possess knowledge. This, however, is not the case, for the conclusions should belong to the same genus as the primary principles.