Commentary Aristotle Posterior Analytics
Lecture 18
Difference between principles and non-principles, common and proper principles
a26. It is hard to be sure whether one knows or not; for it is hard to be sure whether one’s knowledge is based on the basic truths appropriate to each attribute — the differentia of true knowledge. We think we have scientific knowledge if we have reasoned from true and primary premises. But that is not so: the conclusion must be homogeneous with the basic facts of the science.
a31. I call the basic truths of every genus those elements in it the existence of which cannot be proved.
a32. As regards both these primary truths and the attributes dependent on them the meaning of the name is assumed. The fact of their existence as regards the primary truths must be assumed; but it has to be proved of the remainder, the attributes. Thus we assume the meaning alike of unity, straight, and triangular; but while as regards unity and magnitude we assume also the fact of their existence, in the case of the remainder proof is required.
a37. Of the basic truths used in the demonstrative sciences some are peculiar to each science, and some are common, but common only in the sense of analogous, being of use only in so far as they fall within the genus constituting the province of the science in question.
a40. Peculiar truths are, e.g., the definitions of line and straight; common truths are such as ‘take equals from equals and equals remain’.
a42. Only so much of these common truths is required as falls within the genus in question: for a truth of this kind will have the same force even if not used generally but applied by the geometer only to magnitudes, or by the arithmetician only to numbers.
b2. Also peculiar to a science are the subjects the existence as well as the meaning of which it assumes, and the essential attributes of which it investigates, e.g., in arithmetic units, in geometry points and lines. Both the existence and the meaning of the subjects are assumed by these sciences; but of their essential attributes only the meaning is assumed. For example arithmetic assumes the meaning of odd and even, square and cube, geometry that of incommensurable, or of deflection or verging of lines, whereas the existence of these attributes is demonstrated by means of the axioms and from previous conclusions as premises. Astronomy too proceeds in the same way. For indeed every demonstrative science has three elements: (1) that which it posits, (the subject genus whose essential attributes it examines); (2) the so-called axioms, which are primary premises of its demonstration; (3) the attributes, the meaning of which it assumes.
b16. Yet some sciences may very well pass over some of these elements; e.g., we might not expressly posit the existence of the genus if its existence were obvious (for instance, the existence of hot and cold is more evident than that of number); or we might omit to assume expressly the meaning of the attributes if it were well understood. In the way the meaning of axioms, such as ‘Take equals from equals and equals remain’, is well known and so not expressly assumed. Nevertheless in the nature of the case the essential elements of demonstration are three: the subject, the attributes, and the basic premises.
It is hard to be sure. After showing that demonstration does not proceed from common but from proper principles, the Philosopher, to elucidate this point, decides questions concerning proper and common principles.
With respect to this he does two things.
First (76a26), he shows the need for such a determination, saying that it is difficult to discern whether we know from proper principles, which alone is truly scientific knowing, or do not know from proper principles. For many believe that they know scientifically if they possess a syllogism composed of things true and first. But this is not so; indeed, to know in a scientific manner it is required that the principles be proximate to the things to be demonstrated (here they are called 'first', just as above they were called “extremes”); or proximate to the first indemonstrable principles.
Second, at: I call the basic truths (76a31), he determines concerning proper and common principles.
And in regard to this he does two things.
First, he determines concerning common and proper principles.
Second, he shows how the demonstrative sciences are related to such principles, at: the law that it is impossible to affirm (77a10; [167]).
Concerning the first he does two things.
First, he distinguishes principles from non-principles.
Second, he distinguishes the principles from each other, at: of the basic truths used (76a37; [153]).
First (76a31), he shows what the principles are and says that the principles in any genus are those which, since they are true, happen not to be demonstrated, either not at all, if they are first principles, or at least not in that science in which they are accepted as principles. And he says, "the existence of which", (that is, whose truth), cannot be proved, to distinguish from the false which are not demonstrated in any science.
Second, at: as regards both these primary truths (76a32), he shows the points of agreement and of difference between principles and non-principles.
For principles agree with non-principles in the fact that with respect to each, that is, with respect both to the first, namely, the principles, and to the things that proceed from them, that is, things assumed from the principles, one must accept the supposition of what each signifies, because the quod quid est [that which the definition signifies] of a thing pertains properly to the science concerned with substance, that is, to first philosophy, from which all other sciences accept this.
