COMMENTARY ON THE
POSTERIOR ANALYTICS OF ARISTOTLE

However, it has also been proved in the Prior Analytics that in figures other than the first, namely, in the second and third, one cannot form a circular syllogism, i.e., one through which each of the premises can be syllogized from the conclusion; or if one is formed, it is done not by using the premises already used but by using propositions other than those which appear in the first syllogism.

That this is so is obvious. For the second figure always yields a negative conclusion. Consequently, one premise must be affirmative and the other negative. However, it is true that if both are negative, nothing can be concluded; and if both are affirmative, a negative conclusion cannot follow. Therefore, it is not possible to use the negative conclusion and the negative premise to obtain the affirmative premise as a conclusion. Hence, if this affirmative is to be proved, it must be proved through propositions other than the ones originally used. Again, in the third figure the only conclusion ever obtained is particular. However, at least one premise must be universal; furthermore, if either premise is particular, a universal cannot be concluded. Hence it cannot occur that in the third figure each of the premises can be syllogized from the conclusion and the remaining premise.

For the same reasons it is obvious that such a circular syllogism (through which each premise could be concluded) cannot be formed in the first figure except in the first mode, which is the only one that concludes to a universal affirmative. Furthermore, even in this mode the only case in which a circular syllogism could be formed such that each of the premises could be concluded, is when the three terms employed are equal, i.e., convertible. The proof is this: The premise must be concluded from the conclusion and the converse of the other premise, as has been stated. But such a conversion of each premise is impossible (for each is universal), except when the terms happen to be equal.

Lecture 9
(7341-34)
HOW SOMETHING IS SAID TO BE PREDICATED OF ALL

a2l. Since the object of pure— a25. and as a preliminary— a28. I call ‘true in every— a32. There is evidence for

After showing what a demonstrative syllogism is, the Philosopher in this section begins to show the nature and characteristics of the things that comprise a demonstration. Concerning this he does three things. First, he connects this with what has already been established. Secondly, he explains certain matters that must be understood first (73a25). Thirdly, he establishes what he had in mind, namely, to show what and of what sort are the things that constitute a syllogism (74b5) [L. 12].

He says therefore first (73a21), that since the definition of scientific knowledge given above spoke of that which cannot be otherwise, that which is scientifically known through demonstration will be necessary. Then he explains what it is to know something in a scientific way through demonstration, saying that demonstrative science is “what we possess in having a demonstration,” i.e., what we acquire through demonstration. Consequently, it follows that the conclusion of a demonstration is necessary.

Now although the necessary could be syllogized from the contingent, it is not possible through a contingent middle to obtain scientific knowledge of the necessary, as will be proved later. Furthermore, because the conclusion. of a demonstration is not only necessary, but, as has been said, is known through demonstration, it follows that a demonstrative syllogism proceeds from necessary things. Consequently, we must establish from what and from what sort of necessary things a demonstration proceeds.

Then (73a25) he interjects certain things that must be understood as preliminaries to the matters to be discussed. Apropos of these he does two things:

First, he states his intention (73a25), saying that before determining specifically the nature and characteristics of the things that form a demonstrative syllogism, we must indicate what is meant when we say, “of all,” and “per se,” i.e., in virtue of itself, and “commensurately universal.” For if we are to understand the nature of the things that form a demonstration, we must know what these terms mean, because they describe things that must be observed in demonstrations. For in the propositions of a demonstration it is required that something be predicated universally~ which he signifies by the term “said of all”—and “per se,” i.e., in virtue of itself, and “first”—which he signifies by the words, “commensurately universal.” But these three things are related by adding something to the previous one. For whatever is predicated per se is predicated universally [i.e., of all], but not vice versa. Again, whatever is predicated first is predicated per se, but not vice versa. This, therefore, shows why they are arranged as they are.

But why there are three and wherein they differ are explained by the fact that something is said to be predicated “of all” or universally in relation to things contained under the subject. For, as it is stated in Prior Analytics, something is said “of all,” when there is nothing under the extension of ,the subject that does not receive the given predicate. But it is in relation to the subject that something is said to be predicated per se, because the subject is mentioned when this predicate is defined, or vice versa, as will be explained below. Finally, something is said to be predicated of another thing “first” in relation to items that are prior to the, subject and embrace or include it, as the more universal includes the less. Thus to have three angles equal to two right angles is not predicated “first” of isosceles, because it is previously predicated of something prior to isosceles, namely, of triangle.

Secondly (73a28), he establishes his proposition. And his treatment is divided into three parts. First, he shows what is meant by “said of all.” Secondly, what is meant by “said per se,” i.e., in virtue of itself (73a34) [L. 10]. Thirdly, what is meant by “commensurately universal” (73b27) [L. II].

Concerning the first he does two things. First (73a28), he states what it is to be “said of all.” And it should be noted that the phrase, “said of all,” is taken here in a sense somewhat different from the sense it has in Prior Analytics, where it is taken in a very general sense so as to accommodate both the dialectician and the demonstrator. Therefore, no more is mentioned in its definition than that the predicate be found in each of the things included under its subject. But that might be verified only at a given moment—which is the sense in which the dialectician sometimes uses it; or it might be verified absolutely and at all times—which is the sense to which the demonstrator must always limit himself.

Accordingly, two things are mentioned in the definition of “said of all”: one is that there is nothing within the extension of the subject that the predicate does not apply to. And he indicates this when he says, “not of one to the exclusion of others”; the other is that there is no time in which the predicate does not belong. And this he indicates when he says, “not at this or that time only.” And he gives the example of “man” and “animal,” saying that “animal” is predicated of every man; and of anything of which it is true to say that it is a man, it is true to say that it is an animal, and whenever it is a man, it is an animal. The same is true between line and point: for a point is in every line and always in every line.

Secondly (73a32), he explains this definition, using as evidence the techniques employed in rebuttals. For a universal proposition is not rebutted unless one or other of things it states is not verified. For when we are asked whether something is said “of all” in a demonstration, we can say, “No,” for two reasons, i.e., either because it is not true of each instance of the subject, or because now and then it is not true. Hence it is clear that “being said of all” signifies each of these.

Lecture 10
(73a34-b26)
HOW SOMETHING IS SAID TO BE PREDICATED PER SE OF A THING

a34. Essential attributes are— a38. (2) such that, while— b5. Further (a) that— b10. In another sense again— b16. So far then as concerns— b25. Thus, then, we have

After determining about “said of all,” the Philosopher now determines about “said per se” [i.e., said in virtue of itself] and does three things. First, he shows the number of ways something is said per se. Secondly, how the demonstrator makes use of these ways (73b16). Thirdly, he summarizes (73b25).

In regard to the first it should be noted that this preposition per [“in virtue of” or “by”] denotes a causal relationship, although sometimes it also signifies a state, as when someone is said to be per se, i.e., by himself, when he is alone. But when it designates a relationship to a cause, sometimes the cause is formal, as when it is stated that the body lives in virtue of the soul; sometimes the relationship is to a material cause, as when it is stated that a body is colored in virtue of its surface, i.e., because the surface is the subject of color; again, it might even designate a relationship to an extrinsic cause, particularly an efficient cause, as when it is said that water is made hot in virtue of fire. But just as this preposition per designates a relationship to a cause, when something extrinsic is the cause of that which is attributed to the subject, so also when the subject or something pertaining to the subject is the cause of that which is attributed to the subject. This latter is what per se, i.e., in virtue of itself, signifies.

Therefore, the first way of saying something per se (73a34) is when that which is attributed to a subject pertains to its form. And because the form and essence of a thing are signified by its definition, the first mode of that which is per se is when the definition itself or something expressed in the definition is predicated of the thing defined. This is what he means when he says, “Essential attributes are such as belong to their subject as elements in its essential nature,” i.e., included in the definition which indicates what it is, whether those elements are stated in the nominative case or in one of the oblique cases. Thus, “line” is stated in the definition of triangle. Hence “line” is in triangle per se. Again, in the definition of line, “point” is mentioned; hence “point” is per se in line. And the reason why they are mentioned in the definition is stated when he says, “for the very being or substance” [i.e., the essence, which the definition signifies] “of triangle and line is composed of these elements,” namely, of lines and points. However, this does not mean that a line is formed out of points, but that “point” is involved in the very notion of line, just as “line” is involved in the very notion of triangle. And he asserts this in order to exclude things which are part of a thing’s matter and not of its species: thus, “semicircle” is not mentioned in the definition of circle, or “finger” in the definition of man, as it is stated in Metaphysics VII.

He states further that all those items which are found universally in the definition expressing what a thing is are attributed to it per se.

The second mode of saying per se is when this preposition per implies a relationship of material cause, in the sense that that to which something, is attributed is its proper matter and subject. For it is required, when defining an accident, to mention its proper subject in one of the oblique, cases: thus when an accident is defined abstractly, we say that “aquilinity” is a curvature of a nose,” but when it is defined concretely, the subject is put in the nominative case, so that we say that “the aquiline is a curved nose.” Now the reason for this is that since the being of an accident depends on its subject, its definition—which signifies its being—must mention that subject. Hence it is the second mode of saying per se, when the subject is mentioned in the definition of a predicate which is a proper accident of the subject.

And this is what he means when he states (73a38), “essential attributes are those such that while they belong to certain subjects,” i.e., to subjects of accidents, “the subjects to which they belong are contained in the attribute’s own defining formula,” i.e., in the expression which describes what the accident is, i.e., in the definition of the accident. “Thus straight and curved belong to line per se.” For “line” is mentioned in their definition. For the same reason “odd” and “even” belong per se to number, because “number” is mentioned in their definition. Again, prime and

compound are predicated per se of number, and “number” is mentioned in their definition. (For a prime number, for example, seven, is one which is exactly divisible by no other number but “1”; but a compound number, for example, nine, is one which is exactly divisible by some number greater than “1.” Again, “isoplural,” i.e., equilateral, and scalene, i.e., having three unequal sides, belong per se to triangle, and “triangle” is mentioned in their definition. Accordingly, he adds that their respective subjects belong to each of the aforesaid accidents and are mentioned in the expression which states what each is, i.e., in the definition: thus “line” belongs to some of them, and “number” to others.

In each of these subjects that have been mentioned, I say that its accident is in it per se. But those predicates which are neutral, i.e., of such a nature as not to be mentioned in the definition of their subjects, nor the subjects in their definition, are accidents, i.e., are predicated per accidens: for example, “musical” and “white” are predicated per accidens of animal.

Then (73b5) he sets down another mode of that which is per se, i.e., the sense in which it signifies something in isolation. Thus something which is a singular in the genus of substance and which is not predicated of any subject is said to be per se. The reason for this is that when I say, “walking” or “white,” I do not signify either of them as something isolated or apart, since something else which is walking or white is understood. But this is not the case with terms which signify a “this something,” i.e., with terms that signify first substance. For when I say, “Socrates” or “Plato,” it is not to be supposed that there is something else, over and above what they really are, which would be their subject. Therefore, things which are thus not predicated of any subject are per se, but things which are predicated of a subject, as being in the subject, are accidents. However, not all things predicated of a subject, as universals of their inferiors, are accidents.

It should be noted, however, that this mode is not a mode of predicating, but a mode of existing; hence at the very start he said that they exist per se and not that they are said per se.

Then (73b10) he gives the fourth mode, according to which the preposition per designates a relationship of efficient cause or of any other. Consequently, he says that whatever is attributed to a thing because of itself, is said of it per se; but whatever is not so attributed is said per accidens, as when I say, “While he was walking, it lightened.” For it is not the fact that he walks that causes lightning, but this is said by coincidence. Butt if the predicate is in the subject because of itself, it is per se, as when we say, “Slaughtered, it died.” For it is obvious that because something was slaughtered, it died, and it is not a mere coincidence that something slaughtered should die.

Then (73b16) he shows how the demonstrator uses the aforesaid modes. But first it should be noted that, since science bears on conclusions, and understanding [intuition] bears on principles, the scientifically knowable are, properly speaking, the conclusions of a demonstration wherein proper attributes are predicated of their appropriate subjects. Now the appropriate subjects are not only placed in the definition of attributes, but they are also their causes. Hence the conclusions of demonstrations involve two modes of predicating per se, namely, the second and the fourth.

And this is what he means when he says that the predications “in the scientifically knowable in the strict sense,” i.e., in the conclusions of demonstrations are per se in the sense of something contained in the predicates, i.e., in the way that subjects are contained in the definition of accidents which are predicated of the former; or are present on account of them, i.e., in the way that predicates are in a subject by reason of the subject itself, which is the cause of the predicate.

Then he shows that such scientifically knowable things are necessary, because it is impossible for a proper accident not to be predicated of its subject. But this can occur in two ways: sometimes it is absolute, as when the accident is convertible with its subject, as “having three angles equal to two right angles” is convertible with triangle, and “risible” with man. At other times, two opposites stated disjunctively are of necessity in the subject, as “straight or oblique” in line, and.”odd or even” in number. He shows that the reason for this is the fact that contrariety, privation and contradiction are in the same genus. For privation is nothing more, than a negation in a determinate subject. Again, a contrary is equivalent to a negation in some genus, as in the genus of numbers, odd is the same as “not even” by way of consequence. Therefore, just as it is necessary, either to affirm or deny, so it is necessary that one of two things that belong per se, be in its proper subject.

Then (73b25) he summarizes, and the text is clear.

Lecture 11
(73b27-74a3)
HOW SOMETHING IS SAID TO BE PREDICATED AS COMMENSURATELY UNIVERSAL

b27. I term ‘commensurately— b28. from which it clearly— b29. The essential attribute— b32. An attribute belongs— b34. Thus, e.g., (1) the— a1. and the demonstration

After determining about “said of all” and “said per se,” the Philosopher here determines concerning the “universal.” This treatment falls into two parts. In the first he shows what the universal is. Secondly, how error occurs in our understanding of it (74a4) [L. 12]. Concerning the first he does two things. First, he shows what the universal is. Secondly, how the demonstrator uses the universal (74a1). Concerning the first he does two things. First, he shows that the universal contains within itself the attributes of “being said of all” and of “being said per se.” Secondly, he shows what the universal adds to them (73b33).

To understand what is being said here it should be noted that “universal” is not to be taken here in the sense that anything predicated of several is a universal, as when Porphyry treats of the five universals; rather “universal” is taken here according to a certain correspondence or commensurateness of the subject with the predicate, so that the predicate is not found outside the subject nor is the subject without the predicate.

With this in mind, it should be noted that he does three things with respect to the first point. First (73b27), he says that the universal, namely, the predicate, is both verified of all, i.e., is predicated universally of its subject, and is said per se, i.e., is in and belongs to the subject according to the essential nature of the subject. For many things are said universally of certain things to which they do not belong per se and as such. Thus, every stone is colored, but not precisely as stone, but as it has a surface.

Secondly (73b28), he draws a corollary from this and says that since the universal is something which is per se in a thing, and since it has been shown that whatever things are in something per se are in it of necessity,

it is obvious that universal predicates, as they are being taken here, are necessarily present in the things of which they are predicated.

Thirdly (73b29), lest anyone suppose that “per se” and “precisely as such,” both of which were mentioned in the definition of the universal, are different, he shows that they are the same. Thus, “point” is per se in line in the first way, and “straight” in the second way. For each is in line precisely as it is a line. In like manner, “two right angles” belongs to triangle precisely as triangle, i.e., its angles are equal to two right angles, which is per se in triangle.

Then (73b33) he shows what “universal” adds to the notions, “being said of all” and “being said per se.” In regard to this he does two things:

First, he says that a predicate is “universal,” when it is not only in each thing of which it is asserted, but it is demonstrated to be first or primarily in the thing which receives that predicate.

Secondly (73b34), he clarifies this with an example and says that “having three angles equal to two right angles” is not found in just any figure in general, although this could be demonstrated of some figure, because it is demonstrated of triangle, which is a figure; yet it is not found in any random figure, nor is just any figure used when it is demonstrated. For a rhombus is a figure, but it does not have three angles equal to two right angles. But an isosceles, i.e., a triangle with two equal sides, always has its three angles equal to two right angles. Nevertheless, isosceles is not the primary thing to which this belongs, for it belongs basically to triangle, and belongs to isosceles precisely as it is a triangle. Therefore, whatever is demonstrated basically to have its three angles equal to two right angles (or whatever else be thus demonstrated), the universal predicate is present in it primarily, as in triangle.

Then (74a1) he shows how a demonstrator uses the “universal,” saying that demonstration is concerned per se with such a universal, but with other things qualifiedly and not per se. For a demonstrator demonstrates a proper attribute of its proper subject; and if he demonstrates it of anything else, he does so only insofar as it pertains to that subject. Thus, he proves that some property of triangle belongs to a figure and to an isosceles precisely as some figure is a triangle, and as the isosceles is a triangle. But the reason why “having three” is not in isosceles primarily is not because it is not predicated of it universally, but because it is found more frequently, i.e., in more things than in isosceles, since this is common to every triangle.

Lecture 12
(74a4-b4)
HOW ERROR OCCURS IN TAKING THE UNIVERSAL

a4. We must not fail to observe— a6. We make this mistake— a13. Case (3) may— a17. An example of (1)— a18. An instance of (2)— a25. Hence, even if one— a33. When, then, does our knowledge— a35. ‘But’, it will be asked

After specifying what the universal is, the Philosopher here shows how one might err in understanding the universal. In regard to this he does three things. First, he says that sometimes one might err in this matter. Secondly, he tells in how many ways (74a6). Thirdly, he gives the criterion for knowing whether the universal is being employed correctly (74a35).

He says therefore first (74a4), that in order to avoid mistakes in demonstrating, one should be aware of the fact that quite often something universal seems to be demonstrated, which is not being demonstrated.

Secondly (74a6), he indicates the ways in which this mistake can occur. And in regard to this he does two things.

First, he enumerates these ways and says that there are three possible errors in understanding a universal. The first is likely to occur when under some common genus there is nothing else to take as the thing to which the universal initially applies than this singular, to which it is incorrectly applied. For example, if man were the only animal existing, and “sensible,” which is initially and per se in animal, were to be assigned as a primary universal to man. (It should be noted that singular is being used here in a wide sense for any inferior, in the way that a species might be called a singular contained under a genus). Or we might say that it is not possible to find a genus with only one species: for a genus is divided into species through opposing differences. But if one contrary is found in nature, so must the other, as the Philosopher explains in On the Heavens II. Therefore, if one species is found, another will be found. However, one species is divided into distinct individuals by the division of matter. But it sometimes happens that all the matter proportionate to a given species is comprehended under one individual, so that in that case there is only one individual under one species. Hence it is significant that he did say, “singular.”

The second way is when it is possible to take several inferiors under something common which is verified in things that differ in species, but that common item has no name. For example, if “animality” had no name, and “sensibility,” which is proper to animal, were to be assigned to the inferiors of animal (either collectively or distributively) as their primary universal.

The third way is when that of which something is demonstrated to be its primary universal is related to what is demonstrated of it, as a whole is related to a part. For example, if the power to see were assigned as a primary universal to animal: for not every animal can see. In this case the demonstration, i.e., what should have been demonstrated, is “in those things which are in part,” i.e., in some but not all of the things included under the subject; furthermore, it will be a demonstration “of all,” but not of all that the demonstration mentions. For the power to see can indeed be demonstrated universally of something, but not of animal universally, as of that to which it belongs primarily. And he explains why he says, “primarily,” namely, because a demonstration bears on what is both universal and first.

Secondly (74a13), he gives examples of each of these ways. First, of the third way, saying that if someone were to demonstrate of two straight lines that they do not intersect, i.e., that they do not meet, it might seem that we have a demonstration of this sort, i.e., one that bears on a primary universal, on the ground that “not to meet” is true of certain straight lines, and not that this happens only because the straight lines are equal, i.e., equally distant. But if the lines should be equal, i.e., equidistant, then “not to meet” belongs to any and all of them, because it is universally true that lines which are straight and equally distant, even should they be lengthened ad infinitum, will not meet.

Secondly (74a17), he gives an example of the first way, saying that if there were no triangle but the isosceles which is a triangle having two equal sides, it might seem that what is true of triangle as triangle should be true of isosceles as isosceles. But this would not be so.

Thirdly (74a18), he gives an example of the second way. Now it seems that he saved this for last because he wished to spend more time on it. Concerning it he does three things. First, he gives the example. Secondly, he draws a corollary from what he has said (74a25). Thirdly, he gives a reason for the aforesaid (74a33).

Concerning the first it should be noted that a ratio is a relation of one quantity to another, as 6 is related to 3 in the ratio of 2 to 1. The com. paring of one ratio to another is a proportion, which, if it is disjoint, has four terms: for example, as 4 is to 2, so 6 is to 3. But if it is joint, it has three terms, one of which is used twice: for example, as 8 is to 4, so 4 is to 2.

Now it is obvious that in a proportion two terms are antecedents and two are consequents. For example, in the proportion that 6 is to 3 as 4 is to 2, 6 and 4 are the antecedents, 3 and 2 the consequents. Again, a proportion is alternated by bringing the antecedents together and the consequents together. For example, when I say: As 4 is to 2, so 6 is to 3; therefore, as 4 is to 6, so 2 is to 3.

What he is saying, therefore, is that alternate proportion is verified in lines, numbers, solids, (i.e., bodies), and times. But just as this is established separately for each of them, namely, for numbers in arithmetic, for lines and solids in geometry, and for times in natural philosophy or astronomy, so it is possible to prove it of all of them in a single demonstration. But the reason why a separate demonstration was employed to prove alternation for each was that the one feature they had in common was unnamed. For although quantity is a feature common to all of them, quantity includes other things besides them; for example, speech and other things that are accidentally quantitative.

Or, better still, it was because alternate proportion does not belong to quantity precisely as quantity, but as compared to another quantity according to a fixed ratio. That is why at the very start he spoke of alternate proportion. But there is no general name for the aforesaid things precisely as they are proportional. Furthermore, when alternate proportion is demonstrated separately for each of them, it is not a universal that is being demonstrated. For “to be alternately proportional” is not in numbers and lines according to what each of them is, but according to something common. Besides, those who demonstrate alternate proportion of lines look upon this attribute as a universal predicate of line precisely as it is a line, or of number precisely as it is number.

Then (74a25), he draws a corollary from the above and says that for the same reason that something universal is not being demonstrated when a common unnamed attribute is proved to be a universal predicate of each species, so too when something is demonstrated this way of a common attribute which does have a name. For example, if someone uses the same demonstration for each species of triangle and proves that each has three angles equal to two right angles, or if he uses different demonstrations, say one for isosceles and another for scalene, even then he does not on that account know that a triangle has three angles equal to two right angles except in a sophistical way, i.e., per accidens: for he does not know it of triangle as triangle, but as equilateral or as having two equal sides or as having three unequal sides.

Furthermore, one who demonstrates in this way does not know the universal of triangle, i.e., he does not have a knowledge of triangle in a universal way (even if there happens to be no other triangle besides those of which he knew this: and this because he had not the knowledge of triangle as triangle but under the aspect of its various species). Again, strictly speaking, he did not know every triangle: for although he knew every triangle according to number (if there was none he did not know), yet according to species he did not know each one. For something is known universally according to species, when it is known according to the notion of the species, but according to number and not universally, when it is known according to the multifarious things contained under the species. And in this matter there is no difference whether we compare species to individuals or genera to species. For triangle is the genus of equilateral and isosceles.

Then (74a33) he assigns the reason for the aforesaid, first asking when does one know universally and absolutely, if one who knew in the aforesaid way did not know universally. And he answers that obviously if the essence of triangle in general and of each of its species (each taken sepai-ately or all taken together) were the same, then one would know universally and absolutely about triangle, when he knew about any one species of it separately or about all of them together. But if the essence is not the same, then it will not be the same but something different to know triangle and to know its several species. And by knowing something of the species one does not know it of triangle precisely as triangle.

Then (74a35) he gives the rule on how the universal can be properly understood, saying that whether something belongs to triangle precisely as triangle or to isosceles precisely as isosceles, and when it is that what is demonstrated is first and universal in the given subject precisely as this, i.e., precisely according to the subject—all this will be clear from what I shall say.

For whenever some item is removed from the subject and the original universal still applies to what remains, then it is not the first universal of that subject. For example, upon the removal of isosceles or of brazen there still remains the triangle which has three angles equal to two right angles; hence the possession of three angles equal to two right angles is not a first universal of isosceles or of brazen triangle. But if the item “figure” be removed, nothing which has three angles equal to two right angles remains, nor again if you remove “bounded” which is more universal than figure, since a figure is something enclosed by a bound or bounds. Nevertheless, the attribute in question [namely, having three angles equal to two right angles] does not belong first to figure or to bounded, because it does not belong to either of them universally.

To what then will it be first? Obviously to triangle, because it belongs to the others (both superiors and inferiors) precisely as they are triangles For it belongs to figure to have three angles equal to two right angles only because a triangle is some figure, and similarly to isosceles, only because it is a triangle, and it is of triangle that “having three...” is demonstrated. Hence, it is to triangle that it belongs as a first universal

Lecture 13
(74b5-75a17)
DEMONSTRATION PROCEEDS FROM NECESSARY THINGS

b5. Demonstrative knowledge— b6. Now attributes attaching— b13. We must either state— b18. That demonstration proceeds— b22. This shows how naive— b27. A further proof— b32. Or again, if a man— a1. When the conclusion— a13. To sum up, then:

Having finished his treatment of that which is said “of all” and “per se” and “universally,” all of which we use in demonstration, the Philosopher now begins to discuss the items from which demonstration proceeds. And there are two parts. In the first he shows what demonstration of the reasoned fact [demonstration propter quid] proceeds from. In the second what demonstration of the fact [demonstration quia] proceeds from (78a22) [L. 23]. The first is divided into two parts. In the first he shows the sort of things from which a demonstration proceeds. In the second, what the principles of demonstration are (76a26) [L. 18]. The first is divided into three parts. In the first he shows that demonstration issues from necessary things. In the second that it issues from things that are per se (76a26) [L. 14]. In the third that it proceeds from proper principles (75a38) [L. 15]. Concerning the first he does two things. First, he shows that demonstration should proceed from necessary things. Secondly, he proves certain things he had presupposed (74b27). In regard to the first he does three things. First, he makes a connection with what went before. Secondly, he proves his proposition (74b6). Thirdly, he draws a conclusion from the aforesaid (74b22).

He says therefore first (74b5), as an inference from what has already been established, that if there be demonstrative science, i.e., if science is acquired through demonstration, it must issue from necessary principles. The necessity of this inference is clear, because that which is known scientifically cannot be otherwise than it is, as was pointed out in the definition of scientific knowing.

Then (74b6) he shows that demonstration issues from necessary things. First, with a reason. Secondly, with a sign (74b19).

With respect to the first he gives two reasons. The first (74b6) is this: Things predicated per se are in a thing necessarily. (And he manifests this in two of the modes of predicating per se: in the first mode, because the things predicated per se are “in that which is what,” i.e., in the definition of the subject. But whatever is put in the definition of anything is predicated of it necessarily. In the second mode, because certain subjects are put in “that which is the what of things predicated of them,” i.e., in the definition of their predicates. And if these are opposites, one or other of them is necessarily in the subject: thus, either “odd” or “even” is in number, as we showed above). Now it is clear that it is from principles of this kind, namely, per se, that a demonstrative syllogism proceeds. (And this is proved by the fact that whatever is predicated, is predicated either per se or per accidens, and that things predicated per accidens are not necessary). Therefore, since it is not a demonstration but rather a sophistic syllogism that issues from things that are per accidens, it follows that demonstration proceeds from necessary things.

Furthermore, it should be noted that, since in a demonstration a proper attribute is proved of a subject through a middle which is the definition, it is required that the first proposition (whose predicate is the proper attribute, and whose subject is the definition which contains the principles of the proper attribute) be per se in the fourth mode, and that the second proposition (whose subject is the subject itself and the predicate its definition) must be in the first mode. But the conclusion, in which the proper attribute is predicated of the subject, must be per se in the second mode.

Then (74b14) he sets forth the second reason and it is this: “Demonstration is concerned with the necessary and the demonstrated,” i.e., the conclusion of the demonstration cannot be other than it is. This statement is to be taken as the principle proving our proposition that demonstration proceeds from necessary things. And from what has been said, the truth of this principle is obvious. From this principle one argues thus: a necessary conclusion cannot be scientifically known save from necessary principles; but a demonstration makes a necessary conclusion scientifically known: therefore, it must proceed from necessary principles.

Herein lies the difference between a demonstration and other syllogisms. For in the latter it is enough if one syllogizes from true principles. Nor is there any other type of syllogism in which it is required to proceed from the necessary: in a demonstration alone must this be observed. And this is proper to a demonstration, i.e., to proceed from the necessary.

Then (74b18) he proves the same thing in the following way, using a sign: Suit is brought against an argument solely on the ground that something is wanting that should have been observed in that argument. But against one who believes that he has demonstrated, we bring the charge that the things from which he proceeded are not necessarily true, whether we are convinced that they could be otherwise than stated, or whether we bring a charge of reason, i.e., for the sake of disputing. Therefore, demonstration should proceed from necessary things.

Then (74b22) he draws a conclusion from the foregoing, saying that “this,” namely, the fact that a demonstration must conclude from necessary things, “shows” that they are obtuse who assumed that they rightly chose the principles of demonstration, if the proposition they chose was probable or true, as the Sophists do, i.e., those who appear wise but are not. For scientific knowing consists in nothing less than having science, namely, from demonstration. But when something is probable or improbable, it is not certain whether it is first or not first, whereas that on which a demonstration bears must be first in some genus and be true. This does not mean that a demonstrator may take anything that is first: it must be first in that genus with respect to which he is demonstrating. Thus, arithmetic does not choose what is first in regard to magnitude, in regard to number.

It should be noted that Sophists are not taken here as they are in the book of Sophistical Refutations, i.e., who proceed from things that seem probable but are not, or seem to syllogize but do not. For just as some are called Sophists, i.e., seeming to be wise but not really so, inasmuch as they appear by their arguments to be scientific knowers but are not, so dialectical arguments, if they seem to prove demonstratively but do not, are sophistic.

Then (74b27) he proves something he had presupposed. And concerning this he does two things. First, he shows that a necessary conclusion cannot be scientifically known from non-necessary principles. Secondly, that although the necessary cannot be scientifically known from the nonnecessary, it can nevertheless be syllogized (75al).

He proves the first with two reasons, one of which (74b27) is this: If one does not have an argument which shows the cause why [i.e., propter quid], he does not know scientifically, even though a syllogism be had; because to know scientifically is to know the cause of a thing, as we stated above. But an argument which infers a necessary conclusion from nonnecessary principles does not show the cause why.

To clarify this he gives an example, using general terms. Assume that the conclusion, “Every C is A,” is necessary, and that it is demonstrated by a middle, B, which is not a necessary middle but a contingent one, so that, for example, one of the propositions, “Every B is A,” and “Every C is B,” or both, are contingent. Now it is obvious that through this contingent middle, namely, B, one cannot obtain scientific knowledge propter quid of the necessary conclusion, “Every C is A.” For if the cause on which something depends is removed, the effect must be removed. But this middle, being contingent, can be withdrawn, whereas the conclusion, since it is necessary, cannot be withdrawn. It follows, therefore, that a necessary conclusion cannot be scientifically known through a contingent middle.

