LESSON 5

Answers to Questions Raised in Book III about Principles of Demonstration

Chapter 3: 1005a 19-1005b 8

             319. Moreover, it is necessary to state whether it is the office of one science or of different sciences to inquire about those principles which are called axioms in mathematics, and about substance.

             320. Now it is evident that it is the office of one science--that of the philosopher--to investigate these.

             321. For these principles apply to all beings and not to some class distinct from the others. And all men employ them, because they pertain to being as being; for each class is being. But they employ them just so far as to satisfy their needs, i.e., so far as the class contains the things about which they form demonstrations. Hence, since it is evident that these principles pertain to all things inasmuch as they are beings (for this is what they have in common), the investigation of them belongs to him who considers being as being.

             322. Hence no one who is making a special inquiry attempts to say anything about their truth or falsity, neither the geometer nor the arithmetician.

             323. However, some of the philosophers of nature have done this, and with reason; for they thought that they alone were inquiring about the whole of nature and about being. But since there is one kind of thinker who is superior to the philosopher of nature (for nature is only one class of being), the investigation of these principles will belong to him who studies the universal and deals with first substance. The philosophy of nature is a kind of wisdom, but it is not the first.

             324. And whatever certain ones of those who speak about the truth attempt to say concerning the way in which it must be accepted, they do this through ignorance of analytics. For they must know these principles in order to attain scientific knowledge and not be seeking them when they are learning a science.

             325. It is evident, then, that it is also the business of the philosopher, i.e., of him who investigates all substance insofar as its nature permits, to investigate all syllogistic principles.

COMMENTARY

             588. Here he answers another question raised in Book III (198:C 387): whether it belongs to this science to consider the first principles of demonstration. This is divided into two parts. In the first (319:C 588) he shows that it belongs to this science to make a general study of all these principles; and in the second (326:C 596) he shows that it also belongs to it to make a special study of the first of these principles ("And it is fitting").

             In regard to the first he does three things. First, he raises the question whether it belongs to one or to different sciences to consider substance and the principles which are called axioms in the mathematical sciences. He assigns these principles more to the mathematical sciences because such sciences have more certain demonstrations and use these self-evident principles in a more manifest way inasmuch as they refer all of their demonstrations to them.

             589. Now it is evident (320).

             Second, he answers this question by saying that a single science investigates both of the foregoing things, and that this is the philosophy with which we are now concerned.

             590. For these principles (321).

             Third, he proves his proposed answer, and in regard to this he does two things. First (321:C 590), he proves it. Second (325:C 595), he introduces his main conclusion ("It is evident").

             Now he proves his proposed answer in two ways. He does this, first, by an argument; and second (322:C 592), by an example ("Hence no one").

             The argument is as follows: whatever principles pertain to all beings, and not just to one class of beings distinct from the others, belong to the consideration of the philosopher. But the above-mentioned principles are of this kind. Therefore they belong to the consideration of the philosopher. He proves the minor premise as follows. Those principles which all sciences use pertain to being as being. But first principles are principles of this kind. Therefore they pertain to being as being.

             591. The reason which he gives for saying that all sciences use these principles is that the subject genus of each science has being predicated of it. Now the particular sciences do not use the foregoing principles insofar as they are common principles, i.e., as extending to all beings, but insofar as they have need of them; that is, insofar as they extend to the things contained in the class of beings which constitutes the subject of a particular science, about which it makes demonstrations. For example, the philosophy of nature uses them insofar as they extend to changeable beings and no further.

             592. Hence no one (322).

             Then he proves what he had said by using an example. First, he introduces the proof; and second (323:C 593), he rejects a false notion held by some men ("However, some").

             He accordingly says, first (322), that no one whose chief intention is to hand down scientific knowledge of some particular being has attempted to say anything about the truth or falsity of first principles. Neither the geometer nor the arithmetician does this even though they make the greatest use of these principles, as was said above (319:C 588). Hence it is evident that the investigation of these principles belongs to this science.

             593. However, some (323).

             Here he rejects the false notion held by some men, and in regard to this he does two things. First, he rejects the false notion of those who occupied themselves with these principles even though they did not concern them. Second, (324:C 594), he rejects the false notion of those who wanted to deal with these principles in a different way than they should be dealt with ("And whatever").

             He accordingly says, first (323), that even though none of the particular sciences ought to deal with the above-mentioned principles, nevertheless some of the natural philosophers have dealt with them; and they did so not without reason. For the ancients did not think that there was any substance besides the changeable corporeal substance with which the philosophy of nature is concerned. Hence they believed that they alone established the truth about the whole of nature and therefore about being, and thus about first principles, which must be considered along with being. But this is false, because there is still a science which is superior to the science of nature. For nature itself, i.e., natural being, which has its own principle of motion, constitutes in itself one class of universal being. But not every being is of this kind, because it has been proved in the Physics, Book VIII, that an unchangeable being exists. Now this unchangeable being is superior to and nobler than changeable being, with which the philosophy of nature is concerned. And since the consideration of common being belongs to that science which studies the primary kind of being, then the consideration of common being belongs to a different science than the philosophy of nature. And the consideration of common principles of this kind will also belong to this science. For the philosophy of nature is a part of philosophy but not the first part, which considers common being and those attributes which belong to being as being.

             594. And whatever (324).

             Then he rejects the other false notion, which concerns the way in which such principles should be treated. For some men investigated these principles with the aim of demonstrating them. And whatever they said about the truth of these principles, i.e., how they must be accepted as true by force of demonstration, or how the truth found in all these principles must be reached, they did through ignorance of, or lack of skill in, "analytics," which is that part of logic in which the art of demonstration is treated. For "they must know these principles in order to attain scientific knowledge"; i.e., every science acquired by demonstration depends on these principles. But "those who are learning," i.e., the pupils who are being instructed in some science, must not seek these principles as something to be demonstrated. Or, according to another text, "those who have scientific knowledge must attain science from these principles"; i.e., those who attain knowledge by demonstration must come to know common principles of this kind and not ask that they be demonstrated to them.

             595. It is evident (325).

             He draws the conclusion primarily intended, namely, that it will be the function of the philosopher to consider every substance as such and also the first syllogistic principles. In order to make this clear it must be noted that self-evident propositions are those which are known as soon as their terms are known, as is stated in Book I of the Posterior Analytics. This occurs in the case of those propositions in which the predicate is given in the definition of the subject, or is the same as the subject. But it happens that one kind of proposition, even though it is self-evident in itself, is still not self-evident to all, i.e., to those who are ignorant of the definition of both the subject and the predicate. Hence Boethius says in the Hebdomads that there are some propositions which are self-evident to the learned but not to all. Now those are self-evident to all whose terms are comprehended by all. And common principles are of this kind, because our knowledge proceeds from common principles to proper ones, as is said in Book I of the Physics. Hence those propositions which are composed of such common terms as whole and part (for example, every whole is greater than one of its parts) and of such terms as equal and unequal (for example, things equal to one and the same thing are equal to each other), constitute the first principles of demonstration. And the same is true of similar terms. Now since common terms of this kind belong to the consideration of the philosopher, then it follows that these principles also fall within his scope. But the philosopher does not establish the truth of these principles by way of demonstration, but by considering the meaning of their terms. For example, he considers what a whole is and what a part is; and the same applies to the rest. And when the meaning of these terms becomes known, it follows that the truth of the above-mentioned principles becomes evident.