But they are unlike in the fact that in regard to the principles one must accept the supposition that they are, whereas in regard to things from principles one is required to demonstrate that they are. Thus in mathematics one accepts by supposing both what unity is (which is a principle) and what straight and triangle are (which are not principles but proper attributes): but the fact that unity is or that magnitude is, the mathematician accepts as principles; but the other things, namely, things that are from principles, he demonstrates. For he demonstrates that a triangle is equilateral, or that an angle is right, or even that this line is straight.
Then, at: of the basic truths used (76a37), he distinguishes principles from one another.
First, the proper from the common.
Second, the common, one from the other, at: that which expresses necessary (76b23; [159]).
First, he divides proper and common principles.
Second, he settles something which might be doubtful, at: yet some sciences (76b16; [158]).
First (76a37), he lays down a division, saying that of the principles which we use in demonstrative sciences, some are proper to each single science but others common. And because this statement might seem contrary to what he established above, namely, that demonstrative sciences do not proceed from common principles, he hastens to add that the common principles are taken in each demonstrative science according to an analogy, that is, as proportionate to that science. And this is what he means when by way of explanation he further states that it is useful to accept such principles in the sciences insofar as they pertain to the genus of the subject which is investigated in that science.
Second, at: peculiar truths (76a40), he gives examples of each, saying that proper principles are, for example, that a line or a right angle is such and such. For in the sciences the definitions, both of the subject and of the proper attribute, are held as principles. But common principles are, for example, that 'if equals be subtracted from equals, the remainders are equal', and other common conceptions in the mind.
Third, at: only so much of these common truths (76a42), he shows how the demonstrative sciences use the aforesaid principles.
First, in regard to the common principles, he says that it suffices to accept each of those common ones, so far as it pertains to the generic subject with which the science is concerned. For geometry does this if it takes the above-mentioned common principle not in its generality but only in regard to magnitudes, and arithmetic in regard to numbers. For the geometer will then be able to reach his conclusion by saying that 'if equal magnitudes be taken from equal magnitudes, the remaining magnitudes are equal', just as if he were to say that 'if equals are taken from equals, the remainders are equal'. The same must also be said for numbers.
Second, at: also peculiar to a science (76b2), he shows how the demonstrative sciences employ proper principles. And he says that proper principles are things supposed in the sciences as existing, namely, the subjects, whose proper attributes are investigated in the sciences. In this way arithmetic considers unities, and geometry considers signs, that is, points, and lines. For they suppose these things to be and to be this, that is, they suppose of them that they are and what they are.
But in regard to the proper attributes they suppose what each signifies. Thus arithmetic supposes what even and odd are, and what a square or cubic number is. Similarly, geometry supposes what is rational in lines. (For a rational line is one about which we can reason, the line being given. For example, a rational line is any line commensurable with the given lines; but one which is incommensurable with it is called irrational and surd). In like manner, geometry supposes what a reflex or what a curved line are. However, these sciences demonstrate concerning all the above-mentioned proper attributes that they are, and they do so through common principles and principles demonstrated from the common principles. And what has been said of geometry and arithmetic should also be understood of astronomy. For every demonstrative science is concerned with three things: one is the generic subject whose per se attributes are investigated; another is the common dignities (axioms) from which, as from basic truths, it demonstrates; the third are the proper attributes concerning which each science supposes what their names signify.
Then, at: yet some sciences (76b16) he clarifies something about which there might be doubt.
For since he had said that the sciences suppose concerning the principles that they are and concerning the proper attributes what they are, but concerning the subject both that it is and what it is, someone might believe that he should have made special mention of all these. Hence he removes this by saying that nothing hinders certain sciences from neglecting some of the aforesaid, that is, from making express mention of them, as for example, not mentioning that it takes the existence of its generic subject for granted, if it is already obvious that it does exist. For we do not have the same evidence in all cases that they do exist, as we do in the case of number and in the case of hot and cold, the one being close to reason and the other to sense. Again, certain sciences do not suppose what the proper attributes signify in the sense of making express mention of them, just as they do not think it necessary always to make express mention of the common principles, because they are known. Be that as it may, the three above-mentioned items are naturally to be supposed in each science.