The second reason (74b32) is this: “If a man is without scientific knowledge now, then even though he possesses the same argument he had up to now, and he has been kept sound,” i.e., has not ceased to be, “and the thing known has not changed; and if he has not forgotten, then obviously he has not known scientifically.” In this passage the Philosopher mentions four ways in which a person loses the science he once had: one way is when there slips from his mind the argument through which he formerly knew scientifically. Another is through the destruction of the knower. A third is by a change occurring in the thing known: thus if, while you are sitting, I know that you are sitting, this knowledge perishes when you are not sitting. The fourth way is by forgetting. Hence if none of these ways has occurred, then, if a person does not know a thing scientifically now, he never did know it. Now one who holds a necessary (onclusion in virtue of a contingent middle, no longer knows it scientifically when the middle ceases to be, for the middle no longer exists, cven though he retains the same argument and the fact remains the same, ;tnd he has not forgotten anything. Therefore, even when the middle had not ceased to be, he did not know scientifically.

But that a contingent middle is liable to perish he proves by the fact that whatever is not necessary must at some time perish. And even though the middle has not perished, nevertheless, because it is not necessary, it is obvious that it is liable to perish. Now when something contingent is put forward, its result is not impossible but possible and contingent. But what followed in our case was impossible, namely, that one should have had science of something and later not have had it, all the conditions described above prevailing. And this follows from the fact that a contingent middle has perished, which, although it may not be true [that it has], is nevertheless liable to, as has been said.

Then (75a1) he shows that although a necessary conclusion cannot be scientifically known through a contingent middle, nevertheless a necessary conclusion can be syllogized from a non-necessary middle. Hence he says that nothing hinders a necessary conclusion from being obtained through a non-necessary middle, as witness the case in which something is syllogized with a dialectical syllogism, although not with a demonstrative syllogism which causes science. For the necessary happens to be syllogized from the non-necessary, just as the true happens to be syllogized from the untrue—although the converse does not occur. For when the middle is necessary, the conclusion too will be necessary; similarly, from true premises something true is concluded.

But that the necessary is always concluded from necessary premises he proves thus: “Let A be necessary of B,” i.e., let this proposition, “Every B is A,” be necessary, “and the latter of C,” i.e., let “Every C is B” also be necessary. Now from these two necessary things follows a third necessary thing, namely, the conclusion, “Every C is A.” For it has been proved in the Prior Analytics that “from two propositions of necessity follows a’ conclusion of necessity.”

Then by way of consequence he shows that if the conclusion were not necessary, the middle could not be necessary. For example, suppose that this conclusion, “Every C is A,” is not necessary, but that the two premises are necessary. According to our previous doctrine it follows that the conclusion is necessary—which is contrary to the supposition in our example, namely, that the conclusion not be necessary.

Then (75a13) he infers from the aforesaid the conclusion originally intended, saying that since a thing must be necessary if it is made known by way of demonstration, it is clear from the foregoing that a demonstration must rest on a necessary middle. For otherwise it would not be scientifically known that the conclusion is necessary, neither propter quid nor quia, since the necessary cannot be known through the non-necessary, as we have shown. But if someone rests on an argument based on a non-necessary middle, he will be in one of two states. For since he does not actually know in a scientific way, he will either believe that he does know in a scientific way, if he assumes a non-necessary middle as necessary, or he will not presume that he knows in a scientific way, i.e., if he believes that he does not have a necessary middle. And this is to be universally understood both of scientific knowledge quia, in which something is known through mediate principles, and of science propter quid, in which something is known through immediate principles. The difference between these two will be explained later.

Lecture 14
(75al8-37)
DEMONSTRATION BEARS UPON AND PROCEEDS FROM THINGS WHICH ARE PER SE

a18. Of accidents that are— a2l. A difficulty, however,— a24. The solution is that— a28. Since it is just those

After showing that demonstration proceeds from necessary things, the Philosopher then shows that it is concerned with things that are per se and proceeds from things that are per se. In regard to this he does three things. First, he shows that demonstration is concerned with things that are per se, i.e., that the conclusions of demonstrations are per se. Secondly, he raises a problem (75a21). Thirdly, he shows that demonstration proceeds from things that are per se, i.e., that the principles of demonstrations must be per se (75a28).

He says therefore first (75a18) that demonstrative science cannot bear on accidents that are not per se in the way that per se was explained above, namely, that a per se accident is one in whose definition the subject is mentioned, as “even” or “odd” is a per se accident of number. But “white” is not a per se accident of animal, because animal is not mentioned in its definition.

That there cannot be demonstration bearing on accidents that are not per se he proves in the following way: An accident which is not per se might happen not to be present (and this is the accident under discussion). Therefore, if a demonstration were to bear on an accident which is not per se, it would follow that the conclusion of the demonstration would not be necessary: which is contrary to what has been established.

That an accident which is not per se does not inhere necessarily can be obtained from the following: If an accident inheres necessarily and always in a subject, it must have its cause in the subject—in which case the accident cannot but inhere. Now this can occur in two ways: in one way, when it is caused from the principles of the species, and such an accident is called a per se attribute or property; in another way, when it is caused from the principles of the individual, and this is called an inseparable accident. In either case every accident which is caused from the principles of the subject must, if it is defined, be defined in such a way as to mention the subject in its definition: for a thing is defined in terms of its proper principles. Thus it is clear that an accident which necessarily inheres in its subject must be a per se accident. Therefore, those that are not per se do not inhere of necessity.

It seems, however, that Aristotle is using the circular demonstration, which he previously repudiated. For he had proved earlier that a demonstration is concerned with necessary things, because it is concerned with, things which are per se; but now he shows that demonstration is con-: 1, cerned with things which are per se, because it is concerned with things which are necessary. The answer is that above Aristotle proved that demonstration is concerned with necessary things not only because it is concerned with things which are per se but also because of the definition of scientific knowing—and this was a true way to demonstrate. But the proof that demonstration is concerned with necessary things because it is concerned with things which are per se, is not a demonstration but an indication directed to a person who already knows that demonstration is concerned with things which are per se.

Then (75a21) he raises a problem in regard to which he does two things. First, he states the problem, saying that someone may perhaps, object that if the conclusion which follows from contingent things or things which are per accidens is not necessary, why are inquiries made concerning contingent things or things which are per accidens, so that from such data one may reach a conclusion, since it is required of the’ syllogism that the conclusion follow of necessity. That inquiries touching contingent things or things which are per accidens do occur is evident from his next statement, namely, that it makes no difference whether one inquires into contingent things and then gives his conclusion. As if to say: A conclusion can be drawn from contingent things that have been investigated and verified, just as from necessary things. For each employs the same syllogistic form.

Secondly (75a24), he solves the difficulty and says that questions solved by contingent premises are not of such a nature that the conclusion is necessary absolutely, on the basis of the things investigated, i.e., in virtue of the contingent premises, but because it is necessary for the one who admits the premises to admit the conclusion and to admit truth in the conclusion, if the things premised are true. It is as though he were saying that although a conclusion which is necessary with absolute necessity does not follow from contingent premises, yet it follows with the necessity of consequence according to which the conclusion follows from premises.

Then (75a28) he shows that demonstration proceeds from things which are per se, using the following argument: Demonstration proceeds from necessary things and bears on necessary things—and this because it is scientific, i.e., makes one know scientifically. But things which are not per se are not necessary, for they are per accidens and, being such, are not necessary, as has been stated above. On the other hand, those things bear with necessity on a genus which are per se and belong to it according to what it is. It follows, therefore, that demonstration cannot be from anything or of anything but what is per se.

He further shows that even if the premises were always both necessary and true but not per se, one would not know the cause why [propter quid] of the conclusion. This is clear in syllogisms which prove through signs, for although the conclusion be per se, one does not know it per se nor propter quid. For example, if someone were to prove that every element

corruptible on the ground that it is seen to grow old. This would be a proof through a sign but neither per se nor propter quid, because to know propter quid one must know through the cause. Therefore, the middle must be the cause of that which is concluded in the demonstration. And this is obvious from the premises: for the middle must inhere in the third causatively, i.e., per se, and likewise the first in the middle. Here he calls the two extremes the first and the third.

Lecture 15
(7508-b20)
DEMONSTRATION DOES NOT SKIP FROM ONE GENUS TO AN ALIEN GENUS

a38. It follows that we cannot— a39. For there are three— b2. The axioms which are premises— b8. so that if the demonstration— b10. If this is not so,— b13. That is why it cannot— b17. Geometry again cannot

After showing that demonstration is from things that are per se, the Philosopher here concludes that demonstration is from principles which are proper and not from extraneous or common principles. His treatment falls into two parts. In the first he shows that demonstration proceeds from proper principles. In the second he establishes which principles are proper and which common (76a26) [L. 18]. The first is divided into two parts. In the first he shows that demonstration does not proceed from extraneous principles. In the second that it does not proceed from common principles (75b37) [L. 17]. The first is divided into two parts. In the first, from the preceding, he shows that demonstration is not from extraneous principles. In the second, also from the preceding, he shows that demonstrations do not bear on corruptible things but on eternal (75b2I) [L. 16]. Concerning the first he does three things. First, he states his proposition. Secondly, he proves it (75a39). Thirdly, he concludes to what he intended (75b13).

He says therefore first (75a38) that inasmuch as demonstration is from things that are per se, it is plain that demonstration does not consist in descending or skipping from one genus to another, as geometry, demonstrating from its own principles, does not descend to something in arithmetic.

Then (75a39) he proves this proposition and does three things. First, he lays down the things that are necessary for a demonstration, saying that “three things are necessary in demonstrations. One is that which is demonstrated,” namely, the conclusion which, as a matter of fact, contains within itself that which inheres in its genus per se. For through demonstration a proper attribute is concluded of its proper subject. Another is the dignities [maxims or axioms] from which demonstration proceeds. The third is the generic subject whose proper attributes and per se accidents demonstration reveals.

Secondly (75b3), he shows which of the three aforesaid can be common to various sciences and which cannot, saying that one of the three, namely, the dignities, from which demonstration proceeds, happens to be the same in diverse demonstrations and even in diverse sciences. But in those sciences whose respective generic subject is diverse, as in arithmetic which is concerned with numbers, and in geometry which is concerned with magnitudes, it does not occur that a demonstration which starts with the principles of one science, say, of arithmetic, descends to the subjects of another science, say to magnitudes, which pertain to geometry; unless perchancp the subject of one science should be contained under the sub., ject of the other, for example, if magnitudes should be contained under, numbers. (How this occurs, namely, how the subject of one science may be contained under the subject of another, will be discussed later). For magnitudes are not contained under numbers except perhaps in the sense that magnitudes are numbered. In any case, the subjects of diverse demonstrations or sciences are diverse. For an arithmetical demonstration always has its proper genus in respect to which it demonstrates, and so do the other sciences.

Thirdly (75b8), he proves his proposition and does two things. First he brings in the main intent after the manner of a conclusion on the ground that it can be obtained from the aforesaid, saying: Hence it is clear that it is necessary either that the genus with which the principles and conclusions deal be absolutely the same (in which case there is no descent or skipping from one genus to another); or if the demonstration is to descend from one genus to another, the two must be one thus, i.e., somehow. For otherwise it is impossible for a conclusion to be demonstrated from principles, since there is not the same genus either absolutely or in a qualified sense. However, it should be noted that a genus absolutely the same is being taken when on the part of the subject no determinate difference is admitted which is alien to the nature of that genus: for example, if someone using principles verified of triangle should proceed to demonstrate something about isosceles or about any other species of triangle. But a genus is one in a qualified sense when a difference alien to the nature of the genus is admitted of the subject, as “visual” is alien to the genus of line, and “sound” to the genus of number. Therefore, number absolutely, which is the generic subject of arithmetic, and sonant number, which is the generic subject of music, are not one genus absolutely speaking. The same goes for line absolutely, which geometry considers, and the visual line considered in optics. Hence it is clear that when matters pertaining absolutely to the line are applied to the visual line, a descent is being made to another genus—which is not the case when matters pertaining to triangle are applied to isosceles.

Secondly (75b10), he manifests the proposition in this way. In demonstration the middles and the extremes must belong to the same genus. Now the extremes are contained in the conclusion: for the major extreme in the conclusion is its predicate, and the minor its subject. But the middle is contained in the premises. It is required, therefore, that the principles and conclusion be taken with respect to the same genus. When we add to this the fact that diverse sciences are of necessity concerned with subjects generically diverse, it follows that from the principles of one science, something in another science not under it may not be concluded.

That the middles and extremes in a demonstration must be of one genus he proves thus: Suppose that a middle belongs to a genus other than that of the extremes which might be, for example, “triangle” and “have three angles equal to two right angles.” Now a proper attribute concluded of triangle is in it per se but is not in “brazen” per se; or, conversely, if the proper attribute is per se in “brazen,” say, “high-sounding” or the like, it is obviously in triangle per accidens.

This example makes it plain that if the subject of the conclusion and the middle are in entirely different genera, then the proper attribute is either not in the middle per se or else not in the subject per se. Consequently, it is in one of them per accidens. If it is in the middle per accidens, something will be in the premises per accidens. But if it is in the subject per accidens, something will be per accidens in the conclusion: and this on the part of a proper attribute. But either way something will be per accidens in the premises, inasmuch as the subject is subsumed under the middle, for example, triangle under brazen, or vice versa. However, it has been previously established that in a demonstration both the conclusion and the premises are per se and not per accidens. It is required, therefore, in demonstrations that the middle and the extremes be of the same genus.

Then (75b13) he infers two conclusions from the foregoing. The first conclusion is that no science demonstrates anything about the subject of another science, when this other science is more common than o entirely disparate from the first. Thus, geometry does not demonstrate that one and the same science deals with the both of two contraries: for contraries pertain to the common science, namely, to first philosophy or to dialectics. In like manner, geometry does not demonstrate that “two cubes are one cube,” i.e., that the product of one cube and another cube is a cube. “Cube” here means the number which results from multiply., ing a number by itself twice: for example, 8 is a cube, for it is the result 11of multiplying 2 by itself twice, for 2 times 2 times 2 are 8. Similarly,, 27 is a cube, whose root is 3, because 3 times 3 times 3 make 27. Now if 8 be multiplied by 27 the product is 216, which is a cube whose root is 6, because 6 times 6 times 6 make 216. Therefore, arithmetic, not, geometry has: the power to prove this. Similarly, that which pertains to one science cannot be proved by another science unless the one happens~, to be under the other, as optics is under geometry, and consonance or harmony, i.e., music, is under arithmetic.

The second conclusion (75b17) states that a science cannot prove just any random accident of its subject, but the accidents proper to its genus Thus, if something belongs to lines not as lines or not according to th proper principles of lines, geometry does not demonstrate it of lines: for example, that a straight line is the most beautiful of lines, or whether straight line is contrary or not to the curved. For these matters are outside the proper genus of line and belong to something more general. For beauty and contrary transcend the genus of line.

Lecture 16
(75b21-36)
DEMONSTRATION IS NOT OF PERISHABLE BUT OF ETERNAL MATTERS

b2l. It is also clear that— b24. The proof can only be— b30. The same is true of— b32. Demonstration and science

After concluding from the above that demonstration does not conclude from extraneous principles, the Philosopher intends to conclude something else from the above, namely, that demonstration is not of destructible things. Concerning this he does two things. First, he shows that demonstration is of eternal and not of destructible things. Secondly, he shows how it is of things that occur now and then (75b32). Concerning the first he does two things. First, he shows that demonstration is not of perishable but of eternal things. Secondly, he shows that the same is true of definition (75b30). In regard to the first he does two things. First, he proposes the intended conclusion. Secondly, he sets down the reason proving it (75b24).

Therefore first (75b21), he sets down two conclusions, one of which follows from the other. The first is that the conclusion of this demonstration which we are now discussing and which we can call demonstration in the full sense must be eternal: which, of course, follows from what has been stated so far, namely, that the propositions of a syllogism should be universal: which he signified by “said of all.” The second conclusion is that neither demonstration nor science is of destructible things, i.e., absolutely speaking, but only accidentally.

Then (75b24) he presents an argument to prove the proposed conclusions. It is this: It is not the character of a destructible non-eternal conclusion to contain what is universally so, but what is so for a time and in certain instances. For it has been established above that “said of all” contains two things, namely, that it is not such as to be so in one case and not in another, or so at one time and not at another. But in destructible things we find that at one time something is so and at another not so. Hence it is clear that “said of all” or “said universally” are not found in destructible things. But where a conclusion is not universal, at least one of the premises is not universal. Therefore, a destructible conclusion must have followed from premises, one of which is not universal. Accordingly, when we add to this the fact that demonstration absolutely must be from what is universal, it follows that a demonstration cannot have a destructible conclusion, but must have an eternal one.

Then (75b30) he shows that definition, too, is not of destructible but of eternal things. The reason for this is that demonstration, both as to its principles and conclusions, is of eternal and not of destructible things. But a definition is either a principle of a demonstration, a conclusion of a demonstration, or a demonstration with a different ordering of its terms. Therefore, a definition is not of destructible, but of eternal things.

For a better understanding of this passage it should be noted that it is possible to give different definitions of the same thing, depending on the different causes mentioned. But causes are arranged in a definite order to one another: for the reason of one is derived from another. Thus the reason of matter is derived from the form, for the matter must be such as the form requires. Again, the agent is the reason for the form: for since an agent produces something like unto itself, the mode of the form which results from the action must be according to the mode of the agent. Finally, it is from the end that the reason of the agent is derived, for every agent acts because of an end. Consequently, a definition which is formulated from the end is the reason and cause proving the other definitions which are formulated from the other causes.

Therefore, let us lay down two definitions, one of which is formulated from the material cause, for example, that a house is a shelter composed of stones, cement and wood; the other being formulated from the final cause, namely, that a house is a shelter protecting us from the rain and heat and cold. Now the first definition can be demonstrated from the second in the following way: Every shelter protecting us from rain, heat and cold should be composed of wood, cement and stones; but a house is such a thing: therefore...

Thus it is clear that the definition formulated from the end is the principle of the demonstration, whereas the one formulated from the matter is the conclusion of the demonstration. However, the two can be combined in the following way to form one definition: A house is a shelter composed of the aforesaid, protecting us from rain, cold and heat. But such a definition contains all the elements of a demonstration, namely, a middle and a conclusion. Accordingly, such a definition is a demonstration differing merely in the arrangement of its terms, because the only way it differs from the demonstration is that it is not arranged in a syllogistic mode and figure.

Here we might remark that because demonstration, as well as definition, is not of destructible but of eternal things, Plato was led to posit “Ideas.” For since sensible things are destructible, it appeared that there could be neither demonstration nor definition of them. As a consequence it seemed necessary to postulate certain indestructible substances concerning which demonstrations and definitions could be given. These eternal substances are what he calls “Forms” or “Ideas.”

However, Aristotle opposed this opinion above, when he said that demonstration is not of destructible things except per accidens. For although those sensible things are destructible as individuals, nevertheless in the universal they have a certain everlasting status. Therefore, since demonstration beais on those sensible things universally and not individually, it follows that demonstration is not of destructible things except per accidens, but of eternal things per se.

Then (75b32) he shows how there can be demonstration of things that occur now and then, saying “that there are science and demonstrations of things that occur frequently, as the eclipse of the moon,” which is not always. For the moon is not always being eclipsed, but only now and then. Now things that occur frequently, so far as they are such, i.e., so far as demonstrations are given concerning them, are always; but they are particular, so far as they are not always. But demonstration cannot be of particulars, as we have shown, but only of universals. Hence it is clear that these things, insofar as there is demonstration of them, are always. And as in the case of the eclipse of the moon, so in all kindred matters.

However, there are certain differences to be noted among them. For some are not always with respect to time, but they are always in respect to their cause, because it never fails that under given conditions the effect follows, as in the eclipse of the moon. For the moon never fails to be eclipsed when the earth is diametrically interposed between sun and moon. But others happen not to be always even in respect to their causes, i.e., in those cases where the causes can be impeded. For it is not always that from a human seed a man with two hands is generated, but now and then a failure occurs, owing to a defect in the efficient cause or material cause. However, in both cases the demonstration must be so set up that a universal conclusion may be inferred from universal propositions by ruling out whatever can be an exception either on the part of time alone, or also of some cause.

Lecture 17
(75b37-76a25)
DEMONSTRATION DOES NOT PROCEED FROM COMMON PRINCIPLES, BUT FROM PRINCIPLES PROPER TO THE THING DEMONSTRATED

b37. It is clear that if— b40. Such proofs are like— a8. The only exceptions— a17. It is no less evident— a19. This is so because— a23. But, as things are,

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. Secondly, he draws a conclusion from what he has said (76a17).

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, i.e., 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.

Secondly (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, i.e., 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. Bu 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, i.e., 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, i.e., according to what comes per se from its principles, as “having three angles equal to two right angles” belongs per se to triangle, i.e., 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, i.e., proximate according to genus, to the conclusion.

Thirdly (760), 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, (i.e., 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 [i.e., 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 [i.e., 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, i.e., 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 (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, i.e., 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.

Secondly (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.

Thirdly (7643), he returns to the principal conclusion and 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, i.e., music.

Lecture 18
(7646-b22)
DIFFERENCE BETWEEN PRINCIPLES AND NON-PRINCIPLES, COMMON AND PROPER PRINCIPLES

a26. it is hard to be sure— a3l. I call the basic truths— a32. As regards both these— a37. Of the basic truths— a40. Peculiar truths are,— a42. Only so much of these— b2. Also peculiar b16. Yet some sciences

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.

Secondly (76a3l), he determines concerning proper and common principles. And in regard to this he does two things. First he determines concerning common and proper principles. Secondly, he shows how the demonstrative sciences are related to such principles (77a10) [L. 20]. Concerning the first he does two things. First, he distinguishes principles from non-principles. Secondly, he distinguishes the principles from each other(76a37).

In regard to the first he does two things. First (76a3l), 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,” (i.e., whose truth), “cannot be proved,” to distinguish from the false which are not demonstrated in any science.

Secondly (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, i.e., with respect both to the first, namely, the principles, and to the things that proceed from them, i.e., things assumed from the principles, one must accept the supposition signify, because the quod quid est [that which the definition of what they signifies] of a thing pertains properly to the science concerned with substance, i.e., 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 (76a37) he distinguishes principles from one another. First, the proper from the common. Secondly, the common, one from the other (76b23) [L. 19]. The first is divided into two parts. First, he divides proper and common principles. Secondly, he settles something which might be doubtful (76b16).

In regard to the first he does three things. 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.” Then, 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,” i.e., 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.

Secondly (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.

Thirdly (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.

Secondly (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,” i.e., points, and lines. For they suppose these things to be and to be this, i.e., 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” is and what “odd” is 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 6f 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 (axioms) dignities from which, as from basic truths, it demonstrates; the third are the proper attributes concerning which each science supposes what their names signify.

Then (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, i.e., 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 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.

Lecture 19
(76b23-77a9)
HOW COMMON PRINCIPLES DIFFER FROM ONE ANOTHER

b23. That which expresses— b27. That which is capable— M. The definitions— b39. Nor are the geometer’s— a3. A further distinction— a5. So demonstration does not

After dividing common principles from proper principles, Aristotle now distinguishes among the common principles.

His treatment f Is 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 (76b35). In the third he excludes an error (77a5). Concerning the first he does two things. First, he distinguishes the common conceptions in the mind from postulates or suppositions. Secondly, he distinguishes among the latter (76b27).

With respect to the first (76b23) 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, 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, i.e., 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 (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 thein without demonstrating, although they are demonstrable.

Then (76b35) he distinguishes definitions from suppositions with two reasons, the second of which begins at (77a3). In regard to the first he does two things.

First, he presents this reason: Every postulate or supposition declares something to be or not to be; but terms, i.e., 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 Js made from them in regard to what they are, i.e., in virtue of the premises.

Secondly (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 (77a3) he presents the second reason which is this: Every supposition or postulate is in whole or in part, i.e., 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 (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,” i.e., “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.

Lecture 20
(77a10-35)
RELATION BETWEEN DEMONSTRATIVE SCIENCES AND COMMON PRINCIPLES

a10. The law that it is impossible— a22. The law that every predicate— a26. In virtue of the common— a29. and in communion with them

After determining about proper and common principles, the Philosopher now shows how the demonstrative sciences behave in regard to the common and to the proper principles. His treatment falls into two parts. In the first he shows how the demonstrative sciences are related to common principles. In the second he shows how they are related to proper principles (77a36) [L. 21]. Concerning the first he does two things. First, he shows how the demonstrative sciences are related to the first of the common principles. Secondly, how they are related generally to all the common principles (77a29). Concerning the first he does two things. First, he shows how the demonstrative sciences behave relative to the principle that one should not affirm and deny the same thing. Secondly, how they behave relative to the principle that there is either true affirmation or true negation of each thing, for, as is proved in Metaphysics IV, these two principles are the first of all principles (77a22).

He says therefore first (77a10) that no demonstration makes use of the principle that one does not affirm and deny at one and the same time. For if a demonstration were to use it to show some conclusion, it would have to use it in such a way as to assert that the first, i.e., the major, extreme is affirmed of the middle and not denied; because if it were to admit affirmation and negation on the part of the middle, it would make no difference whether it was this way or that. And the same reasoning holds for the third, i.e., minor, extreme in relation to the middle. For example, take “animal” as the first, “man” as the middle’ and “Callias” as the third. If someone wished to use this principle in a demonstration, he would have to argue in the following way:

For when we say that every man is an animal, it matters not whether it is also true that a non-man is an animal, or whether it is not true. Similarly, in the conclusion it matters not, Callias being an animal, whether nonCallias is an animal or not.

Now the reason for this is that the first is not limited to being said only *of the middle but can also be said of something else diverse from the middle and described by a negation of the riliddle (since the first is sometimes said of things other than the middle, as “animal” is said of things other than man; for it is said of horse, which is not man). Hence if the middle is taken the same and not the same, i.e., if an affirmative and a negative middle be taken, as when I say, “man” and “non-man” are animals, it contributes nothing to the conclusion. However, if the affirmation and negation are taken on the part of the major extreme, it does make a difference both as to the conclusion and as to the truth of the premises. For if man were not an animal, it would not be true that man is an animal, nor would it follow that Callias is an animal. Yet nothing more is verified by stating that man is an animal and is not a non-animal than by merely stating that man is an animal, for the same thing is conveyed by each. And thus it is clear that demonstrations do not use the principle that affirmation and negation are not simultaneously true, either on the part of the predicate or on the part of the subject.

Then (77a22) he shows how demonstrative sciences use the principle that “of anything there is either true affirmation or negation.” And he says that this principle is utilized in a demonstration leading to the impossible. For in this demonstration something is proved to be true by the fact that its opposite is false. (This of course would never happen, if it were possible for the two opposites to be false).

Nevertheless such a demonstration does not always employ this principle, for sometimes that opposite which is shown to be false is not a negation but an immediate contrary. For example, if a number were shown to be even on the ground that its opposite, namely, “it is odd,” is false, and this were done by leading to the impossible. Neither does it use this principle universally, i.e., in its universality, namely, under the terms “being” and “non-being,” but only so far as is sufficient for a genus or so far as it is narrowed to a generic subject. And I mean the first genus involved in the demonstration. For example, in leading to the impossible in geometry the terms would be “straight” and “non-straight” in the case where it is shown that some line is straight on the ground that it is false to state that it is not straight.

Then (77a26) he shows how the sciences as a community function relative to all common principles. In regard to this he does two things. First, he says that all the sciences share alike in the common principles in the sense that they all use them as items from which they demonstrate-which is to use them as principles. But they do not use them as things about which they demonstrate something, i.e., as subjects, or as things which they demonstrate, i.e., as conclusions.

Secondly (77a29), he shows that certain sciences employ the common principles in a manner other than has been described. For dialectics is concerned with common things, and a certain other science is concerned with common things, namely, first philosophy, whose subject is being and which considers the things which follow upon being as the proper attributes of being. Yet it should be noted that dialectics is concerned with common things under an aspect different from logic and first philosophy. For first philosophy is concerned with common things because its consideration is focused on those common things, namely, on being and on the parts and attributes of being. But because reason occupies itself with all things that are, and logic studies the operations of reason, logic will also be concerned with matters common to all things, i.e., with reason’s intentionalities which bear on all things, but not in such a way that logic has these common things as its subject-for logic considers as its subject the syllogism, enunciation, predication, and things of that type.

But although the part of logic which is demonstrative is engaged in teaching about common intentionalities, the use of a demonstrative science does not consist in proceeding from common intentionalities to show anything about the things which are the subjects of the other sciences. But dialectics does this, for it goes from common intentions and argues to things that pertain to other sciences, whether they be proper or common things, but mainly common things. Thus, it argues that hatred is in the concupiscible appetite, because love is, on the ground that contraries are concerned with a same thing. Consequently, dialectics is concerned with common things not only because it treats concerning the common intentionalities of reason, which is common to all logic, but also because it argues about common characteristics of things. But any science that argues about the common characteristics of things must argue about the common principles, because the truth of the common principles is made manifest from the knowledge of common terms, as “being” and “non-being,” “whole” and “part” and the like.

It is significant that he says, “and any science which might attempt,” because first philosophy does not demonstrate the common principles, since they are absolutely indemonstrable, although some have unwittingly attempted to demonstrate them, as is stated in Metaphysics IV. Or else, because even though they cannot, strictly speaking, be demonstrated, the first philosopher attempts to uphold them in a way that is possible, namely, by contradicting those who deny them, appealing to things that must be conceded by them, though not to things which are more known.

It should also be noted that the first philosopher demonstrates them not only in this way, but also shows something about them as about subjects: for example, that “it is impossible for the mind to think their opposites,” as is clear from Metaphysics IV. Therefore’ although both the first philosopher and the dialectician debate about these first principles, yet one does so in one way and the other in another way. For the dialectician neither proceeds from demonstrative principles nor takes only one side of a contradiction but is open to both. For each side might happen to be probable or be upheld by probable statements, which the dialectician utilizes: and that is why he asks [his questions in terms of two alternatives]. But the demonstrator does not ask [in that way], because he is not open to opposites. And this is the difference between the two, as was laid down in the treatment of the syllogism, namely, in the beginning of the Prior Analytics. Therefore, first philosophy in treating common principles proceeds after the manner of a demonstration and not after the manner of a dialectical disputation.

Lecture 21
(77a36-77b15)
THE QUESTIONS, RESPONSES AND DISPUTATIONS PECULIAR TO EACH SCIENCE

a36. If a syllogistic question— a42. only those questions will— b2. Of these questions the— b6. There is a limit, then— b8. if, then, in controversy

After showing how demonstrative sciences function in relation to common principles, the Philosopher now shows how they employ proper principles. And his treatment falls into two parts. In the first he shows that in each science there are questions, responses and disputations peculiar to each. In the second he shows how in each science there are deceptions, peculiar to each (77b16) [L. 22]. Concerning the first he does two things. First, he, shows that in each science there are its own questions. Secondly, that each science employs the responses and disputations proper to it (77b6). Concerning the first he does two things. First, he shows that in each science there are questions peculiar to each. Secondly, what these are (77a42).

He shows the first (77a36) in the following way: A syllogistic question and a proposition which takes one definite side of a contradiction are the same as to content, but they have not the same function. For the answer which one gives to the question is made to serve as a proposition in some syllogism. But in each of the sciences, the propositions from which a syllogism is formed, are peculiar to it, for it has been shown that every science proceeds from things proper to it. Therefore, in each science there are questions peculiar to it. Hence not any random question is pertinent to geometry or to medicine or to some one of the other sciences.

However, it should be noted that questioning occurs one way in demonstrative sciences and another way in dialectics. For in dialectics not only the conclusion but also the premises are open to question; but in demonstrative sciences the demonstrator takes premises as per se known or proved by such principles. Hence, he asks only about the conclusion. And when he has demonstrated it, he uses it as a proposition to demonstrate some other conclusion.

Then (77a42) he explains which questions are peculiar to each science. First, insofar as they are taken as propositions from which the demonstrator proceeds. Secondly, when taken as conclusions (77b2).

He says therefore first (77a42) that geometric questions are ones from which something is demonstrated pertaining to matters of geometry or pertaining to matters demonstrated from the principles of geometry—for example, the ones from which something is demonstrated in optical science, i.e., perspective, which proceeds from the principles of geometry. And what is said of geometry applies to other sciences, namely, that a proposition or question is peculiar to a science, if it is one from which a demonstration in that science or in a subalternate science proceeds.

Then (77b2) he analyzes geometric questions insofar as they are conclusions, saying that solutions to geometric questions must be proved by demonstrating their truth from geometric principles and from conclusions already demonstrated by those principles. (For sometimes the proofs of certain geometric demonstrations are not drawn from the first principles of geometry, but from matters concluded from those first principles). Hence when it is a matter of questions which are always conclusions in demonstrative sciences, the proof can be obtained in the science itself, but the proof of the principles cannot be drawn from geometry qua geometry. The same also applies to other sciences. For no science proves its own principles, as we have explained above. And he says, “from geometry qua geometry,” because it may happen that a science proves its own principles, insofar as that science assumes the principles of another science, as a geometer proves his own principles insofar as he assumes the role of first philosopher, i.e., of metaphysician.

Then (77b6) he shows that each has its own responses and disputations. First, that each has its own responses. Secondly, its own disputations (77b8).

He says therefore (77b6) that from the foregoing it is clear that each scientific knower does not ask just any question whatever. Hence it is also clear that he will not answer just any random question but only those which conform to his own science on the ground that a question and its answer pertain to the same science.

 Then (77b8) because disputations are concerned with a question and its answer, he shows that there are disputations - peculiar to each science, saying that if geometer disputes with geometer precisely as geometer, i.e., in matters pertaining to geometry, then obviously the disputation goes well so long as the disputation not only concerns a point of geometry but proceeds from the principles of geometry. But it does not go well, if the disputation in geometry does not proceed along these lines. For if someone disputes with a geometer in matters alien to geometry, he does not argue, i.e., does not convince, except accidentally: for example, if the dispute concerns music, and the geometer accidentally happens to be a musician. Hence it is clear that one should not dispute in geometry about matters not geometric, because it will not be possible to judge by the principles of that science whether the dispute went favorably or unfavorably: and the same holds for other sciences.

Lecture 22
(77b16-78a21)
EACH SCIENCE HAS ITS OWN DECEPTIONS AND AREAS OF IGNORANCE

b16. Since there are ‘geometrical’— b17. Further, in each special— b19. Again, is the erroneous— b2l. Or, perhaps, the erroneous— b23. whereas the notion that— b27. In mathematics the formal— b34. If a proof has an inductive— b40. On the other hand— a5. Sometimes, no doubt,— a8. If false premises— a10. Reciprocation of premises— a13. A science expands

After showing that each science has its own questions, responses and disputations, the Philosopher shows that each science has its own deceptions and errors. And his treatment is divided into two parts. In the first he raises certain questions. In the second he solves them (77b21).

Accordingly, he poses three questions, the first of which (77b16) is this: Since there are geometric questions, as we have shown, are there not also non-geometric ones? And what is asked of geometry can be asked of every other science.

Then (77b17) he poses the second question, namely: May questions which arise from ignorance bearing on some particular science be called geometric; and likewise for questions proper to any other science? (Questions arising from ignorance bearing on some science are those which ask about matters contrary to the truths of that science).

Then (77b19) he poses the third question, namely: In each science it is possible to be deceived by a syllogism which he calls “according to ignorance.” But deception through a syllogism can occur in two ways: in one way when it fails as to form, not observing the correct form and mode of a syllogism. In another way, when it fails in matter, proceeding from the false. Now there is a difference between these two ways, because one that fails in matter is still a syllogism, since everything is observed that pertains to the form of a syllogism. But one that fails in form is not even a syllogism, but a paralogism, i.e., an apparent syllogism. In dialectics, deception can occur in both these ways. Hence in Topics I Aristotle speaks of the contentious, which is a syllogism, and of the one defective in form which is not a syllogism but an apparent one. Hence the question is this: Whether or not a syllogism of ignorance which is used in the demonstrative sciences is a syllogism based on matters opposed to the science, i.e., one that proceeds from false premises, or a paralogism, namely, one that fails in form, which is not a syllogism, but an apparent one?

Then (77b21) he solves these questions. First, he solves the first Secondly, the second (77b23). Thirdly, the third (77b27).

He says therefore first (77b21) that a completely non-geometric question is one which is formed entirely from another art, say, music. For example, if one asks in geometry whether a tone could be divided into two equal semi-tones, such a question would be entirely non-geometric, because it concerns matters which do not pertain at all to geometry.

Then (77b23) he solves the second question, saying that a question about geometry, i.e., about matters pertinent to geometry, when a person is asked about something which is against the truth of geometry (for example, if the questions concerned parallels meeting, i.e., equidistant lines coming together), would be geometric in one sense and non-geometric in another. For just as arrythmon, i.e., without rhythm or sound, can be taken in two senses: in one sense for that which has no sound at all, as wool, and in another sense for that which does not give a good sound, as a poorly-sounding bell, so a question is called non-geometric in two ways. In one way, because it is completely alien to geometry, as a question about music. In another way, because it mistakenly holds something in the field of geometry, namely, because it holds something contrary to geometric truth. Therefore, the question about the convergence of parallel lines is non-geometric not in the first way, since it touches on a point of geometry, but in the second way, because it is mistaken about some point of geometry. Such ignorance, namely, which consists in wrongly using the principles of geometry is contrary to the truth of geometry.

Then (77b27) he solves the third question. And he does two things. First, he shows that in demonstrative sciences there is no paralogism in language. Secondly, nor apart from language (77b34).

Now although paralogism. in language occurs in any of six sophistical ways, he takes one of them, namely, the paralogism which proceeds by way of equivocation, and shows that such,a paralogism cannot occur in demonstrative sciences, being easier to detect. He says, therefore, that “formal paralogism,” i.e., a syllogism defective in form, as in dialectics, “does not occur in the disciplines.” For in a demonstrative syllogism the middle must always be the same in two ways, i.e., the same middle must be compared to the two extremes: for the major extreme is predicated universally of the middle, and the middle is predicated universally of the minor extreme, even though when it is predicated, we do not say “every,” i.e., the sign of universality is not applied to the predicate.

But in the fallacy of equivocation the middle is the same according to vocal sound but not according to reality. Consequently, it escapes notice when it is proposed orally; but if it is demonstrated to the senses, the deception cannot succeed. For example, the name “circle” is said equivocally of a figure and a poem. Therefore, in reasons, i.e., in argumentations, the ambiguity may go unnoticed, i.e., deception is possible, as for example, if one were to say: “Every circle is a figure; Homer’s poem is a circle: therefore, Homer’s poem is a figure.” But if a circle is drawn for someone to look at, there cannot be deception. For it will be obvious that songs are not circles.

Now just as in this case the deception is prevented by presenting the middle to the senses, so in demonstrative syllogisms, deception is prevented by the fact that the middle is shown to the intellect. For when something is defined, it is as plain to the intellect as something sensibly drawn is plain to the sight. Hence he says that in demonstrative syllogisms “these,” i.e., things defined, “are seen by the intellect.” But in demonstrations one always proceeds from definitions. Hence, deception through equivocation has no place there, much less through any of the other fallacies of language.

Then (77b34) he shows that in demonstrative syllogisms paralogism according to fallacy outside of language cannot occur. And because a paralogism of this kind is frequently challenged by citing an objection, through which the defect in the form of syllogizing is shown: therefore: First, he shows how an objection should be presented in demonstrative matters. Secondly, he shows that in demonstrative matters there cannot be paralogism according to fallacy outside the language (77b40).

He says therefore first (77b34) that “one should not bring an objection against it,” i.e., against a paralogism. by citing an inductive, i.e., particular, proposition. (For an induction proceeds from particulars as a syllogism proceeds from universals). The reason for this is that in demonstrative matters no proposition is admitted unless it is verified in the greater number of cases, for if it is not verified in the greater number it will not be in all. But a demonstrative syllogism must proceed from universals. Therefore, it is obvious that in demonstrative matters an objection must be universal, because the propositions and the objections must be the same. For in dialectical and in demonstrative matters that which is taken as an objection is later used as a proposition to syllogize against the one who proposed.

Then (77b40) he shows that in demonstrative matters deception through paralogisms outside of language does not occur. And just as above he showed that there was no paralogism in language in demonstrative matters by showing it for one, namely, for the paralogism which employs the fallacy of equivocation, so now he shows that in demonstrative matters there is not paralogism outside of language by showing it of the one which relies on fallacy of consequent. For it is obvious that paralogism. in demonstrative matters cannot occur according to the other fallacies outside of language: not the fallacy according to accident, because demonstration proceeds from things that are per se; nor according to qualified and absolute, because the statements used in demonstrations are taken universally and always, and without qualification.

Therefore, he does two things in regard to his thesis. First, he shows how paralogism according to fallacy of consequent works. Secondly, that deception does not take place in this manner in demonstrative matters (78a5).

He says therefore first (77b40) that “illogical arguments do sometimes occur,” i.e., do not observe syllogistic form, “because they admit both inherences,” i.e., they accept a middle predicated affirmatively of each extreme, which is the same as syllogizing in the second figure but employing two affirmative propositions, thus committing the fallacy of consequent. This is what the philosopher Caeneus did in order to show that “fire is in multiple proportion,” i.e., that fire is generated in greater quantity than was the body from which it is generated, on the ground that fire, being the most rarified of bodies, is generated from other bodies through rarefaction. Hence it is required that the matter of the previous body be spread out under larger dimensions when assuming the form of fire. To prove this he used the following syllogism: “Whatever is generated in multiplied proportion is generated quickly; but fire is generated quickly: therefore, fire is generated in multiplied proportion.”

Then (78a5) he shows that deception arising from this form of reasoning does not occur in demonstrative sciences. In regard to this he does two things. First, he makes it clear that this form of syllogizing does not always end in deception, saying that according to this form of arguing there are cases when one cannot syllogize from the premises, namely, when the terms are not convertible-for it does not follow, if every man is an animal, that every animal is a man. But now and then there are cases when one can syllogize, namely, when the terms are convertible. For just as it follows that if a thing is a man, it is a rational mortal animal, so conversely, if a thing is a rational mortal animal, it is a man. However, it does not seem to follow syllogistically, because the due form of a syllogism is not observed.

Secondly (78a7), he shows that it is possible to syllogize in the abovementioned manner without deception. And he shows this in three ways, the first of which is this: According to the above-mentioned manner of syllogizing, deception occurs because the consequence which was assumed convertible is not converted. But in this situation deception would not occur, if to the extent that the conclusion is true, so are the premises true: for there will then be no deception in converting. For if it is stated of Socrates that he is a man, therefore he is an animal; no deception or falsity follows: but it does if it is argued conversely that he is an animal, therefore he is a man.

But if the premise is true and the conclusion false, then deception occurs in converting. For example, if I were to say: “If an ass is a man, it is an animal; therefore, if it is an animal, it is a man.” Consequently, if it were impossible to deduce the true from the false, and it were always necessary to deduce the true from the true, then it would be easy to analyze a conclusion into its principles without deception, because there would be no falsity if either of the premises were to be inferred from the conclusion. On this supposition the conclusion and premises would be converted of necessity as to truth. For just as the conclusion is true if the premise is true, so vice versa. For suppose that A is, and granting this, suppose that the same things follow as are known by me to be true, say B. Hence since both are true, I can then also infer A from B. And so, one reason why deception through fallacy of consequent does not occur in demonstrative sciences is that it is impossible in demonstrative sciences for the true to be syllogized from the false, as we have explained above.

Then (78a10) he gives the second reason. For in cases of convertible terms, deception through fallacy of consequent does not occur, because in those cases the consequent is converted. Now the things which are used in mathematical, i.e., in demonstrative sciences, are for the most part convertible, because these sciences do not admit as a middle anything predicated per accidens, but only definitions which are principles of demonstration, as we have explained. And this is their point of difference from the statements in dialogues, i.e., in dialectical syllogisms, in which accidens are frequently admitted.

Then (78a13) he gives the third reason and it is this: In demonstrative sciences there are determinate principles from which one proceeds to the conclusions. Hence it is possible to return from the conclusions to the principles as from something determinate to something determinate. That demonstrations do proceed from determinate principles he shows by the fact that “demonstrations are not increased by middles,” i.e., in demonstrations one does not use a series of different middle terms to demonstrate one conclusion. (This, of course, is to be understood of demonstrations propter quid, of which he is speaking). For of one effect there can be but one sole cause why it is.

But although demonstrations are not increased by using several middles of demonstration, nevertheless they are increased in two ways: in one way, “by assuming one after another,” i.e., subsuming one middle under another, say B under A, C under B, D under C, and so on. For example, when “having three angles equal to two right angles” is proved of triangle on the ground that it is a figure having an external angle equal to the two opposite interior angles, and is then proved of isosceles on the ground that it is a triangle. Another way they are increased is laterally, as when A is proved of C and of E. For example, “Every quantified number is either finite or infinite.” Let A be this predicate, i.e., “to be finite or infinite.” But an odd number is a quantified number. Let B represent “quantified number,” and C, “odd number.” It follows therefore, that A (to be finite or infinite) is predicated of C (odd number), i.e., that an odd number is either a finite or an infinite number. And he says that it is possible along the same lines to conclude concerning even number, and this through the same middle.

The passage (78aI3), which begins, “A science expands,” can be introduced in another way. Since he had just said that in demonstrative matters, definitions are used for middles, and since there is one sole definition of one thing, it follows that it is not through middles that, demonstrations are increased.

Lecture 23
(7842-b13)
HOW DEMONSTRATIONS “QUIA” AND “PROPTER QUID” DIFFER IN A SAME SCIENCE
DEMONSTRATION “QUIA” THROUGH AN EFFECT

a22. Knowledge of the fact— a30. Thus you might prove— a39. The major and the middle— b3. Another example is the inference— b10. Again in cases where the cause

After determining about demonstration propter quid, the Philosopher here shows the difference between demonstration quia and demonstration propter quid. And he does two things about this. First, he shows how they differ in the same science. Secondly, in diverse sciences (78b33) [L. 25]. Concerning the first he does two things. First, he states the twofold difference between the two in the same science. Secondly, he clarifies this with examples (78a30).

He says therefore first (78a22) that, as said above, demonstration is a syllogism causing scientific knowledge and proceeds from the causes both first and immediate of a thing. Now this is to be understood as referring to demonstration propter quid. But there is a difference between knowing that a thing is so and why it is so. Therefore, since demonstration is a syllogism causing scientific knowledge, as has been said, it is necessary that a demonstration quia which makes one know that a thing is so should differ from the demonstration propter quid which makes one know why. Consequently, this difference must be considered first in the same science and later in sciences that are diverse.

In one and the same science each of the above is said to differ in regard to the two things required for demonstration in the strict sense—which causes knowledge of the why—namely, that it be from causes and from immediate causes. Hence one way that scientific knowledge quia differn from propter quid is that it is the former if the syllogism is not through, immediate principles but through mediate ones. For in that case the first cause will not be employed, whereas science propter quid is according to the first cause; consequently, the former will not be science propter quid.

It differs in another way, because it is science quia when the syllogism, although not through middles, i.e., mediate, but through immediate things, is not through the cause but through “convertence,” i.e., through effects convertible and immediate. Hence a demonstration of this kind is through the better known, namely, to us; otherwise it would not effect scientific knowledge. For we do not reach a knowledge of the unknown except through something better known. However in the case of two things equally predicable, i.e., convertible, one of which is the cause and the other the effect, there is nothing to preclude that now and then the better known will not be the cause but the effect. For sometimes the effect is better known than the cause both in respect to us and according to sense-perception, although absolutely and according to nature the cause is the better known. Consequently, through an effect better known than the cause there can be demonstration which does not engender propter quid knowledge but only quia.

Then (78a30) he clarifies these differences by examples. And this is divided into two parts. In the first he develops an example of the demonstration quia which is through an effect. In the second of demonstration quia which is through a mediate cause (78b13) [L. 24]. The first is divided into two parts. In the first he gives an example of a syllogism which is through a convertible effect. In the second of a syllogism through a nonconvertible effect (78b10). The first is divided into two parts according to the two examples he gives, the second of which begins at (78b3). Concerning the first he does two things. First, he gives an example of demonstration quia which is through an effect. Secondly, he states when it can be converted into a demonstration propter quid (78a39).

He says therefore first (78a30) that demonstration quia is through an effect if one concludes for example that the planets are near because they do not twinkle. For non-twinkling is not the cause why the planets are near, but vice versa: for the planets do not twinkle because they are near. For the fixed stars twinkle because in gazing at them the sight is beclouded on account of the distance. Therefore, the syllogism might be formed in the following way: “Whatever does not twinkle is near; but the planets do not twinkle: therefore, they are near.” Here we let C be the planets, i.e., let “planets” be the minor extreme, and let B consist in not twinkling, and A “to be near” be the major extreme. Then the proposition, “Every C is B,” is true, namely, the planets do not twinkle. Also it is true that “Every B is A,” i.e., every star that does not twinkle is near. Rowever, the truth of such a proposition must be obtained through induction or through sense perception, because the effect here is better known than the cause. And so, the conclusion, “Every C is A,” follows. In this way, then, it has been demonstrated that the planets, i.e., the wandering stars, are near. Consequently, this syllogism is not propter quid but quia. For it is not because they do not twinkle that planets are iiear but rather, because they are near, they do not twinkle.

Then (78a39) he teaches how a demonstration quia is changed to a demonstration propter quid. And he says that “it is possible to demonstrate the one through the other,” i.e., to demonstrate that they do not twinkle, because they are near. Then the demonstration will be propter quid. Thus let C be the wanderers, i.e., let “wandering star” be the minor extreme; let B consist in being near, i.e., let “to be near,” which was the major extreme above, be the middle term; and let A consist in not twinkling, i.e., let “not to twinkle,” which above was the middle term, now be the major term. Therefore, B is in C, i.e., “Every planet is near”; and A is in B, i.e., “Any planet which is near does not twinkle.” Wherefore, it follows that A is in C, i.e., “A planet does not twinkle.” In this way we have a syllogism propter quid, since it rests on the first and immediate cause.

Then (78b3) he presents another example of this, saying that “in this way” (i.e., by means of a demonstration quia), “one demonstrates that the moon is round because of its phases,” according to which it waxes and wanes every month. They argue thus: “Everything which waxes thus circularly is circular; but the moon waxes thus: therefore, it is circular.” “Put in this form it is a syllogism demonstrating quia. But if the middle be interchanged,” i.e., if “circular” be made the middle term and “waxes” the major term, “it becomes a demonstration propter quid.” For the moon is not circular because it waxes in that way, but because it is circular it undergoes such phases. Therefore, let C be “the moon,” i.e., the minor extreme; let “waxing” be A, i.e., the major extreme, and let “circular” be B, the middle term. This will be the situation in the syllogism propter quid.

Then (78b10) he shows that a demonstration through a non-convertible effect is quid. He says, therefore, that even in those syllogisms in which the middles are not converted with the extremes, and in which an effect rather than a cause is taken as the middle better known in reference to us, even in those cases the demonstration is quia and not propter quid. If the middle be such that it can be converted with the major extreme and it exceeds the minor, then obviously it is a fitting syllogism; for example, if one proves that Venus is near because it does not twinkle. On the other hand, if the minor exceeded the middle, it would not be a fitting syllogism: for one cannot conclude universally of stars that they are near because they do not twinkle. Quite the contrary is true in comparison to the major term: for if the middle is in less things than is the major term, the syllogism is fitting, as when it is proved that someone has a sensible soul on the ground that he is capable of progressive local motion. But if it is in more, than the syllogism is not fitting, for from an effect which can proceed from several causes, one of them cannot be concluded. Thus, one cannot conclude from a rapid pulse that he has a fever.

Lecture 24
(78b13-34)
HOW THERE IS DEMONSTRATION “QUIA” THROUGH THINGS NOT IMMEDIATELY CONNECTED

b13. This also occurs when— b14. For example, the question— b24. A syllogism with this kind— b27. Such causes are like— b32. Thus, then, do the syllogism

After clarifying with examples how there is demonstration quia through effect, the Philosopher here shows how there is demonstration quia through things not immediately connected. First, he amplifies his proposition. Secondly, he shows how the middles relate themselves to the conclusions in this type of demonstration (78b27). Concerning the first he does three things. First, he states his intention. Secondly, he clarifies it with examples (78b14). Thirdly, he puts it in syllogistic form (78b24).

He says therefore first (78b13) that there is demonstration quia and not propter quid not only in those matters that are proved through an effect but also “in matters in which the middle is set outside.” Now a middle is said to be set outside when it is diverse from the major term, as in negative syllogisms; or a middle is said to be set outside when it is outside the genus as being something more common and not convertible with the major term. That something cannot be demonstrated propter quid through such a middle he proves on the ground that demonstration propter quid is through a cause. But the middle in question is not, properly speaking, a cause.

Then (78b14) he clarifies what he had said with an example, saying that if someone attempted to prove that a wall does not breathe because it is not an animal, he would not be demonstrating propter quid or giving the cause. Because if the fact of not being an animal were the cause of not breathing, it would be required that being an animal would be the cause of breathing-which is false. For there are many animals which do not breathe. For it is required, if a negation is the cause of a negation, that the affirmation be the cause of the affirmation, as the fact of not being warm and cold in due measure is the cause of someone’s not getting well, and the fact of being warm and cold in due measure is the cause of someone’s getting well. The converse is also true, namely, that if an affirmation is the cause of an affirmation, the negation is the cause of the negation. But this does not occur in the case at hand, because the affirmation is not the cause of an affirmation, since not every animal breathes.

Then (78b24) he arranges the aforesaid example in syllogistic form and says that it should be arranged in the second figure. And this is so because in the first figure there cannot be a negative conclusion such that the major would be affirmative, as our example requires. For “to breathe,” which is the major extreme, must be joined with “animal,” which is the middle term, according to affirmation. But “wall,” which is the minor extreme, must be joined with “animal,” which is the middle, according to negation. Consequently, the major will be affirmative and the minor negative. But such a thing never occurs in the first figure, but only in the second.

Let “animal,” then, be A, i.e., the middle term, “breathe” be B, i.e., the major extreme, and “wall” be C, i.e., the minor extreme. it therefore follows that A is in every B, because everything that breathes is an animal; but A is in no C, because no wall is an animal. Hence it follows that B, too, is in no C, i.e., that no wall breathes. Furthermore, if the most proximate middle were used, it would be demonstration propter quid: for example, if it were shown that a wall does not breathe because it does not have lungs. For whatever has lungs breathes, and conversely.

Then (78b27) he shows how these middles are related to the conclusion, saying that such far-away causes are compared to what they explain as being too remote, because they go beyond the pale of the conclusion to be proved. Furthermore, a middle of this sort happens to assign what is far-fetched, as in Anacharsis’ proof that there are no flutists among the Scythians because there are no vines there. For this is quite a far-fetched middle. A nearer one would be that they have no wine, and nearer still, that they do not drink wine, from which follows a merry heart which moves one to sing-if fluting is taken to mean singing. Or it might be better to say that fluting is not taken for just any singing but for the song of the grape harvesters, called “celeuma.”

Then (78b32) he summarizes what he had said and declares that these are the differences between the syllogism quia and the syllogism propter quid in the same science, “and according to the position of the same,” i.e., of those that have the same order. This he says to exclude what he will discuss later, namely, that one science is under another.

Lecture 25
(78b34-79a16)
HOW DEMONSTRATION “QUIA” DIFFERS FROM DEMONSTRATION “PROPTER QUID” WHEN THE FORMER PERTAINS TO ONE SCIENCE AND THE LATTER TO ANOTHER

b34. But there is another way— b35. This occurs in the case— a1. (Some of these sciences— a3. Here it is the business— a10. As optics is related— a13. Many sciences not standing

After showing how demonstration quia differs from demonstration propter quid in the same science, the Philosopher shows how they differ in sciences that are diverse. And he does two things:

First, he states his proposition, saying (78b34) that in a way other than the above the propter quid differs from the quia due to the fact that they are considered in diverse sciences, i.e., that the propter quid pertains to one science and the quia to another.

Secondly (78b34) he elucidates his proposition. First, he elucidates it in sciences one of which is under the other. Secondly, in sciences one of which is not under the other (79a13). Concerning the first he does two things. First, he shows how those sciences are related, one of which is under the other and to one of which pertains the quia and to the other the propter quid. Secondly, he shows how in these sciences the quia pertains to one and the propter quid to the other (790). Concerning the first he does two things. First, he shows how such sciences relate to one another as to order. Secondly, how they relate to one another as to agreement (79a1).

He says therefore first (78b35) that these sciences (i.e., to one of which pertains quia and to the other propter quid) are the ones so related that one is under the other. But this occurs in two ways: in one way, when the subject of one science is a species of the subject of the higher science, as animal is a species of natural body—consequently, the science of animals is under natural science; in another way, when the subject of the lower science is not a species of the subject of the higher science but is compared to the latter as material to formal. An example of this latter way of one science being under another is the way “specular,” i.e., optics, is under geometry. For geometry is concerned with lines and other magnitudes, whereas optics is concerned with a line determined to matter, i.e., the visual line. Now the visual line is not, strictly speaking, a species of line any more than wooden triangle is a species of triangle: for wooden is not a difference in respect to triangle. In like manner mechanical engineering, i.e., the science of making machines, is related to stereometry, i.e., the science of measuring bodies. This science is said to be under the other as applying the formal to the material. For the measures of bodies’ absolutely are compared to the measures of wood and other material required for machines as the application of the formal to the material. In like manner, harmonics, i.e., music, is related to arithmetic: for music applies formal number, which arithmetic considers, to matter, i.e., to sounds. And the same is true of “appearance,” i.e., nautical science (which considers the signs indicative of calm or storm), as compared to astronomy, which considers the motions and positions of the stars.

Then (79a1) he shows how the aforesaid sciences relate to one another in point of agreement. And he says that these sciences are almost mutually univocal. And he says, “almost,” because they agree in the name of their genus and not in the name of their species. Thus, all the sciences mentioned above are called mathematical: some, indeed, because they are concerned with a subject which is abstracted from matter, as geometry and arithmetic, which are absolutely mathematical; but others are so through applying mathematical principles to material things, as astronomy is called mathematical and as nautical science is. Similarly, harmonics, i.e., music, is called mathematical and so is acoustics, i.e., practical music, which knows sounds through experience based on hearing. Or it might be said that they are univocal because they also agree in the name of their species, for we speak of nautical astronomy as astronomy and of practical music as music. But because this is not so in all but only in some, he says “almost.”

Then (79a3) he shows how in these sciences the quia pertains to the one science and the propter quid to the other. In regard to this there are two points. First, he shows how those sciences which contain others under them state the propter quid. Secondly, how sciences which are contained under another one state the propter quid of some others (79a10).

One should note, therefore, with respect to the first point that in all the above-mentioned sciences, the ones which are contained under others apply mathematical principles to sensible things, while the ones which contain others under them are more mathematical. Accordingly, the Philosopher first of all says (79a3) that to know the quia pertains to the sensible, i.e., to the lower sciences, which make application to sensible things; but to know the propter quid pertains to the mathematical, i.e., to those sciences whose principles are not applied to sensible things. For these latter sciences are concerned with demonstrating matters which are accepted as causes in the lower sciences.

But because someone might suppose that whoever knows the propter quid also necessarily knows the quia, he dismisses this, saying that very frequently those who know the propter quid do not know the quia. And he gives as an example of this those who, when considering the universal often take no account of the singulars precisely because their speculation does not consider them: thus, one who knows that every mule is sterile, does not know this of the one he does not consider. In like manner, a mathematician who demonstrates propter quid, now and then does not know the quia, because he does not apply the principles of the higher sciences to matters demonstrated in the lower science.

And because he had said that it belongs to mathematics to know the propter quid, he proposes to indicate which genus of cause is used by mathematics. Hence he says that those sciences which receive the propter quid from mathematics “are something else,” i.e., they differ from the mathematical according to subject, i.e., insofar as they make application to matter. Hence these latter “use forms’ “ i.e., formal principles, which they receive from mathematics: “for the mathematical sciences concern forms.” For their considerations do not bear on the subject, i.e., on matter, because although the items which geometry considers exist in matter, for example, the line, plane and so on, nevertheless geometry does not consider them precisely as they are in matter, but as abstracted. For those things that are in matter according to existence, geometry abstracts from matter according to consideration. Conversely, the sciences subalternated to it accept those things which were considered in the abstract by geometry and apply them to matter. Hence it is plain that it is according to the formal cause that geometry states the propter quid in those sciences.

Then (79a10) he shows that even the subalternated sciences state the propter quid, not of its subalternating science but of some other science. Thus, optics is subalternated to geometry, so that if we compare the one with the other, optics states the quia and geometry the propter quid. But just as optics is subalternated to geometry, so the science of the rainbow is subalternated to optics, for it applies to a determinate matter the principles which optics hands down absolutely. Hence it belongs to the naturalist who treats of the rainbow to know the quia, but to the expert in optics to know the propter quid. For the naturalist says that the cause of the rainbow is the convergence of a visual line at a cloud arranged in some relation to the sun; but the propter quid he takes from optics.

Then (79a13) he shows how quia and propter quid differ among sciences that are diverse but not subalternate. And he says that many sciences which are not subalternate are nevertheless related, i.e., in such a way that one states the quia and the other the propter quid. This is true of medicine and geometry. For the subject of medicine is not subsumed under the subject of geometry as the subject of optics is. Nevertheless, the principles of geometry are applicable to certain conclusions reached in medicine: for example, it belongs to the man of medicine who observes it to know quia that circular wounds heal rather slowly; but to know the propter quid belongs to the geometer, whose business it is to know that a circle is a figure without corners. Hence the edges of a circular wound are not close enough to each other to allow them to be easily joined. It should also be noted that this difference of quia and propter quid between sciences that are diverse is contained under one of the modes previously discussed, namely, when the demonstration is made through a remote cause.

Lecture 26
(79a17-b22)
DEMONSTRATIVE SYLLOGISMS BEST MADE IN THE FIRST FIGURE. ON MEDIATE AND IMMEDIATE NEGATIVE PROPOSITIONS

a17. Of all the figures— a23. Thirdly, the first is— a30. Finally, the first figure— a33. just as an attribute— a36. It follows that if either— b5. That the genus of A— b12. If on the other hand— b2l. Hence it is clear

After determining about the material of the syllogism, the Philosopher here determines about its form, showing in which figure chiefly the, demonstrative syllogism is formed. His treatment is divided into two parts. In the first he shows that the demonstrative syllogism is formed first and foremost in the first figure. In the second, because it is possible in the first figure to proceed from negatives and because a demonstration must proceed from things immediate, he shows how a negative proposition happens to be immediate (79a33).

He shows the first with three reasons, the first of which (79a17) is this: In whatever figure the propter quid syllogism is best made, that figure is the best for causing scientific knowledge and for that reason is most suitable for demonstrations, since demonstration is a syllogism which causes scientific knowledge. But a syllogism propter quid is best made in the first figure, and this is evidenced by the fact that the mathematical sciences of arithmetic and geometry and all other sciences which demonstrate propter quid employ the first figure in most cases. Therefore, the first figure is the one which first and foremost causes scientific knowledge and is most suitable for demonstrations.

But the reason why demonstration propter quid is best made in the first figure is this: in the first figure the middle term is both subjected, to the major extreme, which is the predicate of the conclusion, an(I predicated of the minor extreme, which is the subject of the conclusion. Now in demonstrations propter quid the middle must be the cause of the proper attribute which is predicated of the subject in the conclusion. Furthermore, one of the modes of “saying per se” is when the subject is the cause of the predicate, as “being butchered, he died,” as has been explained above; and this is verified in the first figure in which the middle is subject to the major extreme, as has been said.

The second reason (79a23) is this: the quod quid est [i.e., the definition as signifying the essence] plays a most important role in demonstrative sciences because, as has been said, a definition is either the principle of a demonstration or the conclusion or a demonstration differing in the position of its terms. Now when it is a question of formulating a definition the first figure alone is suitable. For that is the only figure in which a universal affirmative is concluded, which alone produces science of the quod quid est. For the quod quid est is known through an affirmation. Furthermore, the definition is predicated affirmatively and universally of the defined: for it is not some man that is a two-legged animal, but every man. Therefore, the first figure is foremost in causing scientific knowledge and is best accommodated to demonstrations.

Then (79a30) he gives the third reason which is this: For purposes of I demonstration the other figures need the first, but the first does not need them. Therefore, the first figure is better equipped for causing scientific knowledge than the others. That the others need the first is evidenced by the fact that if complete scientific knowledge is to be had, the mediate propositions present in demonstrations must be reduced to immediate ones. But this reduction is made in two ways, namely, by condensing middles [i.e., by inserting fresh middles, the movement proceeding from the extremes toward the middle first employed] and by expanding outwards [i.e., by going from the middle toward the remote extreme].

It is done by condensing when the middle actually employed is joined mediately to each of the extremes or only to the minor extreme. Hence when other middles are introduced between the first middle and the extremes, the middles, as it were, close in. Thus, if someone first says, “Every E is C,” [minor], “Every C is A,” [major], and then a middle, say D, is introduced between C and E, and a middle B between C and A. It is done by expanding when the middle is immediate to the minor extreme and mediate to the major: for then it is necessary to introduce several other middles more general than the middle first taken. Thus, if one says, “Every E is D,” “Every D is A,” and the later other middles more general than D are introduced [between D and A].

Now these processes of condensing and expanding can be performed only in the first figure, both because the first figure is the only one which yields a universal affirmative conclusion, and because it is only in the first figure that the middle lies between both extremes. For in the second figure the middle lies outside the extremes, as being predicated of them; in the third figure [the middle] is below each of the extremes, as being subjected to them.

Then (79a33) he teaches how a negative proposition can be immediate. In regard to this he does two things.

First (79a33), he states his intention, saying that “just as it is possible for every A to be in B atomically,” i.e., immediately, “so too not to be,” i.e., so too it is possible for a proposition signifying that A is not in B to be immediate. Then he goes on to explain what it is to be or not to be atomically, namely, when the affirmation or negation does not have middle through which it might be proved.

Secondly (79a36), he elucidates his proposition. In regard to this he does two things. First, he shows when a negative proposition is mediate. Secondly, when immediate (79b12). Concerning the first he does two things. First, he clarifies his proposition. Secondly, he proves something he had presupposed (79b5).

He says therefore first (79a36), that when A, i.e., the major term, or B, i.e., the minor term, is in some whole as a species in its genus, or both are under diverse genera or predicaments, it does not occur that A is not in B first, i.e., it does not occur that this proposition, “No B is A,” is immediate. First, then, he manifests this when A is in some whole, say in Q and B is in no whole. For example, if A is “man,” C is “substance,” and B is “quantity,” it is possible to form a syllogism to prove that A is in no B on the ground that C is in every A and in no B. Thus we get a syllogism in the second figure:

In like manner, if B, i.e., the minor term, is in some whole, say D, but A is not in that whole, it is possible to syllogize that A is in no B. For example if A is “substance,” B “line,” and D “quantity,” we get a syllogism in the first figure:

In the same way, a negative conclusion could be demonstrated if either is in some whole. For example, if A is “line,” C “quantity,” B “whitenem” and D “quality,” it is possible to form a syllogism in the first and in the second figure. In the second figure thus:

And in the first figure thus:

We should understand, however, that a negative proposition is mediate, when both terms exist in some whole which is not the same but different for each. For if they are in the same whole, the proposition will be immediate, as “No rational being is irrational,” or “No biped is a quadruped.”

Then (79b5) he explains something he had presupposed, namely, “on condition that one of the extremes exist in some whole and that the other be not in the same,” saying that it is “clear from the ‘orderings’ of the various predicaments, which are” not mutually interchangeable. In other words, because that which is in one predicament is not in another, it is plain that B happens not to be in the whole in which A is, or vice versa, because one of the terms happens to be taken from one predicament in which the other is not found. Thus, let one ordering of the predicaments be ACD, say the predicament of substance, and another ordering be BEF, say the predicament of quantity. Then if none of those in the ordering ACD be predicated of none in the ordering BEF, while A is in P as in that most general item which is the principle of the whole first ering, it is plain that B is not in P, because then the orderings, i.e., the predicaments, would be interchanged. Similarly, if B is in some whole, say in E, it is plain that A is not in E.

Then (79b12) he indicates how a negative proposition may be immediate, saying that “if neither is in some whole,” i.e., neither A nor B, and A is not in B, it is necessary that this proposition, “No B is A,” be immediate. Because if a middle were taken to syllogize it, then one of them would have to be in some whole, for the syllogism would have to be made either in the first figure or in the second, since in the third figure a universal negative cannot be concluded, as is required for an immediate proposition. However, if it is made in the first, B would have to be in some whole, because B is the minor extreme, and in the first figure the minor proposition must always be affirmative. For a syllogism with the major affirmative and the minor negative cannot be formed in the first figure.

But if it be in the second figure, either may, i.e., A or B, may be in a whole, because in the second figure the first proposition may be negative in some moods and the minor in other moods. Of course it is never permitted, neither in the first nor the second, to have both propositions negative. And so it is required that when either proposition is affirmative, one of the extremes must be in some whole. Thus it is clear that a negative proposition is immediate, when neither of its terms is in some whole. This does not mean, however, that although neither is in some whole, a middle could be found to conclude it, namely, if one were to take a convertible middle, because it is necessary that such a middle be prior and better known. And this is either the genus itself or the definition, which is not without a genus.

Then (79b21) he concludes and summarizes what has been said. Here the text is sufficiently clear.

Lecture 27
79b23-80a8)
HOW IGNORANCE OR DECEPTION BEARING ON FIRST AND IMMEDIATE THINGS CAN BE INDUCED BY SYLLOGISM AND LEAD ONE TO SUPPOSE SOMETHING TO BE WHICH IS NOT

b23. Ignorance–defined not as— b28. The error, however, that— b3l. Now, two cases are— b34. C may quite well— b40. On the other hand— a6. Error of attribution

After determining about the demonstrative syllogism through which science is acquired, the Philosopher here determines concerning the syllogism through which ignorance or deception is produced in us. In regard to this he does two things. First, he shows what sort of ignorance can be induced by a syllogism. Secondly, he shows the mode in which such a syllogism proceeds (79b28).

He distinguishes therefore first (79b23) a twofold ignorance, one of which is negative and the other a positive state. There is ignorance in a negative way, when a man has no scientific knowledge at all about a thing. This ignorance consists in not attaining, as the Philosopher says in Metaphysics IX, and is exemplified in a peasant who knows absolutely nothing about whether a triangle has three angles equal to two right angles. But ignorance is present as a positive state, when one does have a definite opinion but it is unsound. For example, when he falsely thinks something about a thing, either because he thinks something to be which is not, or something not to be which is. And this ignorance is the same as error.

Now the first ignorance is not produced by a syllogism, but the second can be, and then it is called deception. Such ignorance or deception can concern two things: first, those things which are first and immediate principles, namely, when a person opines things opposed to the principles. And although he cannot so opine inwardly in the mind, as has been stated above, because these things do not fall under apprehension, nevertheless he can contradict them orally and according to a false imagination, as is said in Metaphysics IV of some who deny the principles. Secondly, it might be concerned with conclusions which are not first and immediate. Furthermore, the first of these ignorances or deceptions is opposed to the knowledge which is understanding, and the second to the knowledge which is science.

Each of these states of ignorance, whether concerned with things that are first or things not first, can befall a man in two ways: first, straightway, when independently of any process of reasoning he thinks a falsehood by affirming or denying. In another way, when he is brought to his opinion through some syllogized reason. Hence the Philosopher says in Metaphysics IV that some contradict the principles as though persuaded by reasons; but others, not as though persuaded by reasons, but through lack of erudition or through wilfullness which demands to have a demonstration in all matters.

Then (79b28) he shows how these ignorances are caused. First, how the ignorance produced by a syllogism is caused. Secondly, how ignorance befalls a man without syllogizing (81a38) [L. 30]. Concerning the first he does two things. First, he states how ignorance in regard to first and immediate principles is engendered by a syllogism. Secondly, how it is caused in regard to things not first and immediate (80b17) [L. 29]. Concerning the first he does two things. First, he states how the ignorance is caused whereby that which is not is believed to be. Secondly, that whereby that which is is believed not to be (80a8) [L. 28]. Regarding the first he does three things. First, he states how the aforesaid ignorance is generally caused. Secondly, he assigns its various possible forms (79b31). Thirdly, he answers a tacit question (80a6).

He says therefore first (79b28) that the deception according to false opinion which above was said to occur straightway is simple, i.e., is engendered in only one way. For it is not caused by a reason, which might vary, but by the lack of a reason which [lack] is not diversified into various modes each having its own characteristics any more than other negations are.

Yet because false reasons can be various and many, this ignorance, when it is engendered by a syllogism can occur in many ways according to the several ways in which a syllogism can be false. The common way is set forth when he says, “Let A be in no B individually” (79b29), i.e., let this proposition be immediately true, “No B is A.” For two negations are stated in the place of one— for instance if we should say, “No quality is a substance”— according to the doctrine on immediate negative propositions presented above. Therefore, if someone were to conclude the opposite of this with a syllogism, taking C as a middle to show that every B is A, there will be a deception through syllogism.

Then (79b31) he shows in how many ways this can vary. And first of all it should be noted that a false conclusion is not concluded except when the syllogism is false. But a syllogism can be false in two ways: first, because it lacks syllogistic form, in which case it is not a syllogism but appears to be one. In a second way, because it employs false propositions, in which case it is a syllogism as to form, but it is a false one because of the false propositions used. Therefore, in a dialectical disputation, which bears on probables, use could be made of both types of false syllogism, because such a disputation proceeds from common premises. Consequently, error can arise in it both as regards the matter employed, which is common, and also as regards the form, which is common.

But in a demonstrative disputation, which bears on necessary things, the type of syllogism used is the one which could be false only on account of the matter, because as it is stated in Topics I, a paralogism in a given discipline proceeds from things proper to the discipline, but not from true things. Hence, since syllogistic form must be counted among the common things, paralogism within a discipline-which is the matter under discussion-is not defective in form but only in matter, and furthermore in regard to proper and not in regard to common things. And so: First, he tells how such a syllogism might proceed from two false premises. Secondly, how it might proceed from one or the other premise being false (79b40). But since the first can occur in two ways, namely, the false proposition might be contrary to the true or contradictory to it, therefore: First, he shows how such a syllogism might proceed from two false statements that are contrary to true statements. Secondly, how a contradiction is accepted (79b34).

First, then (79b30) he says that in a syllogism causing deception it sometimes happens that both premises are false, and sometimes only one. Let us take”the case of two false premises, each contrary to what is true. For example, suppose that C is so related to A and to B. that no C is A and no B is C. Now if we take the contraries of these, namely, that every C is A and every B is C, these two propositions will be completely false. For example, if I should say: “Every quality is a substance; every quantity is a quality: therefore, every quantity is a substance.”

Then (79b34) he shows how both premises can be false and not contrary, but contradictory, to what is true. For example, if C is so related to A and to B, that it is neither contained totally under A nor is universally in B. Thus one might take C as “perfect” or “actual being,” and proceed in the following way: “Every perfect thing is a substance; every quantity is perfect: therefore, every quantity is a substance. Now although both are false, they are not entirely so, for their contradictories are true, namely, “Some perfect things are not substances,” and “Some quantity is not perfect”; but their contraries are false, namely, “No perfect thing is a substance,” and “No quantity is perfect.”

But that C is not in B universally (i.e., that the statement, “Every quantity is perfect,” which was the minor, is not true), i.e., that “Every B is C,” is not true he proves by the fact that B cannot be contained under any whole that might be predicated universally of it. And this is so because the proposition, “No B is A,” was said to be immediate, which means that A is universally not in B. But, as has been stated above, those negative propositions are immediate, neither of whose terms is under a whole.

Nevertheless, this proof does not seem sufficient, because something can be predicated universally even of that which is not under some whole as a species under a genus. For the genus and the difference are not the only things predicated universally, but a property is too. But it must be said that although the proof in question is not efficacious generally speaking, it is in this case. Because, as it is stated in Topics I, a paralogism within a discipline— with which we are concerned here— proceeds from items that accord with the discipline. Hence he intends to use such middles as a demonstrator uses. But the middle of a demonstration is the definition, as has been stated above. Therefore, even in the syllogism of which he is speaking, he intends to use a definition as the middle. Now a definition contains the genus and difference. Hence that which is predicated universally in this syllogism should contain that in which the subject is, as in a whole.

Again, that A is not in C universally, i.e., that the statement, “Every perfect thing is a substance,” is not universally true is proved by the fact that it is not required of every universal that it be universally in all things that are: because no predicament is predicated of things contained under another predicament nor universally predicated of those items that commonly follow upon being, namely, act and potency, perfect and imperfect, prior and posterior, and the like.

Then (79b40) he shows how the aforesaid syllogism might proceed from one true and one false. And he says that in the aforesaid syllogism we might take one true, i.e., the major, which is AC, and the other false, namely, the minor, which is BC. That the minor proposition, which is BC, is always the false one he proves, as he did above, on the ground that B is in no C as in a whole. But that this proposition AC could be true, while the other is false he proves in terms. Let A be in B and in C individually, i.e., immediately, as a genus in its species; for example, as color is to whiteness and blackness. Now under these conditions it is obvious that the major will be true, namely, “Every C is A,” for example, “Every whiteness is a color,” and the minor false, namely, “Every blackness is whiteness”: for when something is predicated “first” of several, none of the several is predicated of any other of them. For the first predication of a genus is of species which are opposite.

Here, too, a doubt arises, because from the terms used here, not a false but a true conclusion follows. For the conclusion will be that A is in B, it having also been assumed that it is in B individually. But it must be answered that this example was used merely to illustrate how the major could be true and the minor false; although it is of no use in a case where a false conclusion is sought. Hence the Philosopher at once adds, “It is equally the case of AC if not atomic.” However, we could take terms such that A is not in B either individually or in any way, but is rather immediately removed from B. Neither is it necessary that it be in C individually, because it is not necessary that a demonstrator employ only immediate propositions: for he may use ones which are supported by immediate propositions. We can, therefore, take other terms pertaining to the present case, for example, “intellectual substance,” as the middle, for “Every intelligence is a substance,” and take as the false minor, “Every quantity is an intelligence.” Hence a false conclusion follows.

Then (80a6) he answers a tacit question. For someone might request that he exemplify the diversity of such a syllogism in the other figures. But he answers that a deception, which bears on being, i.e., through which someone opines a false affirmative proposition, can be derived only by the first figure, because in the next figure, i.e., in the second, an affirmative syllogism cannot be formed. As for the third figure, it has no bearing on the case, because it cannot conclude a universal, which is principally intended in demonstration and in this syllogism.

Lecture 28
(80a8-b16)
HOW BY SYLLOGIZING IN THE FIRST OR SECOND FIGURE A FALSE NEGATIVE IS CONCLUDED CONTRARY TO AN IMMEDIATE AFFIRMATIVE

a8. On the other hand,— a11. It may occur when a — a14. It is also possible — a20. Or again, C-B — a27. In the second figure — a38. Or one premiss may — b6. The case is similar — b14. It is thus clear

After showing how a false affirmative conclusion contrary to an immediate negative is obtained by syllogizing, the Philosopher here shows how by syllogizing a false negative is concluded contrary to an immediate affirmative. First, in the first figure. Secondly, in the second (8048). Concerning the first he does two things. First (80a8), he states his intention and says that since a universal negative may be concluded in the first as well as in the second figure, we must first show in which moods a syllogism of ignorance is formed in the first figure and under which conditions of truth and falsity in the propositions. Secondly (80a11), he establishes his proposition. First, he shows how such a proposition is formed from two false premises in the first figure. Secondly, how it is formed from one false and one true premise (80a14).

He says therefore first (80a11) that the aforesaid syllogism can be formed from premises, both of which are false. This is clear if A is both in B and in C individually, i.e., immediately. It is thus that a genus is immediately in the proximate species into which it is first divided, as color into blackness and whiteness. For the genus is predicated per se of the species, because the former is placed first in the definition of the latter; and it is predicated immediately of a proximate species, because it is put in its definition immediately and not in the way that a remote genus-which is put in the definition of a defining part-is related to an ultimate species. Therefore, let the terms be “color,” “whiteness” and “blackness.” If, then, we assume that A is in no C, for example, if we say, “No whiteness is a color,” but C is in every B, say “All blackness is whiteness,” both propositions are false, as is the conclusion, “No blackness is a color.”

Then (80a14) he shows how there can be one false and one true premise in the syllogism under discussion. First, he shows how the major can be true and the minor false. Secondly, how it might be the reverse (8040).

He says therefore first (80a14) that a negative syllogism of ignorance can be formed in the first figure no matter which one of the propositions happens to be false. For it might happen that the proposition AC, which is the major, is true, and the proposition BC, which is the minor, is false. That the major proposition could be true he proves by the fact that the term A, whatever it be, need not be in all things, as color is not predicated of all beings. That the minor would be false he proves on the ground that it is not possible to assume a term of which A would be universally denied and which would also be predicated of B: for we are supposing that the proposition, “Every B is A,” is true and immediate. Therefore, if something were universally predicated of B, so that “Every B is C” would be true, then A cannot be universally denied of C. Consequently, this proposition, “No C is A,” which was the major, will not be true. For if every B is A, as we supposed, and every B is C, as we are now assuming, it follows in the third figure that some C is A, which contradicts the major. Therefore the proposition, “No C is A,” will be false. Hence if this is true, which is the major, it is required that this be false, which is the minor, i.e., “Every B is C.”

Then he proves the same thing on the ground that if both premises are true, then as has been proved above, a false conclusion cannot follow—which is out of place in a syllogism of ignorance, which ought to conclude to a false conclusion. But it was given that this is true, namely, “No C is A”’: if then it is also true that “Every B is C,” it follows that the conclusion, “No B is A,” is true, whereas it is supposed to be false, being contrary to this immediate proposition, “Every B is A.”

Then (80a20) he shows how the minor can be true, the major being false. And he says that the proposition CB, namely, the minor, can be true, while the major is false. For since this proposition, “Every B is A,” whose contrary is to be concluded, is immediate, it is necessary that B exist in A as a part in a whole, as “whiteness” in “color.” But it is possible to take something else in which B also exists as in a whole, though not immediately—let this other thing be “quality,” i.e., C. It is necessary, therefore, according to the aforesaid, that as between these two, namely, A and C, one should be under the other, i.e., color under quality. Now if someone assumes that A is in no C and says, “No quality is a color,” the proposition will be false. But the minor will be true, namely, “Every whiteness is a quality.” The conclusion, however, “No whiteness is a color,” will be false and contrary to an immediate proposition. And so it is clear that a negative syllogism of ignorance can be formed in the first figure when either one or both of the premises are false.

Then (8047) he shows how a negative syllogism of ignorance is formed in the second figure. First, when both are false. Secondly, when one or the other is false (80a38).

He says therefore first (8047) that in the second figure it does not happen that both propositions are entirely false. And he calls those propositions entirely false which are contrary to true propositions. He proves this: For since we are trying to conclude a false negative contrary to an immediate affirmative, we must assume that this proposition, “Every B is A,” is true and immediate, say, “Every whiteness is a color.” But with terms so related it is impossible to find a middle term which would be predicated universally of one and universally removed from the other. For suppose that the term C could be universally removed from A and universally predicated of B. Then the proposition, “No A is C,” will be true; consequently, its converse, “No C is A,” will also be true. But every B is C. Therefore, no B is A, the contrary of which was supposed.

Similarly, it cannot be universally removed from B and universally predicated of A. For if it is true that every A is C, the converse, “Some C is A,” will be true. But if it is true that no B is C, its converse, “No C is B,” will be true. So, then, from these two propositions, “Some C is A” and “No C is B,” there follows, “Some B is not A,” which is the contradictory of what was supposed, namely, that “Every B is A.” What remains, therefore, is that it is impossible to find any middle which, A and B being related in the way we have supposed, can be predicated of one and removed from the other. Yet if a syllogism is to be formed in the second figure, the middle must be predicated of one of the extremes and denied of the other. Therefore, if both are totally false, the contraries would have to be true, which is impossible as has been proved.

However, nothing prevents both from being false particularly: thus we may take some middle which is predicated particularly of A and of B, say “male,” which is predicated particularly of animal and of man. Now if C is taken in every A, say “Every animal is male,” and in no B, say “No man is male,” each proposition will be false, not entirely but particularly. And the same holds if, conversely, the major is negative and the minor affirmative, i.e., if we should say, “No animal is male” and “Every man is male.”

Then (80a38) he shows how it happens when one is false. First, in the second mode of the second figure. Secondly, in the first mode (80b6).

He says therefore first (80a38) that in this figure it occurs that either proposition may be false. This is clear from the fact that if A is supposed to be predicated per se and immediately of B, whatever is in every A is in every B, as whatever is predicated universally of animal is predicated universally of man. Therefore, if some middle, C, be taken which is universally predicated of A, say “Every animal is living,” and universally removed from B, say “No man is living,” it is evident that AC, which is the major proposition, will be true, but BC, which is the minor, will be false.

Similarly, he proves that the converse occurs when the major is false. For it cannot be that something be universally removed from B and universally predicated of A, when the terms have that position. For it has been stated that if something is in A universally, it follows that it is also in B. Consequently, if something be removed universally from B, it cannot be that it is predicated universally of A. For example, anything universally removed from “man” cannot be universally predicated of “animal.” Therefore, if something is taken which is universally removed from man, say “irrational,” and you state that “Every animal is irrational” and “No man is irrational,” it follows that the minor proposition is true and the major false. But in these terms the major premise is not totally false. However, one can take a term in which it is totally false, for example, if we should take “inanimate” as the middle.

Then (80b6) he shows the same in the first mode of the second figure, where the major is negative. For it is clear that with the terms A and B so related, as was said, something universally removed from A cannot be in any B. Therefore, if a middle, C, be taken which is universally removed from A and universally predicated of B, the major will be true and the minor false. For example, if the terms are “inanimate,” “animal” and “man.”

Similarly, he shows that the minor can be true and the major false. For it is clear, according to the aforesaid, that that which is universally predicated of B cannot be universally removed from every A, because what is universally predicated of B must be in some A at least. Therefore, if C be taken as a middle which is universally predicated of B, say, “rational” or “living,” and universally denied of A, there will be a true minor proposition, namely, “Every man is rational” or “living.” But the major, namely, “No animal is rational,” is false in part, while “No animal is living,” is false entirely.

Then summarizing (80b14) he concludes that a deceptive syllogism can be formed in immediates, when both propositions are false or only one is false.

Lecture 29
(80b17-81a37)
SYLLOGISM IGNORANCE IN REGARD TO MEDIATE PROPOSITIONS

b17. In the case of attributes— b27. Similarly if the middle— b32. On the other hand— a5. When the erroneous— a16. If the conclusion is— a20. This is equally true— a25. The middle may be— a35. Thus we have made

After showing how a syllogism of ignorance in regard to immediate propositions is made, the Philosopher here shows how it is made in regard to mediate propositions. First, how a false negative proposition is concluded which is opposed to a true affirmative. Secondly, how a false affirmative is concluded which is opposed to a true negative (81a16). Concerning the first he does two things. First, he shows this in the first figure. Secondly, in the second (81a4). In regard to the first he does three things. First, he shows how a syllogism of ignorance is constructed in mediate propositions through a proper middle. Secondly, how one is constructed through a middle which although not proper has a relationship to the terms, a relationship akin to that of a proper middle (80b27). Thirdly, he shows how such a syllogism is constructed through an extraneous middle (80b32).

He says therefore first (80b17) that when a syllogism is constructed which concludes something false in propositions which are not individual, i.e., not immediate, if the “proper middle” through which the syllogism is formed be taken, then both propositions cannot be false, but only the major. He explains what he means by “proper middle.” For since the proposition whose contrary is to be syllogized is mediate, it is required that the predicate be syllogized of the subject through some middle. Therefore, the same middle can be employed to conclude the opposite. Say that the mediate proposition is “Every triangle has three angles equal to two right angles.” The middle through which the predicate is syllogized of the subject is, “A figure having an exterior angle equal to the two opposite interior angles.” Now if we would prove through the same middle that no triangle has three angles equal to two right angles, it will be a syllogism of falsity through a proper middle. Hence he says that a proper middle is one through which the syllogism of contradiction is made, i.e., leading to the opposite conclusion. In the above example, A would be “triangle,” B, “having three...” and C, the middle, “such a figure.” Now in the first figure the minor must be affirmative; therefore, that which was the minor in the true syllogism must remain unconverted and not changed into its opposite in the syllogism of falsity. Hence it must always be true. But the major proposition of the true syllogism is changed into its contrary negative; hence the major must be false. For example, we might say: “No figure having an exterior angle equal to the two opposite interior angles has three angles equal to two right angles; but the triangle is such a figure: therefore, no triangle has three angles equal to two right angles.”

Then (80b27) he shows how the aforesaid syllogism is constructed through a middle which is extraneous but like the proper. And he says that it will be syllogized in like fashion if the middle is taken from another ordering. For example, if A had been demonstrated of B through C, and we were to take in the syllogism of falsity not C but D as the middle, in such a way, however, that D is also contained universally under A and predicated universally of B: say if we took for the middle, “a closed figure of three lines,” because here too the minor proposition DB must remain as it was in the syllogism which concluded the true, although through a proper middle. But the major proposition will have to be changed into its contrary. And so the minor will always be true and the major always false. But as to the mode of the argument, this deception is similar to that which is formed through the proper middle.

Then (80b32) he shows how a syllogism of falsity is made through a middle which is extraneous and unlike the proper. For a middle can be taken such that it is contained universally under A but is predicated of no B. In this case both propositions will have to be false, because in order that the syllogism be formed in the first figure, it will be necessary to take propositions to the contrary, namely, a major which is negative, for example, “No D is A,” and the minor affirmative, for example, “Every B is D.” Clearly then both are false. Now this relationship of terms cannot be found in things convertible, say in a subject and its proper attribute which is concluded of the subject through some middle. For it is obvious that no middle can be taken such that the proper attribute would be universally predicated of it, and that middle be removed universally from the subject. But this relationship can be found when the proposition is mediate, for example, when a higher genus or the proper attribute of a higher genus is predicated of an ultimate species, as when we say, “Every man is living.” For “living” can be concluded of man through the middle, “animal.” Therefore, if we should take something of which “living” would be universally predicated, say “olive,” but would be universally removed from man, the relationship of terms that we are seeking will result. For this will be false, “No olive is living”; and the minor, too, will be false, “Every man is an olive.” Similarly, the conclusion will be false, “No man is living,” which is contrary to the true mediate proposition.

It also happens that the major may be true and the minor false. For example, if we take as middle something which is not contained under A, say “stone”; then the major, AB, will be true, namely, “No stone is living,” because “stone” is not contained under “living,” but the minor will be false, namely, “Every man is a stone.” For if the minor remained true, while the first Was true, then the conclusion would be true, whereas it has been said that it is to be false. However, the converse does not occur, i.e., that the minor be true when the middle is extraneous, because such a middle cannot be predicated universally of B. But in the first figure the minor taken must always be an affirmative statement.

Then (81a5) he shows how a negative syllogism of ignorance is made in the second figure. And he says that in the second figure it cannot occur that both propositions be totally false. For if we are to conclude the false proposition, “No B is A,” contrary to the true, it would have been required that A be predicated universally of B. Hence nothing will be able to be found which would be universally predicated of one and universally denied of the other, as has been established above when we treated concerning the syllogism of ignorance in immediates.

But one ox the other of them can be totally false. And he manifests this first in the second mood of the second figure where the major is affirmative and the minor negative. Thus, let the middle be related to the extremes so that it is predicated universally of both, as “living” is predicated universally of man and of animal. Then if we take the major affirmative, say “Every animal is living,” and the minor negative, . say “No man is living,” the major will be true, the minor false, and the conclusion false. In like manner also, if in the first mood of the second figure we take the major negative, say “No animal is living,” and the minor affirmative, say “Every man is living,” the major will be false, the minor true, and the conclusion false.

Having said these things he sums up and concludes that he has stated when and through which kinds of premises deception can occur, if the deceptive syllogism is privative [negative].

Then (81a16) he shows how an affirmative syllogism of deception is formulated in mediate propositions. First, when it is formulated through a proper middle. Secondly, when it is formulated through a middle similar to a proper middle (8140). Thirdly, when it is formulated through an extraneous middle (81a25).

He says therefore first (8106), that if an affirmative syllogism of deception is to be formulated in mediate propositions, if a proper middle such as explained above be taken, it is impossible that both propositions be false. For since a syllogism of this kind can be formed only in the first figure, both propositions being affirmative, it is required that the minor proposition remain as it was in the true syllogism. Hence the major will have to be changed, namely, from negative to affirmative, so that it will have to be false. For example, if we desire to conclude that “Every man is a quantity,” which is the contrary of the statement, “No man is a quantity,” whose proper middle is “substance,” we will take the false proposition, “Every substance is a quantity,” and the true proposition, “Every man is a substance.”

Then (81a20) he shows how a syllogism of ignorance is formed when a non-proper middle is taken which is not of the same order but from some other ordering. For example, if I say, “Every agent is a quantity; every man is an agent: therefore, every man is a quantity.” For it is necessary in this case for the minor to remain, but the major will have to be changed from negative to affirmative. Hence this deception is similar to the previous deception, as was stated in the privative syllogism.

Then (81a25) he shows how an affirmative syllogism of deception is formulated through an extraneous middle. And he says that if an extraneous middle be taken such that it is contained under the major extreme, then the major will be true and the minor false. For A, the major extreme, can be predicated universally of many things that are not under one another, say “habit” of grammar and virtue. For this is mediate, “No grammar is a virtue.” There we can conclude the contrary of this, namely, “All grammar is virtue,” through a middle which is contained under virtue. Then the major will be true and the minor false. For example, we might say: “All temperance is a virtue; grammar is temperance: therefore, all grammar is a virtue.”

But if a middle is taken which is not under the major extreme, the major will always be false, because it will be affirmative. But the minor may sometimes be false with such a major; then both will be false. For example, if we should say: “Every whiteness is a virtue; all grammar is whiteness: therefore....” But sometimes it can be true. For when the terms are so related, there is nothing to hinder A from being removed from every D, and D from being in every B, as happens in these terms, namely, “animal,” “science,” “music.” For the major extreme, “animal,” is removed universally from all science; hence this proposition which is taken as the major in the syllogism of ignorance is false, namely, “Every science is an animal.” But the minor, namely, “All music is science,” is true. And the conclusion will be false, being contrary to the true mediate negative. It can also happen that A is in no D, and D in no B, as has been said.

Thus it is evident that when the middle is not contained under the major extreme, they may both be false or just one of them, because the major and the minor may be false. However, with the three terms so related, as we have said above, the major cannot be true.

Finally (81a35), he summarizes and concludes that it is plain from the foregoing how many ways and through which alignment of true and false propositions it is possible to construct deceptions through syllogisms, both in immediate propositions and in mediate propositions, which are proved by demonstration.

Lecture 30
(81a38-b9)
CAUSE OF SIMPLE NEGATIVE IGNORANCE

a38. It is also clear— a39. since we learn either

After the Philosopher has determined concerning ignorance through deception which is caused by a syllogism, he now determines concerning the ignorance of simple negation which is produced without a syllogism. First, he shows in which type of person this ignorance is had of necessity. Secondly, he proves his proposition (81a39).

He says therefore first (81a38) that if a person lacks any of the senses, say, sight or hearing, then necessarily the science of the sensible objects proper to those senses will be lacking. Thus, if a person lacks the sense of sight, then of necessity the science of colors will be lacking in him. And so he will have ignorance of negation in regard to colors, being entirely ignorant of colors. However, this must be understood of persons who never had the sense of sight, as a person born blind. But if someone loses the sight he once had, he does not on that account necessarily lack a science of colors, because the memory of colors previously sensed remains in him.

But it is possible that ignorance of negation be had of certain things which can nevertheless be known through a sense we possess.. For example, if someone with sight were always in the dark, he would de facto lack a science of colors, but not of necessity, because he could acquire it by sensing colors-which does not occur in one who lacks the sense of sight. Hence he adds that it is impossible to receive it because one who lacks the power of sight cannot even acquire a knowledge of colors.

Then (81a39) he proves his proposition on the ground that there are two ways of acquiring science: one is through demonstration, and the other is through induction, as we have stated in the beginning of this book. But these two ways differ, because demonstration proceeds from universals, but induction from particulars. Therefore, if any universals from which demonstration proceeds could be known without induction, it would follow that a person could acquire science of things of which he does not have sense experience. But it is impossible that universals be known scientifically without induction. This is quite obvious in sensible things, because we receive the universal aspect in them through the experience which we have in regard to sensible things, as is explained in Metaphysics I. But this might be doubted in things which are abstract, as in mathematics. For although experience begins from sense, as it is stated in Metaphysics 1, it seems that this plays no role in studying things already isolated or abstracted from sensible matter.

Therefore, to exclude this he says that even those things that are abstract happen to be made known through induction, because in each genus of abstract things are certain particulars which are not isolable from sensible matter, so far as each of them is a “this something.” For although “line” is studied in isolation from its sensible matter, nevertheless “this line”, which is in sensible matter, so far forth as it is individualized, cannot be so isolated, because its individuation is from this matter. Furthermore, the principles of abstracted [isolated] things, from which demonstrations regarding them are formed, are not made manifest to us except from certain particulars which we perceive by sense. Thus, from the fact that we see some single sensible whole we are led to know what a whole is and what a part, and we know that every whole is greater than its part by considering this in many. Thus the universals from which demonstration proceeds are made known to us only through induction.

Now men who lack any of the senses cannot make an induction from singulars pertaining to that sense, because sense is the sole knower of the singulars from which induction proceeds. Hence such singulars are utterly unknown, because it does not occur that anyone lacking a sense receives science of such singulars: first, because he cannot demonstrate from universals without induction through which universals are known, as has been said; secondly, because nothing can be known through induction without the sense which is concerned with the singulars from which induction proceeds.

It should be noted that by these words of the Philosopher two positions are excluded: the first is Plato’s, who stated that we do not have science of things except through Forms participated from ideas. If this were so, universals could be made known to us without induction, and we would be able to acquire a science of things of which we have no sense. Hence Aristotle also uses this argument against Plato at the end of Metaphysics I. The second is the position of those who claim that in this life we can know separated substances by understanding their quiddities, which however cannot be known through sensible objects which we know and which are entirely transcended by them. Hence if they were known according to their essences, it would follow that some things would be known without induction and sense perception, which the Philosopher here denies even in regard to abstracted things.

Lecture 31
(81b10-82b20)
THREE QUESTIONS ABOUT PROCEEDING TO INFINITY IN CONFIRMING DEMONSTRATIONS

b10. Every syllogism is effected— b14. It is clear, then,— b18. If our reasoning aims— b24. Thus: since there are— b30. Suppose, then,— b34. The second question is— a2. A third question— a7. This is the equivalent— a9. I hold that the same— a15. One cannot ask the same

After the Philosopher has determined concerning the demonstrative syllogism by showing from what and what sort of things it proceeds and in which figure demonstrations can be formed, he now inquires whether demonstrations can proceed to infinity. First, he raises the question. Secondly, he settles it (82a21) [L. 32]. Concerning the first he does two things. First, he sets down certain prefatory remarks needed for understanding the question. Secondly, he raises the question (81b30). Concerning the first he does two things. First, he prefaces something about the syllogistic form one must observe in demonstrations. Secondly, he re views what the matter of demonstration should be (81b14).

With respect to the first (81b10) he touches on three things, the first of which is common to every syllogism, namely, that every syllogism is formed in three terms, as is indicated in Prior Analytics I. The second however, pertains to an affirmative syllogism whose form is such that it concludes A to be in C, because A is in B and B in C. And this is the form of a syllogism in the first figure in which alone can be concluded a universal affirmative, the chief quest in demonstration. The third pertains to a negative syllogism which of necessity has one affirmative proposition and one negative, but differently in the first figure and in the second, as is clear from what has been shown in the Prior Analytics.

Then (81b14) he reviews what the matter of demonstration should be. In regard to this he does three things. First, he states what this matter is. Secondly, he shows the difference between this matter and the matter of a dialectical syllogism (81b18). Thirdly, he clarifies this difference (81b24).

He says therefore first (81b14) that since a syllogism has three terms from which are formed the two propositions which conclude the third, it is clear that these propositions, from which one proceeds in a demonstrative syllogism according to the aforesaid form, are the principles and suppositions we discussed earlier. For one who accepts such principles demonstrates through them in the syllogistic form we have mentioned, namely, that A is in C is proved through B; and if the proposition AB is mediate, another middle is used to demonstrate that A is in B. The like is done if the minor proposition, BC, is mediate.

Then (81b18), apropos of what has been said, he shows the difference between a demonstrative and a dialectical syllogism. For since the latter aims at producing opinion, the sole intent of a dialectician is to proceed from things that are most probable, and these are things that appear to the majority or to the very wise. Hence if a dialectician in syllogizing happens upon a proposition which really has a middle through which it could be proved, but it seems not to have a middle because it appears to be per se known on account of its probability, this is enough for the dialectician: he does not search for a middle, even though the proposition is mediate. Rather he syllogizes from it and completes the dialectical syllogism satisfactorily.

The demonstrative syllogism, on the other hand, is ordained to scientific knowledge of the truth; accordingly, it pertains to the demonstrator to proceed from truths which are really immediate. Hence if he happens upon a mediate proposition, he must prove it through its proper middle until he reaches something immediate, because he is not content with the probability of a proposition.

Then (81b24) he elucidates what he has said, asserting that his claim that the demonstrator proceeds “to the truth from things that are” is supported by the fact that it is possible to find something “which is not predicated of a thing accidentally.” What this means he explains by showing how in the case of affirmative statements something is predicated accidentally.

For something is predicated accidentally in two ways: in one way, when the subject is predicated of an accident, as when we say, “The white thing is a man”; in another way, when the accident is predicated of the subject, as when we say, “The man is white.” Now this way differs from the first, because when the accident is predicated of the subject, it is stated that the man is white not because something else is white but because the man himself is white. Yet it is a per accidens proposition, because “white” does not belong to man according to the specific nature of man. For neither is placed in the definition of the other. But when it is stated that the white thing is a man, it is not so stated because being a man is in the whiteness, but because being a man is in the subject of whiteness, which subject happens to be white. Hence this way is further removed from per se predication than the first.

But there are certain things which are not predicated per accidens in either of these ways: these are said to be per se. Such are the things from which the demonstrator proceeds. But the dialectician is not so demanding; consequently, the question concerning such things as are predicated per se is not relevant to the dialectical syllogism but only to the demonstrative syllogism.

Then (81b30) he raises the questions he intended. Concerning this he does two things. First, he raises the questions in regard to things to which they are relevant. Secondly, he shows the cases in which they are not relevant (82a15). Concerning the first he does two things. First, he raises questions in affirmative demonstrations. Secondly, he shows that these questions also have relevance in negative demonstrations (82a9). In regard to the first he does two things. First, he raises the questions. Secondly, he shows where such questions are relevant (82a7).

In regard to the first (81b30) he raises three questions corresponding to the three terms of a syllogism. First, he raises a question concerning the major extreme, namely, whether one can go to infinity in ascending order? In this question an ultimate subject is supposed which is not predicated of any other, but other things are predicated of it. Let this subject be C, and let B be in C first and immediately, and let E be in B, as universally predicated of B; furthermore, let F be in E as universally predicated of it. The question is this: Should this ascending process come to a halt somewhere, so that something is reached which is predicated universally of other things but nothing else is predicated of it, or is that not necessary but a process to infinity occurs?

Secondly (81b34) he raises the question on the part of the minor term, namely, whether one can go to infinity in descending. In this question some first universal predicate is supposed which is predicated of other things and nothing is more universal than it so as to be predicated of it. Thus let A be such that nothing is predicated of it as a universal whole of a part, but A is predicated of H both first and immediately, and H of G, and G of B. The question then is this: Is it necessary to come to a halt in this descending process, or may it proceed to infinity?

Then he shows the difference between these two questions. For in the first one we asked: If someone begins from a most particular subject which is in nothing else the way a universal whole is in a part but other things are in it, does an infinite ascending process occur? But in the second we are asking: If someone begins with a most universal predicate, which is predicated of other things as a universal whole of its parts but nothing is predicated of it in this way, does an infinite descending process occur?

Thirdly (82a2) he raises the third question on the part of the middle term. In this question two determinate extremes are supposed, namely, a most universal predicate and a most particular subject. The question is whether under these conditions there can be an infinity of middles. Thus, if A is the most universal predicate and C the most particular subject, and if between A and C there is the middle, B, and again between A and B another middle, and likewise between B and C; furthermore, if there are other middles of these middles between them and the extremes both in ascending and in descending order. The question then is this: May these processes go on to infinity or is that impossible?

Then (82a7) he shows what is the tenor of these questions. And he says that these questions pertain to the matter now under discussion, namely, to demonstrations. He says, therefore, that the attempt to reach true answers to these questions is the same as trying to settle the question whether demonstrations proceed to infinity by ascending or descending. By ascending, i.e., so that each proposition from which a demonstration proceeds would be demonstrable by another prior proposition. This is what he means when he asks: “Is there a demonstration of everything,” i.e., of every proposition? By so thinking, some have erred in regard to principles, as is stated in Metaphysics IV. And by descending, i.e., whether it is possible from any demonstrated proposition to proceed again to another demonstration subsequent to it. Thus, one element of the doubt is whether demonstrations proceed to infinity either by ascending or by descending. The other element is whether demonstrations are mutually limiting, so that one demonstration may be confirmed by another in an ascending process, and from one demonstration another may proceed by a descending process: and this until a limit is reached.

Then (82a9) he shows that these questions are also relevant to negative demonstrations, because a negative demonstration must employ an affirmative proposition in which the subject of the conclusion is contained under the middle and from which the predicate of the conclusion is removed. Therefore, to the extent that there is ascent and descent in affirmative, there must be ascent and descent in negative syllogisms and propositions. For example, if the conclusion of a demonstrative syllogism is, “No C is A,” and the middle taken is B, from which A is removed: the first thing to be considered, therefore, is whether A is removed from B first and immediately, or whether there is another middle G to be taken, from which A would be removed before it would be removed from B. In that case it would be necessary to consider whether A would be removed from something else before G, namely, from H which is predicated universally of G. Therefore in these also the question arises whether one can proceed to infinity in removing (so that something would always remain from which it would have to be removed), or must one stop somewhere.

Then (82a15) he shows the cases in which these questions have no relevance: for in cases in which there is mutual predication and mutual conversion there is no prior and subsequent to be taken in the sense in which the prior [notion] is that with which a subsequent [notion] is not convertible, as universals are prior; because no matter whether the predicates be infinite, so that one might proceed to infinity in predicating, or whether there be infinity on both sides, i.e., on the side of the predicate as well as of the subject, all such infinites bear a like relationship to all, because any of them could be predicated of any other and be the subject of any of the convertibles. However, there can be this difference: one of them might be predicated as an accident and another as a predicament, i.e., as a substantial predicate. And this is the difference between a property, and a definition: although the two are convertible with the subject ,nevertheless the definition is an essential predicate and therefore naturally prior to the property, which is an accidental predicate. That is why in demonstrations we use the definition as the middle to demonstrate a proper attribute of the subject.

Lecture 32
(82a21-b34)
SOLUTION OF SOME OF THESE DOUBTS HINGES UPON SOLUTION OF OTHERS OF THESE DOUBTS

a2l. Now, it is clear— a24. For suppose that A— a30. Nor is it of any effect— a37. Further, if in affirmative— b4. For a negative conclusion— b13. In the second figure— b23. The third figure shows— b28. Even supposing that the proof

After raising the questions, the Philosopher here begins to settle them. And his treatment is divided into two parts. In the first he shows that the solution of some of the doubts is reduced to the solution of others. In the second he settles the doubt as to those items in which the difficulty lies per se as in its source (82b34) [L. 33]. Concerning the first he does two things. First, he shows that the doubt bearing on the middles is reduced to the one which is concerned with the extremes and is solved by the solution of the latter. Secondly, he shows that the doubt bearing on negative demonstrations is reduced to the one which is concerned with affirmative demonstrations (82a37). In regard to the first he does three things. First, he states his intended proposition. Secondly, he proves this proposition (82a24). Thirdly, he excludes a subterfuge (82a30).

He says therefore first (82a21) that it will be plain to anyone who considers the following reason that “an infinity of middles does not occur,” if the predications both upwards and downwards stop at certain terms, namely, at the highest predicate and the lowest subject. And he explains what upward and downward predication consists in, saying that one proceeds upwards when there is movement to the more universal, one of whose marks it is that it be predicated; but one proceeds downwards when there is movement to the most particular, one of whose marks it is that it functions as a subject.

Then (82a24) he shows what he has proposed in the following way: Let the case be that A is the highest predicate and F the lowest subject, and that there is an infinitude of middles, each of which we shall call B. Now since A was the first predicate, it will be predicated of some middle near it, and that middle of another middle below it. Since the middles are infinite, it follows that the predication will proceed downwards to infinity—which is contrary to what we are assuming. For it was assumed that the predications do not proceed downwards to infinity. The result is the same if we start at F, which is the lowest subject, and proceed upwards to infinity before A is reached—this too would be against our assumption. Therefore, if these are impossible, namely, that one may proceed to infinity by ascending or by descending, it will be impossible for the middles to be infinite. Thus it is clear that the question of an infinity of middles is reduced to a question of the infinity of the extremes.

Then (82a30) he excludes an objection. For someone might object saying that the aforesaid proof would hold if ABF, i.e., the middle and the extremes, were so related as to be “had” to one another, i.e., so that there would be no middle between them: for this is the way the Philosopher defines “had” in Physics V, namely, that it is next to something without anything between. And this seemed to be supposed in the above proof, namely, that A is predicated of some middle as though “had” to it, i.e., following it immediately. But one who posits infinite middles will say that this cannot be supposed. For he will say that between any two terms that are taken there is a middle.

But the philosopher says that it makes no difference whether the infinitude be of middles that are “had” to one another, the way discrete things are (for example, in a city. house is consecutive to house, and in numbers, unity to unity), or whether something “had” cannot be found in the middles although between any two middles it is always possible to find another, as happens in continua, in which, between any two signs, i.e., between two points, another can be found between them.

That this makes no difference one way or the other to the matter at hand he manifests in the following way: Granted that between A and F there is an infinitude of middles, each of which is called B, yet no matter which of these I employ, there is either an infinitude of middles between it and A and F, or there is not an infinitude of them between it and one or the other of the extremes. For example, let us suppose that the middles are mutually “had,” as happens in discrete things, and let us take a middle which is “had” to A; then it will be necessary that between that middle and F there is still an infinitude of middles; and similarly, if we assume a certain finitude of middles between that middle and A. And the same reasoning holds if the middle which is taken be joined immediately to F or is distant from it by a finitude of middles. From the fact, therefore, that from any given middle one must take an infinitude of middles to one or other of the extremes, it makes no difference whether it is joined to either extreme immediately, i.e., without a middle, or not immediately, i.e., through other middles: because even if it be joined to one extreme without a middle, it will still be necessary later to find an infinitude of middles in relation to the other. Consequently, it will always be required, if there is an infinitude of middles, to proceed to infinity in predications either by ascending or by descending, as the above proof showed.

Then (82a37) he shows that if there is no process to infinity in affirmadve demonstrations, then neither in negative demonstrations: and thus the question of negative demonstrations is reduced to the question of affirmative ones. He does three things in regard to this point. First, he proposes what he intends. Secondly, he proves what he proposed (82b4). Thirdly, he excludes an objection (82b28).

He says therefore first (82a37) that it will be clear from what follows that if in the predicative, i.e., in the affirmative demonstration, a stop is made at both, i.e., upwards and downwards, it will be necessary that a stop be made in the negative demonstration.

To elucidate what he is proposing he says: Let the case be such that from the ultimate, i.e., from the lowest subject one cannot go in ascending order to infinity toward universal predicates. And he explains that “ultimate” means that which is not in any other as in a less particular, but something else is in it, and let it be F. And let the case also be that one does not go to infinity when proceeding from the first to the ultimate. And he explains that the “first” means that which is predicated of others but nothing else is predicated of it as more universal than it. Thus the “first” is understood to be the most universal, and the “ultimate” the most particular. If, therefore, on both sides there be a stop in affirmative demonstrations, he says that as a consequence, there is also a stop in negative demonstrations.

Then (82b4) he proves his proposition. First, in the first figure. Secondly, in the second (8513). Thirdly, in the third (82b23). For a negative can be concluded in three figures.

He says therefore first (82b4) that there are three ways of demonstrating a negative proposition through which something is signified not to be. In one way in the first figure according to the mode that B is universally in C in the universal affirmative minor, but A is in no B in the universal negative major. Now since we are supposing that there is a stop in affirmatives, both upwards and downwards, it is necessary that the proposition, BC, which is affirmative, if it is not immediate but a space exists with middles between B and C, be reduced to immediates, because that space which exists between the middle term and the extreme is affirmative, in which a stop is supposed. But if we take the other space, which is between B and A, it is clear that if this proposition, “No B is A,” is not immediate, it is necessary that A be removed from something else before being removed from B. Let this be D. But if this D be taken as a middle between A and B, it is necessary that it be universally predicated of B, because the minor must be affirmative. And if this too is not immediate, i.e., “No D is A ‘ “ then A has to be denied of something prior to D, say E, which again will be predicated universally of D for the same reason. Therefore, since there is a stop in affirmatives when we ascend, as supposed, it follows that something is reached of which A should be denied first and immediately; otherwise one would go still further in affirmatives, as is clear from the foregoing.

Then (82b13) he proves the same thing for the negative which is concluded in the second figure. For let the case be that B, which is the middle, is predicated universally of A and denied universally of C, so that the conclusion is “No C is A.” Now if the negative premise needs to be demonstrated because it is mediate, it must be demonstrated either in the first figure in the mode of demonstrating concerning which we have shown that there is a stop, if there is a stop in affirmatives; or it must be demonstrated through this mode, i.e., in the second figure, or through a third mode, i.e., in the third figure. Now it has been established that there is a stop in negatives of the first figure, if there is a stop in affirmatives. Consequently, the same will now be demonstrated as to the second figure.

Therefore, let this proposition, “No C is B,” be demonstrated in such a way that D is universally predicated of B in the universal affirmative major and denied universally of C in the universal negative minor. Now if the proposition, “No C is D,” is mediate, it will be required to take some other middle which will be predicated universally of D and universally removed from C. Continuing thus, it will be necessary to proceed in negative demonstrations just as we do in affirmatives, namely, B will be predicated of A and D of B, and something else of D, and so on to infinity in affirmatives. But because we are supposing an upward stop in the affirmatives, it is also necessary to come to a stop in the negatives according to this mode in which a negative is demonstrated in the second figure.

Then (82b23) he shows the same thing in the third figure. Therefore, let B be a middle of which A is universally predicated, but C is universally denied of it: the conclusion will be a particular negative, namely, C is denied of some A. Now that there is a stop in the affirmative premise, “Every B is A,” is granted by our supposition. Furthermore, that there must be a stop in the negative, “No B is C,” which is the major, is evident, because if it had to be demonstrated, it would be done either “through what was said above,” i.e., through the first and second figure or in the way that the conclusion was concluded, namely, through the third figure, in which case this minor is not affirmed as universal but as particular. “But there is a stop in that way,” i.e., if one proceeds in the first and second figure. But if one proceeds in the third figure to conclude that “Some B is not C,” let a middle, E, be taken such that B is universally affirmed of it but C is particularly denied of it. “Then this happens once more in like manner,” i.e., according to this, one will always proceed in the negative demonstration by accumulating affirmative predications in descending order, because B which was the first middle will be predicated of E, and E of something else, and so on to infinity. But since we are supposing that there is a stop in the descending order in affirmatives, it is clear that there will be a stop in the negatives on the part of C.

Then (82b28) he excludes an objection. For someone could say that it is necessary to stop in negative propositions when there is a stop in the affirmatives, provided that one always syllogizes according to the same figure; but if one demonstrates now in one figure and now in another, one can go to infinity. And he say’~ that “it is obvious” that if one does not limit himself to one figure\in demonstrating but uses all, proceeding now in the first figure and now in the second and third, there must still be a stop in the negatives if there is one in the affirmatives. For these various ways of demonstrating are finite, and each of them will be enlarged not to infinity but finitely by ascending or descending, as was shown. Now if, finite things be taken a finite number of times, the result is finite. Hence it remains that in all the modes there must be a stop in negative demonstrations, if there is a stop in the affirmatives.

Lecture 33
(82b34-83a35)
THAT ONE DOES NOT PROCEED TO INFINITY IN ESSENTIAL PREDICATES IS SHOWN “LOGICALLY”

b34. That in fact the regress— a1. But as regards predicates— a21. It follows from this— a24. Predicates which signify— a27. White is a coincident— a33. The Forms we can dispense

After showing that if there is a stop in the extremes there must be a stop in the middles, and if there is a stop in affirmatives there must be a stop in negatives, the Philosopher now shows that there is a stop in affirmatives both upwards and downwards. And his treatment is divided into two parts. In the first part he shows his thesis “logically,” i.e., through characteristics common to every syllogism, which are based on predicates considered commonly. In the second he shows the same thing analytically, i.e., through characteristics proper to demonstration, which are based on per se predicates which are proper to demonstration (84a8) [L. 35]. The first is divided into two parts. In the first he shows that one does not proceed to infinity in predicates which are predicated in eo quod quid (i.e., pertaining to the essence of the subject]. In the second he shows universally that one does not proceed to infinity in affirmative predicates (83a1).

He says therefore first (82b34) that since we have established that an infinite process does not occur in negatives if there is a stop in affirmatives, our present task will be to show how one speculates through logical reasons that there is a stop “in those,” i.e., in affirmatives. (These reasons are called “logical,” because they proceed from certain common notions that pertain to the considerations of logic).

Now this truth is clear in regard to things predicated as constituting the essence of a thing, namely, the predicates from which the quod quid est, i.e., the definition, is formed. For if such predicates were agreed to be infinite, the result would be that nothing can be defined, and that if something is defined, its definition cannot be known: and all because the infinite cannot be traversed. For a thing cannot be defined, or its definition known, except by reaching the ultimate through descent and the first through ascent. Therefore, if something can be defined, or if a thing’s definition can be known, then in either case this consequence will follow, namely, there is no infinite process in the aforementioned predicates, but there is a stop in them.

Then (83a1) he shows universally that there is not a process to infinity in affirmative predicates. In regard to this he does two things. First, he prefaces certain things needed for establishing the thesis. Secondly, he establishes it (83a36) [L. 34]. In regard to the first he does two things. First, he distinguishes per accidens from per se predicates. Secondly, he distinguishes among the per se predicates (83a21).

He says therefore first (83a1) that since it has been established in regard to certain predicates that there is no infinite process in them, namely, in those which are predicated as pertaining to the essence, our task is to show that this is universally so in all affirmative predicates. And he begins his consideration with per accidens predicates in which there are three modes of true predication. One mode is when an accident is predicated of an accident, as when we say, “Something white is walking.” The second mode is when the subject is predicated of an accident, as when we say, “Something white is wood.” The third mode is when an accident is predicated of a-subject, as when we say, “Wood is white,” or “Man walks.”

Now these modes of predicating are mutually other and diverse, because when its subject is predicated of an accident, say “Something white is wood,” what is signified is that the universal predicate, “wood,” is being predicated of a subject which happens to be wood, i.e., this particular wood in which there is whiteness. For when I say, “Something white is wood,” the meaning is the same as “This wood which happens to be white is wood”; in other words, the sense is not that “white” is the subject of wood.

And he proves this on the ground that it is either according to its totality or according to a part that a subject comes to be that which is predicated of it as of a subject, as a man comes to be white. But neither the white nor its white part, which is really white, i.e., which pertains to the very essence of whiteness, becomes wood: for an accident is not the subject of change whereby wood comes to be from non-wood. But whatever begins to be such and such, comes to be it; therefore, if it does not come to be this, it is not it-unless it is granted that it always was this. But it was not always true to say that the white [object] is wood, because at some time the whiteness and the wood were not together. Therefore, since it is not true to say that the white [object] becomes wood, it is obvious that the white [object], properly and per se speaking, is not wood. Yet if it be granted that something white is wood, it is understood per accidens, namely, because that particular subject, which happens to be white, is wood. This, therefore, is the sense of any predication in which a subject is predicated of an accident.

But when I say, “Wood is white,” predicating an accident of the subject, I do not signify, as I did in the previous mode of predication, that there is something else substantially white, such that to be wood happens to it: which, of course, is signified both in the previous mode, where a subject is predicated of an accident, and in the other mode, where an accident is predicated of an accident, as when I say, “The musician is white”: for this signifies nothing but the fact that this particular man, say Socrates, who happens to be a musician, is white. But when I say, “The wood is white,” I signify that the wood itself has become the subject of white, and not that something other than the wood or other than a section of the wood has become white.

Therefore, there is a difference among these three modes of predicating, because when an accident is predicated of a subject, it is not predicated in virtue of some other subject; but when the subject is predicated of an accident or an accident of an accident, the predication is made in virtue of that which is subjected to the terms acting as the subject, of which another accident is predicated accidentally in the second case, and the species of the subject is predicated essentially in the first case.

Now since in each of the above modes we use the name, “predication,” and since it lies within our power to employ names as well as to restrict them, let us agree so to use our names in the following proof that only those things are said to be predicated which are said in this way, namely, not in virtue of some other subject. Consequently, whatever is said in another way, namely, by reason of another subject, as when a subject is predicated of an accident or an accident of an accident, shall not be said to be predicated—or if it is said to be predicated, it shall be said to be predicated not absolutely but per accidens. Furthermore, that which is after the manner of “white,” let us always take on the part of the predicate, and that which is after the manner of “wood,” let it be taken on the part of the subject. Therefore, in the following proof let us suppose this to be predicated which is predicated not per accidens, but absolutely, of that of which it is predicated. And the reason why we should use the word “predication” in this way is that we are speaking of demonstrative matters, and demonstrations use only such predications.

Then (83a21) he shows the differences among per se predicates. In regard to this he does two things. First, he distinguishes these predicates into diverse genera. Secondly, he shows the differences among them (83a24).

He says therefore first (83a21) that because we are saying that those things alone are being predicated which are predicated not in virtue of some other subject, and this is diversified according to the ten predicaments, it follows that whatever is thus predicated is predicated “either in quod quid est,” i.e., after the manner of an essential predicate, or after the manner of quality or quantity or one of the other predicaments discussed in the Categories. And he adds, “when one is predicated of one,” because if a predicate is not one but several, it cannot be said to be predicated precisely as quid or quale, but perhaps jointly as quale quid, as when I say “Man is a white animal.” Now it was necessary to add this because if several things are predicated of one in such a way that they function as one predicate, predications will be multipliable to infinity according to the infinite modes of combining predicates one to another. Hence when there is question of a stop in things that are predicated, it is necessary to take one thing predicated of one thing.

Then (83a24) he indicates the difference among the aforesaid predicates. In regard to this he does three things. First, he proposes the difference. Secondly, he clarifies it with examples (83a27). Thirdly, he excludes an objection (83a33).

He says therefore first (83a24) that those which signify substance must signify, in respect to that of which they are predicated, “what it truly is or something that it truly is.” This can be understood in two ways: in one way, as showing a distinction on the part of the predicate which might signify either the entire essence of the subject, as a definition does (and he signifies this when he says, “what it truly is”), or part of the essence, as a genus or a difference does (and he signifies this when he says, “or something that it truly is.”) In another way, and better, as showing a distinction on the part of the subject, which is sometimes convertible with an essential predicate, as the definitum with the definition (and he signifies this, when he says, “what it truly is”), and sometimes is a subjective part of the predicate, as man is of animal (and he signifies this when he says, “or something that it truly is,” for man is a certain anima1). But those which do not signify substance but are said of some subject (which subject is not truly, i.e., essentially, that predicate nor something of it), all such predicates are accidental.

Then (83a27) he clarifies the aforesaid difference with examples, saying that when we say, “Man is white,” the predicate is accidental, because man is not that which white truly is, i.e., to be white is not the essence of man “or anything of what white truly is,” as was explained above. But , when it is stated, “Man is an animal,” perhaps man is that which animal truly is. For animal signifies the essence of man, because that which is man is essentially animal. And although predicates that do not signify substance are accidents, they are not predicated per accidens. For they are not predicated of some subject in virtue of some other subject; for when I say, “Man is white,” “white” is not predicated of man by reason of some other subject’s being white in virtue of which man is called white as was explained above in regard to things predicated per accidens.

Then (83a33) he excludes an objection. For someone could say that predicates which signify the substance are not truly and essentially that which they are predicated of, nor are they something of it; and that accidents which exist in individuals as in subjects, do not correspond to any common essential predicates, because such universal predicates signify certain separated essences always subsisting by themselves, as the Platonists say.

But he answers that if Forms, i.e., Ideas, are assumed to exist, they should be happy, because according to the Platonists they have a nobler existence than the material things known to us. For the latter are particular and material, but the former universal and immaterial. For they are “premonstrations,” i.e., certain exemplars, of material things (taking “premonstrations” here as above, when we spoke of something being shown beforehand in order to prove something). Therefore, since they are the premonstrations or exemplars of natural things, it is necessary that in these natural things there be found certain participations of those Forms which pertain to the essences of these natural things. Hence if such separated Forms exist, as the Platonists contend, they have nothing to do with the present matter. For we are concerned with things, the science of which is produced in us through demonstration. And these are things existing in matter and known to us and concerning which demonstrations deal. Consequently, if it be granted that “animal” is something separated, an existing premonstration, as it were, of natural animals, then when I say, “Man is an animal,” in the sense that we use this preposition in demonstrating, “animal” signifies the essence of the natural thing concerning which the demonstration is made.

Lecture 34
(83a36-84a7)
LOGICAL REASONS WHY ONE DOES NOT PROCEED TO INFINITY IN PREDICATES

a36. If A is a quality of B— a37. Therefore A and B— a39. For one alternative— b1. But it has been shown— b8. Hence they will not be— b10. Nor (the other alternative)— b13. On the other hand— b24. Subject to these assumptions— b33. The argument we have given

Having set forth the distinction of predicates from one another as a necessary preliminary to demonstrating his proposition, the Philosopher now undertakes to show his proposition, namely, that there is no infinite process in predicates. And his treatment falls into two parts, according to the two ways in which he shows his proposition, the second part beginning at (83b33). Concerning the first he does two things. First, he shows that one does not proceed to infinity in predicates after the manner of circularity (see Lect. 8). Secondly, that one does not proceed to infinity in a direct line upwards or downwards (83b13). Concerning the first he does three things. First, granting what has gone before, he adds certain things required for showing his proposition. Secondly, from these and other established facts he concludes the proposition (83a37). Thirdly, he proves it (8309).

First, therefore (83a6) he proposes two things: one of these is that since a predicate which signifies an accident signifies some genus of accident, such as quality, it cannot occur that two things be so related to one another that the first is a quality of the second and the second a quality of the first: for the nature of a quality and that of which it is the quality are diverse. The second is that universally it is not possible that a quality have some other quality inhering in it, because no accident is the subject of another accident, absolutely speaking. For to substance alone does the notion of subject properly belong.

Then (83a37) he proposes, as though concluding from these antecedents, what he intends to prove. He says, “If our preliminary rules are true, it is impossible that there be equal reciprocal predication,” i.e., according to any of the above-mentioned ways. But this does not exclude the possibility of one thing’s being truly predicated of another and conversely. For we say truly that man is white and that something white is a man. However, this is not done equally, i.e., according to an equal manner of predicating. And it is the same in essential predicates.

Then (83a39) he shows the proposition. First, in essential predicates Secondly, in accidental ones (83b10). In regard to the first he does three things. First, he lays down a division of essential predicates. Secondly, he recalls something established above (83b1). Thirdly, he proves the proposition (83b8).

He says therefore first (83a39) that in order to show that they are not reciprocally predicated equally, it will be necessary to consider this in essential predicates. For that which is predicated equally will be predicated either as a substance or some other way: and if as a substance, then either as a genus or as a difference. For these are the two parts of a definition which signify the essence.

Then (83b1) he recalls what he had proved above at the beginning of the previous lecture, namely, that predicates of this kind are not infinite; because if they were to proceed to infinity, reciprocity or circularity would find no place therein. He says, therefore, that as has been shown above, an infinite process does not take place in such predicates either by ascending or by descending: for example, if “two-footed” be predicated of man, and “animal” of two-footed, something else of animal, there is no process to infinity either upwards or downwards. Thus, if “animal” were said essentially of man, and “man” of Callias, and this of somone else (supposing that there were a genus containing under it species, one of which would be Callias), it would not be possible to go on in this way to infinity. And he recalls the reason he used earlier to prove this: for every such substance which has something more universal that can be predicated of it and which can be predicated of some inferior is capable of being defined; but the most general genera of which other more universal things are not predicated, and singulars which are not predicated of any inferiors, are not capable of definition. Only the substance between these can be defined. But a substance of which an infinite number of things is predicated turns out to be indefinable, because one who would define it must go through and understand all the items which are substantially predicated of the defined, since all of them occur in the definition either as a genus or as a difference. But the infinite cannot be gone through. Therefore, it is required that neither a universal substance, which is not a supreme genus, nor a lowest subject, can have an infinitude of predicates which are predicated of it substantially. Consequently, there is no infinite process either upwards or downwards.

Then (83b8) he shows that there is no process to infinity after the manner of circularity in substantial predicates. And he says that if certain substantial predicates are predicated as genera of something, they are not predicated of one another equally, i.e., convertibly, i.e., so that one would be the genus of another and vice versa. To prove this he continues, “for the one will be what something truly is.” As if to say: if something is predicated of something as a genus, that of which it is predicated is something which truly is that, i.e., something particular which receives that predication substantially. Therefore, if this be predicated of that as a genus, it will follow that something which belongs to something particularly would conversely receive the predication of it, which is tantamount to saying that a same thing is both a whole and a part in relation to the same thing—which is impossible. And the same reasoning applies to differences. Hence in Topics I it is stated that problems concerning a difference are reduced to problems concerning the genus.

Then (83b10) he shows that there cannot be an infinite circular process in predications in which an accident is predicated of a subject. And he says that there cannot be conversion of a quality with its subject; furthermore, none of the others which are predicated accidentally can have that sort of predication, unless the predication be made per accidens in the sense described above, namely, that accidents are not predicated of their subjects except per accidens. For quality and all these others are accidental to the subject; hence they are predicated of their substances as an accident of a subject.

Then (83b13) he shows universally that in no genus of predication is i liere an infinite process upwards or downwards. And he says that not only is there no infinite process in predications according to circulation, but also that in proceeding upwards or downwards the predicates will not be infinite. To prove this: First, he recalls certain things which were established above. Secondly, from these he concludes his proposition 83b24).

In regard to the first he reaffirms (83b13) that certain things can be predicated of a subject, whether they signify quality or quantity or any other genus of accident, or even items which constitute the substance of a thing, i.e., essential predicates.

Secondly, he reiterates, that the latter, i.e., the substantial predicates, are finite.

Thirdly, he reaffirms that the genera of predicaments are finite, namely, quality, quantity and so on. For if someone were to say that quantity is predicated of substance, and quality of quantity, and so on to infinity, he excludes this on the ground that the genera of predicaments are finite.

Fourthly, he reaffirms that, as stated above, one thing is predicated of one thing in simple predication. And he mentions this because someone might say that one thing may well be first predicated of one thing, as “animal” of man, and this predication will be multiplied until something else can be found predicable of man, and when this is found, two things, will be predicated of one, so that it will be said that man is a white animal. Thus, many more predicates might be found according to various combinations of predicates. And so by continually adding to this number, the predicates will be increased more and more, so that there will be a process to infinity in predicates, just as there is in the succession of numbers. But he excludes this by predicating one of one.

Fifthly, he repeats that we should not say that certain items are predicated absolutely “of things which are not something,” i.e., of accidents, none of which is a subsistent being. For, as shown above, neither the, subject nor an accident is properly predicated of an accident. For all things of this sort that are not substantial are accidents, and nothing is predicated, simply speaking, of such things. Yet they whether they be substantial predicates or accidental, they are predicated per se of their subjects; but if they are subjects or accidents being predicated of an accident, they are predicated in another way, i.e., per accidens. For it belongs to the very notion of all accidents that they be said of a subject; and since an accident is not a subject, nothing can, properly speaking, be predicated of it, because “none of such,” i.e., no accidents, “are stated to be such that they are said to be that which is said,” i.e., stated to be such that they themselves, rather than something else distinct from them, receives the predication of that which is predicated of them, as happens in the case of substances. For man is not called “animal” or “white” because something else is “animal” or “white,” but because the very thing which is a man is animal or white; but a white thing is called “man” or “musician” because something else, namely, the subject of white is the man or musician. But the accident itself is in something distinct from it; and items which are predicated of an a accident are predicated of something other than that accident, namely, of the subject of the accident: it is on this ground that they are predicated of the accident, as has been stated. (Now he mentioned this because if an accident is predicated of a subject and vice versa, and all accidents of a subject can be mutually predicated of one another, it will follow that predication could proceed to infinity, because an infinitude of things can happen to one thing).

Then (83b24) from these premises he shows his proposition, namely, that in a predication in which one thing is predicated of one thing there is no upward or downward process to infinity, because, as the fifth supposition states, all accidents are predicated of items which pertain to the substance of the thing. Furthermore, according to the second supposition, substantial predicates are not infinite; consequently, on the part of the subjects there is no infinite process downwards in these predications. Again, neither of these is infinite in the upward movement, i.e., neither the substantial nor the accidental predicates, both because the genera of accidents are finite, and because there is no infinite process upwards or downwards in any of these genera any more than there is in substantial predicates, because in each predicament the genus is predicated of a species in regard to something essential. Hence we can conclude universally that there must be some first subject of which something is predicated, thus establishing a stop in downward predication; then something else will be predicated of this, but it will come to a stop in the ascending process, so that something will be found which is not found predicated of another, either as the subsequent is predicated of its prior per accidens, or as the prior is predicated per se of its subsequent. This, therefore, is one way of demonstrating the proposition logically, and it is based on the diverse modes of predicating.

Then (83b33) he sets forth the second way of proving. And he says that when a proposition in which something is predicated of a subject is such that certain things can be predicated per prius of that subject, that proposition will be demonstrable: for example, this proposition, “Man is a substance,” is demonstrated by the proposition, “Animal is a substance,” because “substance” is predicated of animal before it is predicated of man. But if a proposition is demonstrable, there is no better way of knowing it than to know it by demonstration, just as we know indemonstrable principles better than by way of demonstration because we know them as self-evident. Besides, we cannot scientifically know such demonstrable propositions except through demonstration, because a demonstration is a syllogism productive of scientific knowledge, as we showed above. Furthermore, if a proposition is known through another one, and if we do not scientifically know the one through which it is known, or know it in the way which is better than knowing scientifically, then we do not scientifically know that proposition which is made known through it.

With these three suppositions in mind he proceeds thus: If one does know something purely through demonstration “and not from something or from supposition,” it is necessary that there be a stop in the predicates which are-taken as middles. (He says, “purely and not from something,” to exclude) demonstrations leading to the impossible, in which one proceeds, against certain positions by arguing from propositions that have been agreed upon. Furthermore, he says, “or from supposition,” to exclude such demonstrations as are formed in subalternate sciences, which suppose the conclusion of higher sciences, as explained above).

Therefore, one knows purely through demonstration when each one of the premised propositions, if it is demonstrable, is known through demonstration; and if it is not demonstrable, is known in virtue of itself. Under these suppositions it is necessary that there be a stop in predications, because if there is no stop but something prior can always be taken, it follows that there is demonstration of everything, as was said above. Therefore, if a. conclusion is demonstrated, each of the premises must be demonstrable. But if we can have knowledge of it in no better way than by knowing it through demonstration, and if it will be necessary to demonstrate it through other propositions, and those through others again, and so on to infinity, then, since it is not possible to go through those infinites, we shall not be able to make it known through demonstration or through the method better than demonstration, since all things are demonstrable. The consequence will be that one knows nothing purely through demonstration, but only from supposition.

Finally, by way of summary he concludes the main proposition.

Lecture 35
(84a8-b2)
‘THAT THERE IS NOT AN INFINITE PROCESS UPWARD OR DOWNWARD IN PREDICATES IS SHOWN ANALYTICALLY

a8. but an analytical process— a10. Demonstration proves— a11. Now attributes may be— a18. In neither kind of— a23. Note, moreover, that all— a25. Attributes which are— a28. If this is so— a29. An immediately obvious

After showing logically that there is no infinite process upwards or downwards in predicates, the Philosopher now shows the same thing analytically. And his treatment falls into two parts. In the first he shows the principal proposition. In the second he infers certain corrolaries from the aforesaid (84b3) [L. 36]. Concerning the first he does two things. First, he proposes what he intends. Secondly, he proves his proposition (84a10).

He says therefore first (84a8) that the fact an infinite process upwards or downwards does not occur in the demonstrative sciences with which we are concerned can be more briefly and quickly manifested analytically than it was logically. Here we might note that analytic, i.e., demonstrative, science which is called judicative, because it resolves to self-evident principles, is a part of logic which even contains dialectics under it. However, it pertains to logic in general to consider predication universally, i.e., as containing under it predication which is per se and predication which is not per se. But predication per se is proper to demonstrative science. Therefore, above he proves his proposition logically, because he showed universally in every genus of predication that there is no infinite process. But here he intends to show it analytically, because he proves it only in things which are predicated per se. And this is a more efficient way; furthermore, it suffices for our purpose, because that is the only mode of predication we use in demonstration.

Then (84a10) he shows his proposition concerning which he does three things. First, he proposes which predication analytic, i.e., demonstrative, science employs, for it uses per se predication. Secondly, he recalls how many modes there are of such predication (84al 1). Thirdly, he shows that there cannot be an infinite process in any mode of per se predication (8408).

He says therefore first (84a10) that demonstration is concerned exclusively with items that are per se in things. For such are its conclusions and from such does it demonstrate, as was established above.

Then (84a 11) he lays down two modes of predicating per se. For in the first place those things are predicated per se which are present in their subjects as constituting their essence, namely, when predicates are placed in the definition of a subject. Secondly, when the subjects themselves are in the essence of the predicate, i.e., when the subjects are placed in the definition of the predicate. And he gives examples of each of these ways: for “odd” is predicated per se of number in the second way, because “number” is placed in the definition of odd. For the odd is a number not divisible by two. Multitude or divisible, however, are predicated of number and are present in its definition; hence these are predicated per se of number in the first way. The other ways, which he mentioned previously, are reduced to these.

Then (84a18) he shows that there must be a stop in each of these modes of per se predication. In regard to this he does three things. First, he shows that there must be a stop both upwards and downwards in both of these modes of per se predication. Secondly, he concludes that there cannot be an infinitude of middles (84a28). Thirdly, he concludes that one cannot proceed to infinity in demonstrations (84a29). Concerning the first he does two things. First, he shows his proposition in regard to the second way of saying per se,” namely, when the subject is placed in the definition of the predicate. Secondly, in the first way, when the predicate is placed in the definition of the subject (84a25). In regard to the first he gives two reasons, in the first of which he proceeds in the following way.

First (84a18) he states ‘his proposition, namely, that in neither of these modes of saying per se does an infinite process occur. Then he proves this of the second mode, as when “odd” is predicated of number. For if one goes further and states that something else should be predicated per se of “odd” according to that mode of saying per se, it follows that “odd” is present in its definition. But “number” is placed in the definition of odd; hence it will follow that “number” is also present in the definition of that third thing which is present per se in “odd.” However, this cannot go on to infinity, so that an infinitude of things would be in the definition of something, as was established above. It remains, therefore, that in such per se predications an upward process to infinity does not occur, i.e., on the side of the predicate.

He presents the second reason (84a23) and says that no matter how fair one advances in these per se predications of the second mode, it will be required that all the predicates taken in order be in their first subject, say in number, as predicated of it, because if “odd” is predicated per se of number, it will be required that whatever is predicated of odd be also predicated of number. And it is further required that “number” be in all of them, because if “number” is placed in the definition of odd, it has to be placed in the definition of all those things which are defined by “odd.” And thus it follows that they are mutually in one another. Therefore, they will be convertible and none of wider extent than another: for this is the way proper attributes are related to their subjects. Hence, even though there be an infinitude of per se predicates in this way, it offers nothing to the purpose of one who intends to establish that there is an infinitude of predicates upwards or downwards.

Then (84a25) he proves his own point in regard to the first mode of saying per se, and he states that those things which are predicated in essence, i.e., as pertaining to the definition of the subject, cannot be infinite; otherwise, definition would be impossible, as we showed above. From this, therefore, he concludes that if all the items predicated in demonstrations are predicated per se, and if there is no infinite upward process in per se predicates, it is necessary that the predicates in demonstrations stop in the upward movement. And from this it also follows that they must stop in the downward movement, because no matter on which side infinity is posited, science and definition are destroyed, as is evident from what has been said above.

Then (84a28) he concludes from the foregoing that if there is a stop upwards and downwards, the middles cannot be infinite. For it has been established above that if the extremes are determinate, there cannot be an infinitude of middles.

Then (84a29) he further concludes that there is no infinite process in demonstrations. And he says that if the above statements are true, it is necessary that there be certain first principles of demonstration that are not demonstrated; consequently, there will not be demonstration of everything, as some claim, as was stated in the beginning of this book.

Then he goes on to show that his consequence follows. For if it is granted that there are certain principles of demonstrations, it is necessary that they be indemonstrable: for since every demonstration proceeds from things that are prior, as has been established above, then if the principles are demonstrated, it will follow that something would be prior to the principles, and this is contrary to the notion of a principle. And so, if not all things are demonstrable, it will follow that demonstrations do not proceed to infinity.

But all these follow from what has been established, namely, from the fact that there is no infinite process in middles, because the position that any of the foregoing statements is true, i.e., that demonstrations proceed to infinity, or that all things are demonstrable, or that there are no principles of demonstrations is tantamount to the position that no distance is immediate and indivisible’ i.e., to the position that the two terms of any affirmative or negative proposition belong together only in virtue of a middle. For if any proposition is immediate, it follows that it is indemonstrable; because when something is demonstrated, it is necessary to take a term by interposing, i.e., by setting it between the subject and predicate, so that the predicate will be predicated of that term before being predicated of the subject-or removed from it. But the middle in demonstrations is not taken by assuming extraneously; for this would be to assume an extraneous middle and not a proper middle-which occurs in contentious and dialectical syllogisms. Therefore, if demonstrations were to proceed to infinity, it would follow that there is an infinitude of middles between two extremes. But this is impossible if, as has been established above, the predications stop in the upward and downward process. But as we have shown, first logically and then analytically, these predications do stop both upwards and downwards, as explained. Therefore, in virtue of this conclusion finally induced, he manifests the intent of the entire chapter and why each proposition was introduced.

Lecture 36
(84b3-85a11)
CERTAIN COROLLARIES FROM PRECEDING LECTURES

b3. It is an evident corollary— b6. Isosceles and scalene— b9. for, were it so,— b15. Yet if the attribute— b19. It is also clear that— b24. Similarly, if A does not— b32. When we are to prove— b38. And as in other spheres— a2. In syllogisms, then,— a3. In the case of negative— a7. If we have to show that— E a10. In the third figure

After showing that a process to infinity does not occur in demonstrations, the Philosopher here adduces certain corollaries from what has been established. In regard to this he does two things. First, he shows that it is necessary to accept certain first propositions. Secondly, how those first things are to be used in demonstrations (8032). Concerning the first he does two things. First, he shows that it is necessary to arrive at a first, when one thing is predicated of several. Secondly, when one thing is predicated of one (84b19). In regard to the first he does four things. First, he states the intended proposition. Secondly, he manifests the proposition (84b6). Thirdly, he proves it (84b9). Fourthly, he excludes an objection (84b15).

He says therefore first (84b3) that having demonstrated the aforesaid, namely, that there is no process to infinity in predications and demonstrations, it is clear that if something is predicated of two things, say A of C and of D, such that one of them is not predicated of the other, i.e., either not at all, as animal is predicated of man and ox, but neither of them is predicated of the other in any way; or not in all cases, as animal is predicated of man and male, neither of which is universally predicated of the other: when the terms, I repeat, are thus related, it is clear that there is no need for that predicate, which is predicated of both, to be in them in virtue of something common to both and this again in virtue of something else, and so on to infinity.

Then (806) he cites an example to clarify what he is proposing. For there are two species of triangle, one of which is scalene (none of whose three sides is equal to any other) and the other is isosceles (having two sides that are equal). But neither of these species is predicated of the other, and yet this proper attribute of having three angles equal to two right angles is present in both. Furthermore, this attribute is present in them in virtue of something common, namely, that each is a certain figure, namely, a triangle. However, this process does not continue forever so that something would always belong to it in virtue of something else and so on to infinity, so that “having three” would belong to triangle in virtue of something else again, and so on to infinity.

Then (84b9) he proves his proposition and says: Let the case be that B is predicated of C and of D in virtue of this common feature A. It will then be clear that B will be in C and in D in virtue of that common feature which is A. Now if A in turn is in them in virtue of something common, which again is in them in virtue of some other common item, there will be an infinite process in the middles. It follows, therefore, that between the two extremes C and B there falls an infinitude of middle terms. But this is impossible. Therefore, if a same thing is in several things, it is not necessary that it be in them always in virtue of something else ad infinitum, because it is necessary to reach certain “immediate distances,” i.e., certain immediate predications, which he calls “distances,” as explained above.

As can be seen from this proof of Aristotle’s, it is not his understanding that it is not always true that when some item is predicated of several, which are not in turn predicated of one another, that item is not in the several in virtue of something common. For this is true in everything predicated as a proper attribute: for if it is in several, it is required that it be in them in virtue of something common which might even be nameless, as we explained above when we treated concerning the universal. But there is no infinite process in that which is common, as this reason introduced by the Philosopher clearly proves. However, if something be taken which is in several as a genus in its species, it will not always be necessary to find something prior in virtue of which it is in them. For example, “living” is in man and in ass in virtue of something prior, namely, in virtue of “animal,” but it is not in animal and plant in virtue of something prior, because these are the first species of “living,” i.e., of animate, body.

Then (84b15) he excludes an objection. For someone could say that it is always taken in virtue of something common, in the sense that something from another genus might be found common to them: for example, if we say that “to be self-movent” is in man and in ass in virtue of this common feature which is “animal” and also in virtue of some other common feature such as “having quantity” or having color or other things of this sort that can be taken ad infinitum.

In order to exclude this he says that it is required of the middle terms that they be taken from the same genus and “from the same atoms,” i.e., indivisibles. (By “atoms” he means those extreme terms between which the middle must be taken, if that common item which is taken as a middle term is to be numbered among things which are predicated per se). Why the middle terms must be taken from the same genus he shows from the fact that, as stated above, a demonstration does not cross from one genus into another.

Then (84b19) he shows that it is necessary to arrive at a first among the predicables in which one thing is predicated of one. First, in affirmatives. Secondly, in negatives (84b24).

He says therefore first (84b19) that it is clear that when A is predicated of B, if there is a middle between them, we can use that middle to demonstrate that A is in B: and these are the principles of this kind of conclusion. And what things soever be taken as middles are the principles of the mediate conclusions which are concluded through them. For the “elements” or principles of demonstrations are none but immediate propositions: and I mean “either all such propositions or universal ones.”

Now this can be understood in two ways: in one way so that universal proposition is taken as set off against singular. For it is not in virtue of a middle that the most special species is predicated of a singular. Hence this proposition, “Socrates is a man,” is immediate, although it is not a principle of demonstration, because demonstrations are not concerned with singulars, since there is no science of such. Consequently, not every immediate proposition, but only one that is universal, is a principle of demonstration. In another way it can be understood in the sense that of all the propositions of any science, the universal propositions are the common propositions, such as “Every whole is greater than its part.” Hence these are absolutely the principles of demonstrations and self-evident to all. But the proposition, “Man is an animal,” or “The isosceles is a triangle,” is not a principle of demonstration throughout the science but only for some particular demonstrations; hence such propositions are not self-evident to all.

Consequently, if there is a middle for a given proposition, one will demonstrate through a middle until something immediate is readied. But if there is no middle for a given proposition, it cannot be demonstrated. But ihis is the way to find the chief principles of demonstrations, namely, to proceed by analysis from mediates to immediates.

Then (84b24) he shows that it is necessary to arrive at a first in negatives, saying that if A is denied of B, if there is a middle to be taken from which A is removed prior to being removed from B, then this proposition, “B is not A,” will be demonstrable. But if there is no such middle to be taken, that proposition will not be demonstrable but will be a principle of demonstration. And there are as many “elements,” i.e., principles of demonstrations as there are terms at which a stop is reached in such a way that there is no further middle to find. For propositions formed of such terms are principles of demonstrations. Thus, if C be predicated immediately of B, and A be immediately removed from B or immediately predicated of it, B is the final term reached among the middles to be taken; hence each proposition will be immediate and a principle of demonstration.

Furthermore, it is clear from the foregoing that just as there are certain indemonstrable affirmative principles in which one thing is predicated of one thing by signifying that this is essentially that (as when a genus is predicated of a proximate species) or that this is in that (as when a proper attribute is predicated of its proper and immediate subject), so there are indemonstrable principles in negatives, namely, by denying that something is an essential predicate or a proper attribute. From this it is evident that there are certain principles of demonstration for demonstrating an affirmative conclusion (which must be concluded from all affirmative propositions), and there are certain principles of demonstration for proving a negative conclusion (to infer which requires that something negative be taken).

Then (84b32) he shows how first propositions are to be used in demonstrating. First, in affirmative demonstrations. Secondly, in negatives (85a3). In regard to the first he does three things. First, he shows how first and immediate propositions should be taken in demonstrations. Secondly, how such propositions are related to demonstrations (8038). Thirdly, he summarizes (85a2).

He says therefore first (84b32) that when one is required to demonstrate an affirmative conclusion, for example, “Every B is A,” it is first necessary to take something which is first predicated of B before A is, and of which A is also predicated, say C. Furthermore, if there is something of which A is predicated before A is predicated of C, and something ahead of that, and so on, then no proposition and no term signifying a being outside of A will be taken in the demonstration, because it will be necessary that A be predicated of it per se in such a way that it is contained under it and is not extraneous to it; but it will always be necessary to condense (crowd together] the middles. (Here he is speaking according to an example of men who appear crowded together when they are sitting on a bench and there is no room for anyone else to sit between any two who are seated; in like fashion, the middles in demonstrations are condensed when no middle falls between the terms taken). And this is his meaning when he says, “the middle is condensed until the spaces become indivisible,” i.e., the distances between two terms are such that they cannot be divided into several such distances, but there is only one space. And this occurs when a proposition is immediate. For it is then that a proposition is one, not only actually but potentially, when it is immediate. For if it is mediate, then although it is actually one (because one thing is predicated of one thing), it is in fact several in potency, because when the middle is taken, two propositions are formed. In the same way, a line which is actually one, inasmuch as it is continuous, is several in potency, inasmuch as it is divisible at an intermediate point. Hence he says that an immediate proposition is one as a simple indivisible.

Then (8038) he shows the relation between an immediate proposition and a demonstration. Here we should note that, as it is stated in Metaphysics X, in every genus there must be one first thing which is the most simple in that genus and is the measure of all the things in that genus. And because a measure is homogeneous to the thing measured, such first indivisibles will vary according to the diversity of genera. Hence these will not be the same in all genera: but in regard to weights one indivisible, the ounce or mina, is taken as the minimum weight (even though it is not the absolute minimum, because every weight is further divisible into smaller weights, but it is taken as a minimum by supposition). Again, in melodies there is taken as the one principle “a tone,” which consists in a proportion of an octave and a half or a “diesis,” which is the difference between a tone and a semi-tone. Similarly, in diverse genera there are diverse indivisible principles.

Now the principles of a syllogism are propositions; hence it is required that the most simple proposition, which is immediate, be the unit which is the measure of syllogisms. But demonstration has, over and above a syllogism, the added feature that it causes science. Now “understanding” and science are related as the indivisible unit is related to the many. For science is effected by going from principles to conclusions, whereas “understanding” is the absolute and simple acceptance of a self-evident principle. Hence “understanding” corresponds to the immediate proposition, and science to a conclusion, which is a mediate proposition. Consequently, the indivisible unit of a demonstration regarded as a syllogism is the immediate proposition. But on the part of the science which it causes, the unit is “understanding.”

Then (85a2) he sums up and concludes what was established above, namely, that the middle in affirmative syllogisms does not fall outside the extremes.

Then (850) he shows how to use immediate propositions in negative syllogisms. First, in the first figure. Secondly, in the second (85a7). Thirdly, in the third (8500).

He says therefore first (85a3) that in negative syllogisms in the first figure, none of the middles taken while proceeding to immediates will fall outside the genus of the terms of an affirmative proposition; for example, if it is to be demonstrated that “No B is A,” and the middle taken is C, we have the following syllogism: “No C is A; but every B is C: therefore, No B is A.” Now if it should be necessary to prove that “No C is A,” one will be required to take a middle of C and A, i.e., one that will be predicated of C and consequently of B; thus it will belong to the genus of the terms of the affirmative proposition. And so it will always turn out that the middles taken will not fall outside the affirmative proposition, although they will fall outside the genus of the negative predicate, i.e., outside the genus of A.

Then (85a7) he shows how it is in the second figure, saying that if one is required to demonstrate in the second figure that “No E is D,” by taking C as middle, so as to form the syllogism, “Every D is C; but No E is C or Some E is not C: therefore, No E is D or Not every E is D,” the middle term taken will never fall outside of E. For if it should be necessary to demonstrate that “no E is C,” one will have to find a middle between E and C, because in the second figure it will always be necessary to prove a negative, since an affirmative cannot be concluded in this figure. Hence, just as in the first figure the middles are always taken on the side of the affirmative proposition, so in the second figure the middles will always be taken on the side of the negative proposition.

Then (85a10) he shows how it is in the third figure, saying that in the third figure the middles which are taken will not be outside the predicate which is denied, nor outside the subject of which it is denied. The reason for this is that the middle is the subject in both propositions, whether affirmative or negative: hence if it is necessary to take yet another middle, it will again have to act as the subject of both, whether affirming or denying. And so the middles which are taken will never be taken outside the predicate which is denied, or outside the subject of which it is denied.

Lecture 37
(85a12-b21)
WHETHER UNIVERSAL DEMONSTRATION IS STRONGER THAN PARTICULAR DEMONSTRATION

a12. Since demonstrations may— a17. Let us first examine— a20. The following considerations— a3l. The universal has not— b4. We may retort— b15. If there is a single— b18. Because the universal has

After determining about the demonstrative syllogism, the Philosopher now treats of the comparison of demonstrations one to another. And because science is caused by demonstration, his treatment is divided into two parts. In the first he treats of the comparison of demonstrations. In the second of the comparison of sciences (87a31) [L. 41]. In regard to the first he does three things. First, he raises a doubt concerning the comparison of demonstrations. Secondly, he lays down the order of procedure (85a17). Thirdly, he deals with the doubts raised (85a20).

He says therefore first (85a12) that demonstration is divided in three ways: for in one way it is divided into universal and particular; in another way into categorical and privative, i.e., affirmative and negative; in a third way into that which demonstrates ostensively and that which leads to the impossible. In each division, therefore, the question arises as to which is the stronger.

Then (85a17) he shows what order should be followed, saying that the comparison of universal to particular demonstration should be treated first. And when this has been done, we shall speak of demonstrations which demonstrate something affirmatively and of those which demonstrate to the impossible, namely, whether the affirmative is stronger and whether the demonstration to the impossible is stronger.

Then (85a20) he deals with the problems he has proposed. First, of the comparison of the particular with the universal. Secondly, of the comparison of the affirmative with the negative (86a32) [L. 39]. Thirdly, of the comparison of the ostensive with that which leads to the impossible (87a1) [L. 40]. Concerning the first he does three things. First, he proposes reasons to show that the particular demonstration is more powerful than the universal. Secondly, he solves them (85b4). Thirdly, he gives reasons to the contrary (85b22) [L. 38].

In regard to the first he sets forth three reasons, remarking that in virtue of these reasons to be set down forthwith, it will perhaps seem to some that a particular demonstration is of more value than a universal one.

The first reason (85a20) is this: That demonstration is stronger through which we know best in a scientific way. And he proves it on the ground that the strength of a demonstration consists in knowing in a scientific way. For the most that a thing can do is called its strength: for the strength of a man able to carry 100 pounds is not that he can carry ten but that he can carry 100, which is the limit of his power, as it is stated in On the Heavens I. Now the most that a demonstration can do is to cause scientific knowledge; consequently, that is the strength of a demonstration. But a thing is more perfect to the extent that it attains the strength appropriate to it, as is clear from Physics VII. Hence this proposition is quite evident, namely, that the better a demonstration causes scientific knowledge, the stronger it is. (He assumes that we know a thing better when we know it according to itself than when we know it according to something else: thus we know more in regard to Coriscus when we know that Coriscus himself is a musician than when we merely know that some man is a musician). And this proposition is true, absolutely speaking, because that which is per se is always prior to and the cause of that which is through something else, as it is stated in Physics VIII.

From these facts the conclusion is gathered that a demonstration which makes one know something according to itself is stronger than one which makes one know something according to something else. But a universal demonstration demonstrates and makes one know something not according to itself, but according to something else, namely, according to the universal, as that a triangle with two equal sides, i.e., an isosceles, “has three,” not because it is isosceles but because it is a triangle. A particular demonstration, on the other hand, demonstrates about a particular thing according to itself. Hence according to this it follows that a particular demonstration is stronger than a universal.

Then (85a31) he gives a second reason. It is this: The universal is not something apart from singulars, as was proved in Metaphysics VII. But a universal demonstration leads one to think, from the very manner of its demonstration, that the universal is “something,” i.e., a certain nature in the realm of beings: for example, when it demonstrates something of triangle apart from particular triangles, and of number apart from particular numbers.

To these two propositions he adds two others: to the first one, which stated that the universal is not something apart from the singulars, he adds this proposition, namely, that a demonstration concerned with being is stronger than one concerned with non-being. But to the second proposition, which stated that a universal demonstration leads one to think that a universal is something existing as a real nature, he adds another proposition, namely, that a demonstration which does not cause error is stronger than one which leads to error.

Then he shows that one is led into error on account of a universal demonstration, because one who proceeds according to a universal demonstration demonstrates concerning some universal as of some analogue, i.e., as of something common which is referred to many things proportionally, as though there existed something common which is neither a line nor a number nor a solid, i.e., a body, nor a plane, i.e., a surface, but “something apart from these,” i.e., a universal quantity; or “something owing to them,” i.e., something which must be posited, if they are to have the formality of quantity.

Thus, therefore, in virtue of two middles, equivalent as it were to two arguments, he concludes to one conclusion, saying that if the universal demonstration is as described, i.e., is less an entity than the particular, and more likely to create a false opinion than the particular, it follows from these two middles that the universal ranks lower than the particular.

Then (85b4) he solves these reasons in order. First, he solves the first one, saying that the first, i.e., the ground on which the first reason rests, is no different in the universal than in the particular, because in both cases we find something which is according to itself and something according to something else. And he shows that something which is according to itself is found in the universal. For “to have three angles equal to two right angles” does not belong to isosceles according to itself, i.e., precisely as isosceles, but according as it is a triangle. Consequently, one who knows that a certain triangle, namely, the isosceles, “has three,” has less knowledge of that which is per se than if he knew that a triangle “has three.” And it must be admitted universally that if there be any characteristic which does not belong to triangle as triangle, but that characteristic is nevertheless demonstrated of it, the demonstration will not be true. But if it is in it precisely as it is a triangle, then by knowing it in a universal way of triangle precisely as of triangle, he has a more perfect knowledge.

From these facts, therefore, he concludes a conditional statement in whose antecedent three things are placed: one is that “triangle” is in more things than isosceles is; the second is that “triangle” is predicated of isosceles and of those others according to the same formality and not equivocally; the third is that “having three angles equal to two right angles” is present in every triangle. With these three suppositions, the consequent is that “the having of three” does not belong to triangle precisely as it is isosceles, but vice versa.

Now the first two were put in the antecedent on the ground that if “triangle” were not the wider term or if it were predicated equivocally of its several inferiors, it would not be compared to “isosceles” as universal to particular. But he added the third, because if “having three” did not belong to every triangle, it would not belong to isosceles precisely as triangle, but in virtue of being a certain triangle; just as the characteristic of “having three” does not belong to every figure precisely as figure, but because it is a certain figure which is a triangle.

From these statements, therefore, he concludes to the opposite of that which the objection presupposed, namely, he concludes that one who knows in the universal knows the thing per se and as such in a better way than one who knows in the particular. And from this he further concludes his chief proposition, namely, that universal demonstration is stronger than particular.

Then (85b15) he answers the second reason, saying that if the universal is predicated of several according to one formality and not equivocally, the universal, so far as its formality is concerned, i.e., so far as science and demonstration are concerned, will not be less of an entity than the particular, but more. For the incorruptible is more a being than the corruptible; but the formality of a universal is incorruptible, whereas particulars are corruptible, in that they are subject to corruption as to their individual principles, although not as to the formality of the species, which is common to all and conserved through generation. And so in regard to that which pertains to their formality, the universals are beings to a higher degree than particulars. Nevertheless, in regard to natural subsistence the particulars, which are called the first and chief substances, have more being.

Then (85b18) he answers the third reason, saying that although in propositions or demonstrations that are universal, something which is one according to itself, say “triangle,” is signified, nevertheless there is no need for anyone to suppose on this account that “triangle” is some one thing apart from the many, any more than there is need in the case of things which do not signify substance but some genus of accident (when we signify them absolutely, as when we say “whiteness” or “fatherhood”), to suppose that such things exist apart from the substance. For the intellect is able to understand one of the things which are joined in reality without actually thinking of some other one; yet the intellect is not false. Thus, if something white is musical, I am able to think of the white and attribute something to it and demonstrate something of it, say, that it disperses the vision, without adverting at all to musical. However, if one were to understand that the white one is not musical, then the intellect would be false. And so when we say or understand that whiteness is a color, no mention being made of the subject, we are saying something true. But it would be false, were we to say that the whiteness which is a color is not in a subject. In like fashion, when we say that every man is an animal, we are speaking truly, even though no particular man is mentioned. But it would be false were we to say that man is “an animal existing apart from particular men.” And if this is so, it follows that demonstration is not the cause of the false opinion according to which someone supposes that the universal is some thing outside the singulars, but it is rather the hearer who understands incorrectly. Hence this does not detract at all from universal demonstration.

Lecture 38
(85b22-86a32)
UNIVERSAL DEMONSTRATION IS STRONGER THAN PARTICULAR DEMONSTRATION

b22. Demonstration is syllogism— b28. Our search for the reason— a3. The more demonstration— a11. Demonstration which teaches— a14. Proof becomes more and more— a2l. The clearest indication— a28. Moreover, commensurately

After answering the arguments which favored the false side, the Philosopher now introduces arguments for the true side, namely, to show that universal demonstration is the more powerful. In regard to this he sets down seven reasons, adding them to the previous solutions from which the proposition can also be concluded, as was clear from the above.

The first reason (85b22), is this: Demonstration is a syllogism showing the cause and propter quid: for this is the way scientific knowing takes place, as stated above. But the universal does this better than the particular. For as was shown in the first solution, something is per se in the universal more than in the particular. But that in which something is per se is the cause of this something: for the subject is the cause of the proper attribute which is in it per se. However, the first thing in which the proper attribute is present is the universal, as is clear from what has been stated above. Hence it is plain that, properly speaking, the cause is that which is universal. From this he concludes the proposition, namely, that universal demonstration is more valuable as better declaring the cause and propter quid.

Then (85b28) he gives the second reason. This reason is taken from the final causes. Here it should be pointed out that something is the end of another thing both in regard to becoming and in regard to being: in regard to becoming, as generation is for the sake of form; in regard to being, as a house is for the sake of habitation.

He says, therefore, that we search for the “reason why” something is done or why something is, until we reach the point where there is nothing further to assign, beyond the point reached, as the reason why that comes to be or is, whose reason why is sought. And when we find this we think that we know the propter quid: the reason being that that which is ultimate in this way, i.e., leaving nothing further to be sought, is truly the end and terminus which is being sought when we seek the propter quid. And he gives this example: if we should ask why someone went out, and the answer is given, “to get money,” and this in order to pay a debt, and “this for this other reason,” namely, lest he be guilty of injustice; and we continue in this way until there is nothing further for the sake of which as for an end—as when we arrive at the ultimate end, which is happiness—we will say that it was for this, as for an end, that he went. And it is the same in all other things that are or are done for the sake of an end: when we arrive at it, we will know the reason why he went.

Now if things are so related in the other causes as they are in the final cause, namely, that we know fully when the last one has been reached, then we will know best in the others, when we shall have arrived at the fact that “this is in this” no longer because of something further. And this happens when we shall have reached the universal.

To elucidate this he offers the following example: If we should ask concerning this particular triangle, why its exterior angles are equal to four right angles, the answer will be that this happens to this triangle because it is isosceles; and it is so for the isosceles, because it is a triangle; and it is so for a triangle, because it is such and such a rectilinear figure. If no further step can be taken, then we know in the best way. But this happens when the universal has been reached. Therefore the universal demonstration is stronger than the particular.

Then (86a3) he gives the third reason, saying that the more one proceeds toward particulars, the nearer one gets to what is infinite; because, as it is stated in Physics III, the infinite is appropriate to matter which is the principle of individuation. On the other hand, the more one proceeds toward the universal, the nearer he gets to what is simple and to the end itself; because the universal reason is taken on the part of the form, ic is simple and which has the character of an end insofar as it terminates the infinitude of matter. Now it is obvious that infinite things as such are not scientifically knowable; rather to the extent that they are finite, to that extent are they knowable; because the principle of knowing a thing is not the matter but the form. Therefore, it is obvious that universals are scientifically more knowable than particulars. Consequently, they are also more demonstrable, because a demonstration is a syllogism that makes one know scientifically. But a demonstration of the more demonstrable is more powerful; for things which are described one in terms of the other grow apace. But demonstration is described in terms of the demonstrable. Consequently, since universals are more demonstrable, universal demonstration will be more powerful.

Then (86a11) he gives the fourth reason and it is this: Since the end of demonstration is scientific knowledge, the more things a demonstration enables one to know, the more powerful it is. And this is what he states, namely, that a demonstration, according to which a man knows one thing and something additional, is preferable to one according to which a man knows only the one. But a man who has knowledge of the universal knows also the particular, so long as he knows that the particular is contained under the universal. Thus, one who knows that every mule is sterile, knows that this animal, which he recognizes to be a mule, is sterile. But one who knows the particular does not on that account know the universal. For if I know that this mule is sterile, I do not on that account know that every mule is sterile. It remains, therefore, that universal demonstration, through which the universal is known, is stronger than particular, through which the particular is known.

Then (86a14) he gives the fifth reason, and it is this: The nearer to the first principle the middle of a demonstration is, the more powerful is the demonstration. He proves this on the ground that if a demonstration which proceeds from an immediate principle is more certain than one which does not proceed from an immediate, but from a mediate, principle, then it is necessary that to the extent that a demonstration proceeds from a middle nearer an immediate principle, the more powerful it is. But a universal demonstration proceeds from a middle nearer the principle which is an immediate proposition. And he exemplifies this with terms. For if one is required to demonstrate A, which is the most universal, of B, which is the most particular, say “substance” of man, and B and C, say “living” and “animal” be taken, such that B is more general than C, as “living” than “animal,” it is obvious that B, which is the more universal, will be immediate to A; consequently, more will be known through it than through C, which is less universal. Hence it remains that universal demonstration is more powerful than particular.

But he remarks that some of the above reasons are “logical,” because, namely, they proceed from common principles which are not proper to demonstration; especially the third and fourth, which take as their middle something which is common to all knowledge. However, the other three, namely, the first, second and fifth, seem to be more analytic, proceeding as they do from principles proper to demonstration.

Then (86a21) he gives the sixth reason and says that the primacy of universal demonstration over particular is evident from the very propositions from which the two demonstrations proceed. For the universal demonstration proceeds from universal propositions, whereas the particular demonstration proceeds from a particular proposition. Now the comparison between universal and particular propositions is such that one who has knowledge of the former, i.e., of the universal, somehow knows the latter, namely, in potency. For the particulars are potentially in the universal, as the parts are potentially in a whole. Thus, if one knows that every triangle has three angles equal to two right angles, he already knows it potentially of isosceles. But one who knows something in a particular way does not on that account know it universally either potentially or actually. For the universal proposition is neither potentially nor actually contained in the particular. Therefore, if that demonstration is more powerful which proceeds from stronger propositions, it follows that universal demonstration is more powerful.

It should be noted that this reason does not differ from the fourth one given above, except that in the fourth one the comparison was made between the conclusions which are known through demonstration, whereas here the comparison is between the propositions from which the demonstration proceeds.

Then (86a28) he gives the seventh reason and it is this: The universal demonstration is intelligible, i.e., is terminated in the intellect, because it finishes in”a universal which is known only by the intellect. On the other. hand, a particular demonstration, although it begins in the intellect is terminated in sense, because it concludes a particular which is directly known by sense, and the reason demonstrating reaches out to the particular through a certain application or reflexion. Now since intellect is more powerful than sense, it follows that universal demonstration is stronger than particular.

Finally, he concludes that this is clear in virtue of all that has been said above.

Lecture 39
(86a32-b40)
AFFIRMATIVE DEMONSTRATION IS STRONGER THAN NEGATIVE

a32. That affirmative demonstration— b10. It has been proved— b12. So we are compelled— b30. The basic truth of— b39. Affirmative demonstration is more

After showing that universal demonstration is more powerful than particular, the Philosopher here shows that affirmative demonstration is more powerful than negative. In support of this he presents five reasons.

In the first reason (86a32) he presupposes that, all else being equal, the better demonstration is the one which proceeds from fewer postulates or suppositions or propositions. How these differ is clear from what has been stated above. For propositions might even be taken to mean those per se known statements which are neither suppositions nor postulates, as stated above. But a supposition is not the same as a postulate: for a supposition is a proposition which is not per se known, but is taken by the learner as opined; a postulate, on the other hand, is a proposition which is not per se known and has not been opined by the learner, nothing being said about whether he has opinions to the contrary or not.

Accordingly, he shows in two ways that, other things being equal, the demonstration which employs fewer of these is better. First, because if it happens that both propositions from which one proceeds are equally known, it follows that one knows more quickly through fewer propositions than through more, because a discourse through fewer propositions comes to an end sooner than one through more. For it is more desirable or more advantageous that a man, learn more quickly. Hence, it remains that a demonstration which proceeds from fewer propositions, provided they are equally known, is better.

Secondly, he proves the same proposition universally without the above supposition, namely, that all the propositions used are equally known. To prove this he assumes this supposition, namely, that the middles which are of one order are equally known, but the middles which are prior are better known. For this must be universally true. Therefore, supposing this, take one demonstration in which it is demonstrated that A is in E in virtue of three middles, B, C, D, so that a conclusion is reached from four propositions, namely, “Every B is A,” “Every C is B,” “Every D is C” and “Every E is D.” Then let the other demonstration be one which concludes the same conclusion, namely, that A is in E, through two middles which are F and G.

Under these conditions, from the fact that the order of knowing is proportionate to the order of middles, since the prior known are the better known, as has been said, it is obvious that the proposition, “Every D is A” in the first demonstration must be as well known as “Every E is A” in the second, because two middles are used for obtaining each. However, it is also obvious that in the first demonstration this proposition, “Every D is A,” is prior and better known than the proposition, “Every E is A,” because this latter is demonstrated from something prior in the first demonstration-and from what has been established above, it is apparent that that through which something is demonstrated is more credible and more known than that which is demonstrated through it. What remains, therefore, is that this proposition, “Every E is A,” so far as it is concluded by the second demonstration, is better known than the same proposition so far as it is concluded by the first demonstration which used more middles. Consequently, a demonstration which proceeds from fewer is better than one which proceeds from more.

Therefore, having proved the major proposition, Aristotle assumes that an affirmative proposition proceeds from fewer things than a negative: not indeed from fewer terms or fewer propositions materially, because every demonstration, whether affirmative or negative, demonstrates through three terms and two propositions; but a negative demonstration is said to proceed from more according to the quality of the propositions. For an affirmative demonstration takes only “being,” i.e., proceeds only from affirmative propositions, whereas a negative demonstration takes “being and non-being,” i.e., uses an affirmative and a negative. Therefore, the affirmative is stronger than the negative.

Then (86b10) he presents the second reason. Now this second reason is introduced to support the first, which might seem deficient on the ground that it was not subsumed under the major proposition in the way in which it was proved. Therefore, to forestall any subterfuge he adds this second reason to confirm the first. For it has been shown in Prior Analytics I that a syllogism cannot be formed out of two negative propositions, but one proposition at least must be affirmative and the other may be negative. This clearly shows that affirmative propositions have greater efficacy for syllogizing than do negatives. Hence it follows that an affirmative demonstration which proceeds from affirmatives only, is more powerful than the negative demonstration which proceeds from a negative and an affirmative.

Then (86b12) he presents the third reason, saying that as a consequence of the foregoing reasons, we can assert that when a demonstration is augmented, i.e., by resolving propositions into their principles, it is necessary that there be several affirmative propositions, but no more than one negative. Thus, let us take the following negative demonstration: “No B is A; Every C is B: therefore, No C is A.” Then let this demonstration be augmented as to both propositions, if each is mediate, by taking a middle for each, letting the middle of the major proposition, “No B is A,” be D, and the middle of the minor proposition, “Every C is B,” be E. Now since this proposition, “Every C is B,” is affirmative and was concluded without benefit of a negative, it is necessary that its middle, namely, E, be affirmative to each extreme. Consequently, there are two affirmative propositions, namely, “Every E is B,” and “Every C is E,” from which “Every C is B” is concluded. But the major proposition is negative, namely, “No B is A.” Now a negative is not concluded from two negatives; but in the first figure, in which demonstration is best made, according to what has been said, that major must be negative and the minor affirmative. Consequently, this middle, D, must be affirmative to B and negative to A, as in the following demonstration: “No D is A; Every B is D; therefore No B is A.”

Hence by augmenting a negative demonstration by resolving the propositions into their principles, there will be four propositions: only one of these is negative, namely, “No D is A,” and the other three will be affirmative, namely, “Every B is D,” “Every E is B,” and “Every C is E.” And the same thing will happen in all other syllogisms, because it is always required that the middle, through which the affirmative propositions are proved, be affirmative to both extremes. But the middle through which the negative proposition is proved must be negative to one of the extremes only. Thus it follows that only one proposition is negative and the others all affirmative.

From this it is clear that it is mainly through affirmatives that a negative proposition is demonstrated. Therefore, if that through which something is demonstrated is more known and more credible than that which is demonstrated through it (since a negative proposition is mainly proved by affirmatives and not vice versa), it follows that the affirmative proposition is prior and more known and more credible than the negative. Accordingly, the affirmative demonstration will be more valuable.

Then (86b30) he presents the fourth reason, saying that the principle of a demonstrative syllogism is an immediate universal proposition, in the sense that the proper principle of an affirmative syllogism is an affirmative proposition and the proper principle of a negative syllogism is a universal negative proposition. But the effect of a nobler principle is itself more noble. Therefore, as an affirmative proposition is to a negative, so is an affirmative demonstration to a negative one. But an affirmative proposition is more powerful than a negative one. (He proves this in two ways: first, because the affirmative is prior and better known, since the negative is proved by the affirmative and not vice versa. Secondly, because affirmation naturally precedes negation, as being is prior to non-being—for although in one and the same thing which passes from non-being to being, the non-being is prior in the order of time, yet in the order of nature, being is prior and, absolutely speaking, is prior even in time, because non-beings are not brought into existence except by something that is). Therefore, it is clear that affirmative demonstration is mo powerful than negative.

Then (86b39) he presents the fifth reason and it is this: That upon which something depends is more principal. But negative demonstration depends on affirmative, because there cannot be a negative demonstration without an affirmative proposition, which is not proved except by an affirmative demonstration. Therefore, affirmative demonstration is more principal than negative.

Lecture 40
(87a1-30)
NEGATIVE OSTENSIVE DEMONSTRATION IS STRONGER THAN DEMONSTRATION LEADING TO THE IMPOSSIBLE

a1. Since affirmative demonstration— a2. We must first make certain— a17. All the same the— a7. Reductio ad impossibile— a13. The order of the terms— a19. For the destructive result

After showing that universal demonstration ranks higher than particular, and affirmative higher than negative, the Philosopher here shows, thirdly, that ostensive demonstration is more powerful than that which leads to the impossible. Concerning this he does three things. First, he proposes what he intends. Secondly, he prefaces certain matters needed for showing the proposition (87a2). Thirdly, he proves his proposition (87a17).

He says therefore first (87a1) that since we have shown that affirmative, demonstration is more powerful than negative, from this it further follows that an affirmative ostensive demonstration is more powerful than one which leads to the impossible.

Then (87a2) he lays down certain things that are necessary for showing his proposition. In regard to this he does three things. First, he shows what a negative demonstration is. Secondly, he shows what a demonstration to the impossible is (87a7). Thirdly, he concludes the comparison of the one with the other (87a13).

He says therefore first (87a2) that in order to manifest the proposition it is necessary to point out the difference between them, i.e., between the negative demonstration and the one leading to the impossible. If, therefore, one assumes that A is in No B, and B is in every C, and concludes that A is in no C, it will be a negative demonstration.

Then (87a7) he elucidates what a demonstration leading to the impossible is. And he says that a demonstration leading to the impossible is as follows: Suppose we are to prove that A is not in B. Let us assume the opposite of what we wish to prove, namely, assume that every B is A; then assume that B is in C, using the proposition, “Every C is B”; from these the conclusion follows that “Every C is A,” which is such that everyone knows and admits that it is impossible. From this we conclude that the first proposition, namely, “Every B is A,” is false. Consequently, it will be necessary either that no B is A or at least that some B is not A.

But it must be understood that “A is not in B” follows, when it is clear that B is in C, because if it is obvious that “Every C is A” is false, but not obvious that “Every C is B” is true, it would not, just in virtue of the inference, be plain that “Every B is A” is false; for the falsity of the conclusion might have proceeded from either premise, as was stated above.

Then (87a13) he concludes the comparison of each of the aforesaid demonstrations. First, he shows wherein they agree, namely, in having a like ordering of their terms. For just as in the negative demonstration, B is taken as the middle between A and C, so too in that which leads to the impossible.

Secondly, he shows wherein they differ, because it makes a difference which negative proposition in each demonstration is better known, namely, whether it be the proposition, “No B is A,” or the proposition, “No C is A”: for in a demonstration leading to the impossible, the proposition “C is not A” is taken to be better known, since from the fact that A is not in C one shows that A is not in B. Consequently, the proposition, “C is not A,” is taken as better known. But when that which is set down as a premise of the syllogism is taken as better known, there is a demonstrative, i.e., an ostensive, negative demonstration.

Then (87a17) he shows his proposition in the following way: The proposition, “B is not A,” is naturally prior to the proposition, “C is not A.” And this is proved by the fact that the premises, from which the conclusion is inferred, are naturally prior to the conclusion. But in the order of the syllogism, “C is not A” is set down as the conclusion, whereas “B is not A” is set down as that from which the conclusion is inferred. Therefore, “B is not A” is naturally prior.

Then (87a19) he removes an objection. For someone could say that even the negative “C is not A” is the one from which one concludes “B is not A” in a demonstration to the impossible. But he excludes this, saying that by the fact that the conclusion is destroyed and from its destruction something in the premises is destroyed, it does not follow that what was first the conclusion is now a principle and vice versa, absolutely and according to nature, but only in a qualified sense. For the relation of conclusion to principle is such that the principle is destroyed by the conclusion’s being destroyed. But that functions as the principle from which a syllogism proceeds, which is related to the conclusion as a whole to a part; while the conclusion is to the principle, as part to whole. For the subject of a negative conclusion is subsumed under the subject of the first proposition. But the propositions AC and AB are not so related to each other that AC is to AB as whole to part. For BA is not subsumed under CA; rather it is just the opposite. Hence it remains that although from the destruction of CA, one concludes to the destruction of BA, nevertheless CA is naturally the conclusion and BA the principle. Consequently, “B is not A” is naturally better known than “C is not A.”

From this, one argues in the following way: That demonstration is the worthier which proceeds from better known and prior principles. But a negative demonstration proceeds from something better known and prior than does a demonstration leading to the impossible. For each causes one to know something in virtue of a negative proposition: but the negative demonstration proceeds to cause belief from the negative proposition, “B is not A”, which is naturally prior. Demonstration to the impossible, on the other hand, proceeds to cause belief from the negative proposition, “C is not A,” which is naturally posterior. What remains, therefore, is that the negative demonstration is more powerful than one which leads to the impossible. Furthermore, as was shown above, the affirmative is stronger than the negative. Therefore, an affirmative ostensive demonstration is much stronger than one which leads to the impossible.

Lecture 41
(87a31-b17)
COMPARISON OF SCIENCE TO SCIENCE FROM STANDPOINT OF CERTAINTY AND OF UNITY AND DIVERSITY

a3l. The science which is— a38. A single science is one— a38. all the subjects— a4l. One science differs from— b1. This is verified when b5. One can have several

After comparing demonstrations one with another, the Philosopher here treats of the comparison of science, which is the effect of demonstration. And his treatment falls into two parts. In the first he compares science to science. In the second he compares science to other modes of knowing (88b30) [L. 44]. Concerning the first he does two things. First, he compares science to science as to certitude. Secondly, as to unity and multiplicity (87a38).

Concerning the first (87a31) he lays down three modes whereby one science is more certain than another. Laying down the first mode, he says that that science is prior and more certain than another, which, namely, makes one know the same things both quia and propter quid. However, that science is not more certain which knows only the quia apart from one which knows propter quid. But this is the relation of subalternating science to subalternate, as has been said above, namely, that the subalternate science in isolation knows quia without knowing propter quid: thus a surgeon knows that circular wounds are healed more slowly, but he does not know why. But such knowledge pertains to the geometer who considers that characteristic of a circle according to which its parts do not lie close enough to form an angle, the nearness of whose sides makes triangular wounds heal more quickly.

He lays down the second mode when he says that a science which is not concerned with a subject is more certain than one which is. Here “subject” is taken to mean sensible matter because, as the Philosopher teaches in Physics II, some sciences are purely mathematical, those, namely, which abstract according to reason from sensible matter, as geometry and arithmetic; but other sciences are intermediate, namely, those which apply mathematical principles to sensible matter, as optics applies the principles of geometry to the visual line, and harmony, i.e., music, applies the principles of arithmetic to sensible sounds. Hence he says here that arithmetic is both more certain and prior to music: it is prior, because music uses its principles for something non-mathematical; it is more certain, because lack of certitude arises from matter’s changes. Hence the closer one gets to matter, the less certain the science.

He lays down the third mode when he says that a science which arises from fewer things is prior and more certain than one which arises from an addition, i.e., than one which results from that addition. And he gives the example that geometry is posterior to and less certain than arithmetic: for the things of geometry are the result of adding to the things which pertain to arithmetic. This is easy to see if one admits the postulates of Plato, according to which Aristotle is proceeding here, using them to prove his point (as he frequently uses the opinions of other philosophers as examples in his logical works in order to explain a point). Now Plato laid it down that “one” is the substance of each thing, because he did not distinguish between “one” which is converted with being and signifies the substance of a thing, and the “one” which is the principle of number and which arithmetic considers.

Accordingly, this “one,” as receiving the added characteristic of occupying a position in a continuum, takes on the guise of a point. Hence he said that “one” is a substance not occupying a position; but a point is a substance that does have a position. Consequently, a point adds something to the notion of “one,” namely, position. And just as all numbers not having position are caused from “one,” so from the point, according to the Platonists, all continuous quantities are caused. For a point in motion makes a line; a line in motion makes a plane, and a plane in motion, makes a body. According to this, then, continuous quantities (which art~l treated in geometry) are the result of additions made to numbers (which are the concern of arithmetic). Hence the Platonists laid it down that numbers are the forms of magnitudes, saying that the point’s form is “one,” and the line’s form is “two,” because it has two endpoints, while a surface’s form is “three,” owing to the first surface’s being the triangle which is terminated by three angles; but the form of bodies is “four,” on the ground that the first solid figure is the triangular pyramid, which is composed of four (non-plane) solid angles, namely, one at the apex and three at the base.

According to this it is obvious that the comparison of the certitude of sciences is here based on two things: for the first mode is taken according to the cause as it is prior to and more certain than the effect; but the other two are taken according to the form as this is more certain than matter, inasmuch as the form is the principle of knowing the matter. But, as it is stated in Metaphysics VII, matter is twofold: one is sensible, according to which the second mode is taken; the other is intelligible, i.e., its continuity, according to which the third mode is taken. And although this third mode was explained according to Plato’s theory, yet even according to Aristotle’s theory, a point results from some addition to “one”: for a point is an indivisible unity in a continuum, abstracting according to reason from sensible matter; but unity abstracts from both sensible and intelligible matter.

Then (87a38) he compares sciences one to another according to unity and diversity. Concerning this he does two things. First, he shows that there is unity and diversity among sciences both according to subject and according to principles. Secondly, he treats concerning both the subjects and principles (87b19) [L. 42]. Concerning the first he does two things. First, he shows what makes for unity and diversity of sciences. Secondly, he explains something which is needed for understanding what makes for multiplicity of sciences (87b5). Concerning the first he does two things. First, he shows what makes for the unity of a science. Secondly, what makes for diversity of sciences (87a41). Concerning the first he does two things. First, he lays it down that the unity of a science is considered from the unity of its generic subject. Secondly, he describes the genus which can be the subject of a science (8708).

He says therefore first (87a38) that a science is said to be one from the fact that it is concerned with one generic subject. The reason for this is that the process of science of any given thing is, as it were, a movement of reason. Now the unity of any motion is judged principally from its terminus, as is clear in Physics V. Consequently, the unity of any science must be judged from its end or terminus. But the end or terminus of a science is the genus concerning which the science treats: because in speculative sciences nothing else is sought except a knowledge of some generic subject; in practical sciences what is intended as the end is the construction of its subject. Thus, in geometry the end intended is knowledge of magnitude, which is the subject of geometry; but in the science of building that which is intended as the end is the construction of a house, which is the subject of this art. Therefore, the unity of each science must be considered in terms of the unity of its subject. But just as the unity of one generic subject is more universal than another, for example, being or substance is more common than mobile being, so one science is more general than another. Thus, metaphysics, which treats of being or substance, is more general in scope than physics, which treats of mobile body.

Then (87a38) he describes the marks of those genera concerning which there can be sciences: and he lays down two marks. The first of these is stated when he says, “All the subjects constituted out of primary entities,” i.e., of those subjects of which there is one genus, there is one science. To understand this it should be noted that, as has been said, the process of science consists in a certain movement of reason passing from one thing to another. But all movement starts from some principle or beginning and is terminated at something definite; hence in the progress of a science, reason must proceed from certain first principles, Therefore, if there be anything which does not have prior principles from which reason can proceed, there cannot be science of such things, if we take science to mean the effect of demonstration, as we do here. Thus, speculative sciences are not concerned with the very essences of separated substances: for we cannot, through demonstrative sciences, know the essences in them, because the essences of such substances are intelligible of themselves to an intellect proportionate to such intelligibility, and a grasp of the essences of such substances is not obtained by any prior conceptions. The only thing that can be known through speculative sciences about these substances is whether they exist and what they are not; anything else about them is known in terms of likenesses to lower things. But in that case we are really using subsequent things as though they were prior in order to understand them, because things that are subsequent according to nature are prior and better known to us. And so it is clear that those things concerning which we have science through what is absolutely prior, are of themselves composed of prior items; but things which are known through subsequent items (but prior in reference to us), even though they be simple in themselves, are nevertheless in our knowledge of them composed of things that are first to us.

He lays down the second mark when he says, “The parts of this total subject and their per se properties.” Here it should be noted that the subject of a science can have two types of parts: first, the parts out of which, as out of first things, it is composed, i.e., the very principles of the subject; and secondly, the subjective parts. And although what is stated here can be applied to either of these types of parts, yet it seems to truer of the first type of parts. For in every science there are the principles of its subject, and these must be considered before all else: for example, in natural science the first consideration is about matter and form, and in grammar about the alphabet. But in every science there is also something ultimate, at which the study of that science terminates, namely, that the properties of the subject be manifested. But each of these, namely, the first parts and the properties, can be attributed to something either per se or not per se. For the per se principles and properties of a triangle are not the per se principles and properties of an isosceles triangle precisely as isosceles, but precisely as triangle. Neither are they the per se principles and properties of brass or white, even though the triangle happens to be of brass or be white. Hence, if there were a science which manifests the properties of triangle from the principles of the triangle, the subject of that science would not be isosceles or white or brass, but triangle—whose per se subjective parts would be isosceles equilateral and scalene.

Then (87a41) he shows the reason for sciences being diverse. First, he lays down this reason. Secondly, he explains it (87b1).

It should be noted in regard to the first point that although he took the reason for the oneness of a science from the oneness of its generic subject, he does not take the reason for their diversity from the diversity of their subject, but from the diversity of principles. For he says (87a41) that one science is distinct from another when their principles are diverse, in the sense that the principles of the two sciences do not proceed from any prior principles, nor the principles of the one science from those of the other; because if both proceed from the same principles, or if one proceeds from those of another, they would not be diverse sciences.

To understand this it should be noted that a material diversity of objects does not diversity habits, but only a formal diversity. Therefore, since something scientifically knowable is the proper object of a science, the sciences will not be diversified according to a material diversity of, their scientifically knowable objects, but according to their formal diversity. Now just as the formality of visible is taken from light, through which color is seen, so the formal aspect of a scientifically knowable object is taken according to the principles from which something is scientifically known. Therefore, no matter how diverse certain scientifically knowable objects may be in their nature, so long as they are known through the same principles, they pertain to one science, because they will not differ precisely as scientifically knowable. For they are scientifically knowable in virtue of their own principles.

This is made clear by an example, namely, that human voices differ a great deal according to their nature from the sounds of inanimate bodies; but because the consonance of human voices and of the sounds of inanimate bodies is considered according to the same principles, the science of music, which considers both, is one science. On the other hand, if there are things which have the same nature but are considered according to diverse principles, it is obvious that they pertain to diverse sciences. Thus, the mathematical body is never really distinct from a natural body; yet because the mathematical body is known through the principles of quantity, but a natural body through the principles of motion, the science of geometry and the science of nature are not the same. It is clear, therefore, that for sciences to be diverse it is enough that the principles be diverse, this diversity of principles being accompanied by a diversity of scientifically knowable objects.

But in order to have one science absolutely, both are required, namely, unity of subject and unity of principles. That is why above he made mention of unity of subject when he said, “whose domain is one genus,” but made mention of principles when he said, “all the subjects constituted out of the primary entities of the genus.”

However, it should be further noted that second principles derive their force from the first principles. Hence for diversity of sciences, diversity of first principles is required. But this will not be verified if the principles of diverse things flow from the same principles, as the principles of triangle and square are derived from the principles of figure, or the principles of one are derived from the principles of another, as the principles of isosceles depend on the principles of triangle.

Yet it should not be supposed that for the unity of a science it is enough that there be unity of first principles absolutely, but unity of first principles in some scientifically knowable genus. Now the genera of the scientifically knowable are distinguished according to the diverse modes of knowing: thus things that are defined with matter are known in one mode, and those defined without matter are known in another. Hence the natural body is one genus of the scientifically knowable, and mathematical body is another genus. Hence there are diverse first principles for each of these genera and, consequently, diverse sciences. Furthermore, each of these genera is distinguished into diverse species according to diverse modes and aspects of intelligibility.

Then (87b1) he explains the reason he laid down, saying that a sign of the fact that sciences are diversified according to their principles is obtained when one resolves a science into principles that are indemonstrable, for they must be of the same genus as the things demonstrated; because, as stated above, one does not demonstrate by proceeding from an alien genus. Now the reason why the reaching of indemonstrable principles of one genus is taken as a sign is that the facts demonstrated by them are in the same genus and are co-generic, i.e., connatural, or generically proximate to them: for these have the same principles. And so it is clear that the unity of a scientifically knowable genus, precisely as it is the scientifically knowable genus from which was taken the oneness of the science, and the unity of the principles, according to which was taken th diversity among the sciences, mutually correspond.

Then (87b5) he shows how one conclusion can be demonstrated through several principles. And this can happen in two ways: in one way, when”, a number of middles are laid down in the same coordination, and one of these middles is used in one demonstration and another one in another, both leading to the same conclusion. This, of course, requires that the middles be non-continuous. Thus, if the two extremes, A and B, are “to have three” and “isosceles” respectively, but the middles, D and Q are coordinate, as “triangle” and “this type of figure,” A could be demonstrated of B with two demonstrations. In the first one, C could be taken as middle, and D in the other, so that in neither demonstration will, there be a middle continuous with the extremes, because in one demonstration a middle continuous with one extreme and not continuous with the other would be taken, and in the other demonstration the converse would be taken.

Or this can happen in a second way, namely, when diverse middles are taken from diverse coordinations; for example, if A, which is the major extreme, were “to be altered,” and B, which is the minor extreme, were “to be pleased,” and the middles taken were independent, say G would be “to be at rest,” and D “to be in motion.” In this case the same conclusion can be reached through diverse middles that are not of one order. Thus one demonstration would be this: “Whatever is at rest is altered, because it belongs to the same thing to be at rest and to be altered; but whatever is pleased is at rest, because rest in the desired good causes pleasure: therefore, whatever is pleased is altered.” The other demonstration would be: “Whatever is in motion is altered; but whatever is pleased is in motion, because pleasure is a movement of the appetitive faculty [power]: therefore, whatever is pleased is altered.” (Now the statement, “Whatever is pleased is in motion, “ is Plato’s opinion and is verified in sensible pleasures which involve movement. The other statement, “Whatever is pleased is at rest,” is true according to Aristotle’s opinion, as is clear from Ethics VII and X. And this is verified particularly in intellectual pleasures.

Then he says that just as this has been proved in the first figure, so in the other figures it is easy to see that the same conclusion can be syllogized with diverse middles. Now the Philosopher mentioned this to show that diverse middles of demonstration sometimes pertain to the same science, as when they are taken from the same coordination, and sometimes to diverse sciences, when they are taken from a different coordination. Thus, astronomy demonstrates that the earth is round, using one middle, namely, the eclipse of the sun and moon; and natural science uses another middle, namely, the motion of heavy objects tending toward the center, as it is stated in Physics II.

Lecture 42
(8719-88a17)
SCIENCE IS NOT CONCERNED WITH THINGS CAUSED BY FORTUNE OR WITH THINGS LEARNED THROUGH SENSE-PERCEPTION

a5. The commensurate universal— b28. Scientific knowledge is not— b34. nay, it is obvious— a11. Nevertheless certain points— b19. There is no knowledge

After assigning the basis of unity and of diversity of sciences on the part of the generic subject and on the part of the principles, the Philosopher here continues with each. First, with the subjects about which there is science. Secondly, with the principles (88a18) [L. 43].

In regard to the first it should be noted that above he laid down two marks of the genus which is the subject of the science: one of these is that it be composed of first principles, and the other is that its parts and properties be per se. Now one of these conditions is lacking in things that happen by fortune, because they do not come about per se, but per accidens and beyond one’s intention, as is proved in Physics II. The other mark is lacking in things that are known through sense, which are first in our knowledge. Therefore, he shows first that science is not concerned with things which come about by fortune. Secondly, that it is not of things that are known through sense (87b28).

He says therefore first (87b19) that demonstrative science cannot be concerned with things that occur through fortune, i.e., things that come about per accidens. He proves this in the following way: Every demonstrative syllogism proceeds either from necessary propositions or from propositions which are true for the most part. Now from necessary propositions a necessary conclusion follows, as has been proved above; in like manner, from propositions which are true for the most part, a conclusion follows which is true for the most part, or perhaps one which is necessary, in the sense that the necessary can follow from the contingent, as the true can follow from the false. However, it never happens that from propositions which are true for the most part, a conclusion will follow which is true in few cases; because that would mean that in certain cases the propositions would be true and the conclusion false, which is impossible, as has been shown. It is necessary, therefore, that the conclusion of a demonstrative syllogism either be necessary or be true in the majority of cases. But that which is by fortune is neither necessary nor is it true in (the majority of cases, but it occurs rarely, as is proved in Physics II. Therefore, demonstrative science cannot be concerned with that which is by fortune.

It should be noted, however, that there happens to be demonstration of things which occur, as it were, for the most part, insofar as there is in them something of necessity. But the necessary, as it is stated in Physics II, is not the same in natural things (which are true for the most part and fail to be true in a few cases) as in the disciplines, i.e., in mathematical things, which are always true. For in the disciplines there is a priori necessity, whereas in natural science there is a posteriori (which nevertheless is prior according to nature), namely, from the end and form. Hence Aristotle teaches there that to show a propter quid, such as if this has to be, say that if an olive is to be generated, it is necessary that this, namely, the olive seed, pre-exist, but not that an olive is generated of necessity from a given olive seed, because generation can be hindered by some defect. Hence if a demonstration is formed from that which is prior in generation, it does not conclude with necessity, unless perhaps we take as necessary the fact that an olive seed is frequently generative of an olive, because it does this according to a property of its nature, unless it is impeded.

Then (87b28) he shows that science is not concerned with things that are known according to sense. About this he does two things. First, he shows that science does not consist in sensing. Secondly, how sense is ordained to science (88a13). Concerning the first he does two things. First, he shows that science is not through sense. Secondly, he ranks science above sense (88a5). Concerning the first he does two things. First, he shows the truth. Secondly, he shows the errors of certain ones (87b34).

He says therefore first (87b28) that just as science is not of things which are by fortune, so neither does it consist in knowledge which is through sense. And he proves this in the following way: It is obvious that sense knows the qualities of a thing and not its substance. For the per se object of sense is not the substance and the essence, but some sensible quality, such as hot, cold, white, black, and so on. But qualities of this sort affect singular substances existing in a definite place and time: hence it is necessary that this which is sensed be a “this something,” i.e., a singular substance, and that it be somewhere now, i.e., in a definite time and place. From this it is clear that that which is universal cannot fall under sense. For that which is universal is not restricted to the here and now, because then it would not be universal. For we call that a universal which is always and everywhere.

However, this is not to be understood according to the way of an affirmation, as though it pertained to the very notion of a universal or of that which is universal, namely, that it be always and everywhere. For if to exist always and everywhere pertained to the very notion of that which is universal, say to the notion of man or animal, it would then be required that each individual man and animal be always and everywhere, because the notion of man and of animal is found in each of its singulars. Or if this were required by the very notion of the universal, as it is required of a genus that it contain species under it, it would follow that nothing would be universal, if it were not found everywhere and always. According to this, “olive” would not be universal, because it is not found in all lands. Hence, the statement under consideration must be understood after the manner of a negation or abstraction, i.e., that the universal abstracts from every definite time and place. Hence of itself, just as it is found in each thing in one place or time, so it is apt to be found in all. Thus, therefore, it is clear that the universal does not fall under sense. Consequently, because demonstrations are mainly universal, as has been shown above, it is obvious that science acquired by demonstration does not consist in knowledge obtained through sense.

Then (87b34) he excludes the errors of those who believed that science consists in sense-perception. And this notion seems to belong to those who did not believe that understanding differs from sense and, consequently, that there is no knowledge except sense-knowledge, as it is stated in On the Soul III and Metaphysics IV. So, to exclude this he says that even if we could perceive through sense that a triangle has three angles equal to two right angles, it would still be necessary to learn this by demonstration if one is to have science; for it could not be scientifically known through sense, because sense bears on singulars, whereas science consists in knowing the universal, as has been shown.

But because he had given the example of something that cannot be perceived through sense, he makes his point clearer with an example taken from things that can be sensed, namely, from an eclipse of the moon which is caused by the opposition of the earth, which is interposed between the sun and moon, so that the brightness of the sun cannot reach the moon because of the earth’s shadow; and the moon is eclipsed when it enters the shadow. Let us suppose, then, that someone were on the moon and with his senses could perceive the shadow caused by the interposition of the earth. This person, then, would sensibly perceive that the moon is darkened by the shadow of the earth, but this would not mean that he would therefore know totally the cause of the eclipse. For the per se cause of an eclipse is that which universally causes an eclipse. But the universal is not known through sense; rather, we receive universal knowledge from many individually observed cases in which the same thing is found to happen. And so it is through the universal cause that we demonstrate something in the universal way that constitutes science.

Then (88a5) he shows that science is nobler than sense. For it is clear that knowledge through the cause is more noble; but the per se cause is the universal cause, as has been said. Therefore, knowledge through the universal cause, such as science is, is more honorable. But because it is impossible to apprehend such a universal cause through sense, it follows that science, which shows the universal cause, is not only more honorable than any sense knowledge, but more honorable than any other intellectual knowledge, provided it concern things which have a cause: for to know something through the universal cause is more noble than any other way of knowing that which has a cause, without knowing the cause.

However, the case is otherwise in regard to first things, which do not have a cause. For these are understood in virtue of themselves; and such knowledge of these things is more certain than any science, because it is from such knowledge that science acquires its certitude.

Finally, he concludes to his chief point, namely, that it is impossible through sense to know something demonstrable, unless perhaps one takes “sense” in an equivocal meaning, calling demonstrative science “sense” on the ground that demonstrative science concerns the determinately one just as sense does: on which account sure calculations are called science.

Then (88a11) he shows how sense is ordained to science. For some problematic doubts are caused by the shortcomings of sense. For there would be no need to investigate certain matters, if we were to see them; not because science consists in seeing, but because from things seen by way of experiment the universal is obtained, concerning which there is science. For example, if we should see a porous pane of glass and observed how the light passes through the openings in the glass, we would have the scientific answer as to why the glass is transparent. (He uses this example because it is the opinion of those who supposed that light is a body and that certain bodies are transparent because of openings called pores. But because they cannot be perceived by sight, since they are so small, we wonder why glass is transparent. A similar example can be given of certain things that have a latent sensible cause).

Finally, because he had said that science of such things is not obtained by seeing, he shows that this is true. For in seeing we know isolated singulars; but to know scientifically we must know them all at once in the universal, if we are to know that such and such is the fact in all cases. For we see each individual glass singly, but we receive science that it is thus for all glass.

Lecture 43
(88a18-b29)
PRINCIPLES OF ALL SYLLOGISMS ARE NOT THE SAME

a18. All syllogisms cannot— a20. for though a true inference— a26. Then again, (2)— a3l. Not even all these— a37. Nor can any of the common— b4. Again, it is not true— b8. and lastly some of the— b10. If, on the other hand,— b15. Nor again can the contention— b20. but if it be— b22. If, however, it is not— b25. But that this cannot be

After his treatment concerning that about which there is science, the Philosopher continues here with the principles of sciences and shows that the principles of all syllogisms are not the same. First, he shows this logically, i.e., through reasons common to all syllogisms. Secondly, he shows it analytically, i.e., through reasons that are peculiar to demonstration (88b10). Concerning the first he does three things. First, he proves his proposition by showing the difference between false syllogisms and true ones. Secondly, by the difference between some false syllogisms and other false ones (88a26). Thirdly, by the difference between some true syllogisms and other true ones (88a31). Concerning the first he does two things. First, he proves his proposition. Secondly, he excludes an objection (88a20).

He says therefore first (88alS) that if we first speculate this matter in a logical way, it is clear that the principles of all syllogisms cannot be the same, inasmuch as some syllogisms are false, i.e., conclude false statements, and some are true, i.e., conclude to the truth. Now the principles of false and of true syllogisms are diverse: for the principles of true syllogisms are true, but the principles of false syllogisms are false. Therefore the principles of all syllogisms are not the same.

Then (88a20) he excludes an objection. For someone might say that there are false principles even in true syllogisms, because it is possible to syllogize the true from the false. But he excludes this, saying that although it happens that the true is syllogized from the false, this occurs only once in the first syllogism, in which the true is concluded from the false. But if one is forced to adduce other syllogisms to support the propositions in the premises, then those syllogisms would have to proceed from what is false, because from what is true the false cannot be concluded. Consequently, it is only in the first syllogism that the true is concluded from the false.

Then he gives an example of this: Let the proposition, “Every C is A,” be true, and take for each of these extremes a false middle, B, so that A is not in B nor B in C. Now if other middles be taken to prove these propositions, all the propositions of the false syllogism will be false, because every false conclusion is concluded from the false, but a true conclusion can be concluded from the true. Hence, when the propositions from which the true is concluded are true, one need not reach anything false. Thus, therefore, since some propositions are true and some false, it follows that the principles of true syllogisms are not the same as the principles of false syllogisms.

Then (88a26) he shows that not even the principles of false syllogisms are the same. For it occurs that some false conclusions are contrary to one another and incompatible with other false conclusions. Thus, the conclusion, “Justice is injustice,” is incompatible with the conclusion, “Justice is fear,” since both are false. For just as fear differs generically from justice, so from injustice. In like fashion, the two conclusions, “Man is a horse,” and “Man is a cow,” are contrary and incompatible. Again, these two propositions are incompatible, “Something equal is greater,” and “Something equal is less.” For it is necessary to arrive at such conclusions from principles, from which, when they are laid down, other things follow. Therefore, since they are contrary and incompatible, so too were the principles from which they were concluded.

Then (88a31) he shows that not even in the case of true syllogisms are the principles the same. And he gives four reasons, the first of which is based on the differences among proper principles. Hence he says that not even all these,” i.e., true conclusions, “are inferred from the same basic truths.” For the principles of diverse genera are themselves diverse: thus the pripciples of magnitudes are points, and of numbers unities; and these are not exactly the same, for numbers do not have position, but points do.

Furthermore, if all principles of syllogisms were in agreement, they would have to meet at some same middle either by ascending upwards to the major term or by descending downwards toward the minor term: because in a syllogism it is necessary that the terms be assumed either within or without. They are assumed within, when a multiplicity of syllogisms is used to prove the propositions presented. For then it is necessary to take middles that are between the predicates of the propositions and their subjects. Thus, if we form the syllogism, “Every B is A, Every C is B, therefore, Every C is A”; if it is necessary to prove “Every B is A,” we must take a middle between B and A, say D. Again, if it is necessary to prove the minor, we must take another middle between C and B, say E. In this process the terms assumed are always within.

However, they are assumed from without, when the major term is taken as middle by ascending, or the minor by descending: thus, if A is concluded of C through B, and C is then concluded of B through A, and so on; similarly, by descending, if we conclude B of E through C.

Therefore, in syllogisms sharing principles in common, it is necessary either that the middle of one syllogism be taken above the propositions of another syllogism or that the extremes of one be taken above or below the extremes of the other syllogism. But this cannot occur in things whose principles are diverse, because points cannot be taken as middles or extremes in syllogisms in which something about number is concluded, nor unities in syllogisms in which something about magnitudes is concluded. What remains, therefore, is that the principles of all syllogisms cannot be the same.

The second reason is presented at (88a37) and it is based on common principles. He says that there cannot be certain common principles from which alone are syllogized all conclusions, as this common principle, “Of each thing there is affirmation or negation,” which is universally true in every genus. Nevertheless, it is impossible that all things be syllogized exclusively from such common principles, because the genera of beings are diverse. Thus, the principles which pertain only to quantities are diverse from those which pertain exclusively to qualities. Such principles must be co-assumed with common principles, if one is to reach a conclusion in each matter. For example, if one wishes in quantities to syllogize from the aforesaid common principle, it is necessary to admit that since it is false that a point is a line, it must be true that a point is not a line; in like manner, in qualities, it is necessary to co-assume something peculiar to quality. Hence what remains is that it is impossible that the principles of all syllogisms be the same.

Then (88b4) he gives the third reason which is based on a comparison of premises with conclusions. And he says that the principles are not much fewer than the conclusions. They are as a matter of fact fewer, because although two principles are needed to infer one conclusion, i.e., two propositions are required, because one conclusion is not concluded immediately except from two premises, nevertheless you can use one proposition to infer a number of conclusions, insofar as many things can be taken under the subject or under the predicate. However, the principles are not much fewer than the conclusions, because most of the facts which are co-assumed with the principles to obtain other conclusions are themselves conclusions. (Here propositions are being called .,principles”). But propositions are formed of terms either added or interposed, i.e., propositions in syllogisms are multiplied either by assuming terms extrinsically (either above the major term and below the minor, as has been explained above), or by accepting terms which are in the middle.

To this must be added the fact that conclusions are infinite. For anything can be concluded of anything else either affirmatively or negatively. And lest this seem to conflict with the earlier statement that predications do not proceed to infinity, he adds that the terms are finite: and this is why it was stated above that a stop must be made in predications. Nevertheless, an infinitude of conclusions can be derived from a finite number of terms according to diverse combinations, if we take “conclusions” in a general sense, as including those that are per se and those that are per accidens. For we are now speaking of syllogisms in general. Therefore, if the conclusions are infinite and the principles are not much fewer than the conclusions, it follows that principles of syllogisms are also infinite. Therefore, the principles of all syllogisms are not the same.

Then (88b8) he gives the fourth reason and it is based on the difference between the necessary and the contingent. And he says that some of the principles which we use in the syllogism are contingent and some necessary, as is clear from Prior Analytics I, where he taught how to syllogize from necessary and from contingent premises. However, the necessary and the contingent are not the same. Therefore, the principles of all syllogisms are not the same.

What he concludes from these last two reasons is that in view of what has been established, and because conclusions are infinite, it is impossible that the principles of all syllogisms either be the same or even be finite.

Then (88b10) he shows the same thing analytically, namely, through reasons proper to the principles by which sciences demonstrate. And he gives three reasons. Concerning the first of these he says that if someone does not say that the principles of all syllogisms are the same, but says something somewhat different, namely, that some are principles of geometry and some of logic-which are called the principles of syllogisms or reasonings—and some are principles of medicine, and so on for the principles of all sciences, so that in this sense the principles of all demonstrations are the same, this does not support his claim that principles are the same: because from the facts that are given, nothing follows except that each science has its own principles.

The contention that the principles of one science are the same as those of another-which would be required if the principles of all scientific syllogisms were the same-is impossible and ridiculous, because according to this it would follow that everything in the sciences would be the same and hence that all the sciences would be the same. For things that are the same as a same thing are the same. But the principles of each science are somehow the same as the conclusions, because they belong to the same genus: for one may not demonstrate by passing from one genus into another, as has been shown above. Therefore, if the principles are the same, it will follow that everything in the sciences would be the same.

Then (88b15) he gives the second reason and it is this: If a person who contends that the principles of all are the same, intended to say that anything is demonstrated from anything, then he is saying something foolish, because this is possible neither in “evident mathematicals nor in analysis.” (By “evident mathematicals,” i.e., evident considerations or disciplines he means those cases in which a conclusion is drawn immediately from evident premises; but by “analysis” he means those cases where the assumed propositions are not evident but must be analyzed into others which are more evident). And that this is impossible he proves on the ground that in each of these two cases the principles of the demonstrative syllogism are immediate propositions, which are either immediately assumed in evident mathematicals or doctrines, or are reached by analysis. But we see that a different conclusion is demonstrated, when a different immediate proposition is co-assumed. Consequently, it cannot be that anything is demonstrated from just anything.

Then (88b20) he excludes a certain objection. For someone could say that there are two genera of immediate propositions: for some are first immediate propositions and some secondary, taking the order of immediate propositions according to the order of the terms. For those immediate propositions which consist of first and common terms, as “being” and “non-being,” “equal” and “unequal,” “whole” and “part,” are first and immediate propositions, as “It does not occur that the same thing both is and is not,” and “Two things which are equal to the same thing are equal,” and so on. But the immediate propositions that are concerned with later and less common terms are secondary in relation to the first, as that “a triangle is a figure,” or that “man is an animal.” Therefore, someone might say that secondary propositions are co-assumed for demonstrating diverse conclusions, but that the first immediate propositions are the same in all demonstrations.

To exclude this he says that if anyone should assert that those first immediate propositions are the ones from which all things are demonstrated, he ought to consider that in each genus there must yet be one principle or one immediate proposition which is first in that genus, even though it is not absolutely first; and that from one which is absolutely first, together with a co-assumed principle proper to a particular genus, one should demonstrate in that genus. Consequently, not all things can be demonstrated from common principles alone, but it is necessary to accept proper principles which are diverse for diverse genera.

Then (88b22), having excluded the injudicious interpretation of the position against which he is disputing, he concludes to his point. And he says that if it is not admitted that anything is demonstrated from just anything—as we must, in view of the foregoing—it follows, that one conclusion is not derived from some principle in such a way that a different conclusion can be derived from it; otherwise just anything is demonstrated from just anything. Hence it is necessary that the principles of diverse sciences be diverse. If it is required that the principles of each science be of the same genus as everything demonstrated from them, it will be required that from these principles be demonstrated these conclusions, and “those from those,” i.e., from diverse principles, demonstration being made in diverse sciences which treat of diverse genera.

Then (88b25) he presents the third reason which states that it is clear from another angle that this does not occur, namely, that the principles of all the sciences are the same; because it has been shown above that the principles of diverse genera are themselves generically diverse. Hence since diverse sciences are concerned with diverse genera, it follows that the principles of diverse sciences are diverse.

But because the common principles which all sciences use are in some way the same, he distinguishes among the principles and says that principles are twofold: some of the first principles from which one demonstrates are as the first dignities; for example, “Being and non-being are not the same.” Again, there are other principles, namely, those with which the sciences are concerned, namely, the subjects of the sciences, because we use the definitions of the subjects as principles in demonstrations. Therefore, the members of the first group of principles from which we demonstrate are common to all the sciences; but the principles with which the sciences are concerned are proper to each science, as number to arithmetic, and magnitude to geometry. But the common principles must be applied to these proper principles, if there is to be demonstration. And because one does not demonstrate only from common principles, it cannot be said that the principles of all demonstrative syllogisms are the same, which is what he intended to prove.

Lecture 44
(88b30-89b20)
SCIENCE COMPARED WITH OTHER MODES OF KNOWING

b30. Scientific knowledge— b3l. in that scientific knowledge— b33. So though there are— a5. This view also fits— a7. Besides, when a man— a11. In what sense, then,— a16. so that, since the— a23. The object of opinion— b6. Further consideration of modes— b10. Quick wit is a faculty

After comparing sciences with one another from the viewpoints of certitude, of unity and of diversity, the Philosopher here compares science to everything else that pertains to knowledge. And his treatment falls into two parts. In the first he treats of the comparison of science to opinion, which bears on the true and false. In the second, of the comparison of science to those habits of knowledge which are concerned exclusively with the true (89b6). The first part is divided into two parts. In the first he determines the truth. In the second he excludes a doubt (89a11). Concerning the first he does three things. First, he lays down the difference between science and opinion. Secondly, he shows what pertains to science (88b31). Thirdly, what pertains to opinion (88b33).

He says therefore first (88b30) that science differs from opinion; in like manner the scientifically knowable, which is the object of science, differs from the opinable, which is the object of opinion.

Then (88b31) he shows what pertains to science, setting down two things which pertain to it. One of these is that it is of the universal, for science is not concerned with singulars which fall under sense. (He explained this above). The other is that science is obtained in virtue of necessary things. And he explains that the necessary is that which cannot be otherwise. This, too, was explained above, namely, that demonstration proceeds from necessary things.

Then (88b33) he shows what pertains to opinion, namely, that it is concerned with things that could be otherwise, either universally or particularly. This he proves in three ways. First, by way of a division, saying that in addition to necessary truths, which cannot be otherwise, there are non-necessary truths, which could be otherwise. Now from what has been said above, it is clear that science is not concerned with the latter, because it would follow that contingent things would be such as could not be otherwise. For science is concerned with such things, as has been said. Similarly, it cannot be said that understanding is concemed with them. “Understanding” is taken here not as a faculty [power] of the soul, but as being the principle of science, i.e., as the habit of first principles, from which demonstration proceeds in causing science.

Hence in explaining what that understanding is which is the principle of science he adds, “Nor of indemonstrable knowledge” (88b37), i.e., not of facts that could be otherwise. As if to say: understanding is nothing more than a science of the indemonstrable. For just as science implies certainty of knowledge acquired through demonstration, so understanding implies certitude of knowledge without demonstration; not because of a shortcoming in some demonstration, but because that about which this certainty is had, is indemonstrable and per se known. Therefore, to explain this he adds that demonstrative science is nothing else than the certain intuition of an immediate proposition. But that understanding is science of the indemonstrable is clear from the fact that he says that it is the principle of science. For since science is concerned with the necessary, and the necessary is not concluded except from the necessary, as has been proved above, it is required that understanding, which is the principle of science, not be concerned with contingent things.

Having shown, therefore, that neither science nor understanding is concerned with contingent things, he sets down a division and says that understanding and science and opinion happen to be true as “that does which is stated by them,” i.e., which is verbally stated by understanding and science and true opinion. For the true is found both in the intellect’s act of composing and dividing and in the outward expression of this act insofar as it signifies the inward truth of the opinion or science or understanding. Therefore, if every truth belongs either to understanding or science or opinion, and there are some contingent truths, and they belong neither to science nor understanding, what is left is that opinion is concerned with them, whether they be actually true or actually false, provided that they could be other than they are.

In order to elucidate what opinion is, he adds that opinion is the acceptance, i.e., the grasping, of a proposition that is immediate and not necessary. And this can be understood in two ways: in one way, so that the immediate proposition in itself is indeed necessary, but it is accepted by opinion as non-necessary; in another way, so that it is in itself contingent. For an immediate proposition is one that cannot be proved through a middle, whether it be a necessary proposition or not. For it has been shown above that there is no process to infinity in predications, neither on the part of the middles nor on the part of the extremes. And this was shown not only analytically in demonstrations, but also logically in general as to syllogisms.

Therefore, if there be a contingent proposition which is mediate, it must be reduced to others that are immediate. However, it is not reduced to necessary immediate propositions, because necessary things are not the proper principles of contingent things, nor can contingent things be concluded from necessary things. Hence what remains is that some contingent, propositions are immediate and some mediate. Thus, “The man does not run,” is mediate, for it can be proved through this middle, “The man is not in motion,” which is also contingent, albeit immediate. The acceptance of such immediate contingent propositions is opinion. Yet this does not mean that the accepting of a mediate contingent proposition is not opinion. For opinion is related to contingent things, as science and understanding to necessary things.

Secondly (89a5), he proves the same point by means of that which is commonly observed, saying that what has been said, mainly, that opinion is concerned with the contingent is confessed, i.e., in agreement with things that are observed. For opinion seems to denote something weak and uncertain and to be of such a nature that weakness and uncertainty are in it.

Thirdly (89a7), he proves the same point by experiment. For no one, when he thinks that it is impossible for a thing to be otherwise, supposes that he is forming an opinion; rather he considers himself to know scientifically. But when he thinks that something is such and such and that there is nothing to prevent its being otherwise, then it is that he considers himself to have opinions. This would indicate that opinion concerns such things, i.e., contingent things, but science necessary things.

Then (89a11) he raises a doubt concerning the aforesaid. First, he raises the doubts. Secondly, he solves them (89a16).

Concerning the first (89a11), he presents two doubts, one of which is concerned with the opinable and the scientifically knowable. If opinion is concerned with the contingent and science with the necessary, and the contingent is not the same as the necessary, a doubt remains as to how a man can opine and scientifically know the same thing.

The second doubt concerns science and opinion, namely, why opinion is not science; supposing, of course, that one claims that there can be opinion about every known thing. For in regard to everything known a man can opine that it could be otherwise (except perhaps in the case of first principles per se known, whose contraries cannot be conceived, and concerning which there is not science but understanding). But in regard to all mediate truths of which there is demonstration and science, someone can suppose that it is possible that they could be otherwise. Consequently, they can fall under opinion. For opinion is not concerned exclusively with things contingent in their very nature, because then a man could not have opinion about everything he knows. Rather opinion is concerned with that which is accepted as possible to be otherwise, whether it is that or not. Therefore, on this supposition, it remains that science and opinion are the same, because both the scientific knower and the man of opinion acquire science and opinion through middles until they reach things which are immediate, as is clear from the foregoing. Hence if someone proceeds through middles to immediate propositions he has science. Now this is what the man of opinion does, because just as it is possible to have an opinion that something is so, it is also possible to have an opinion why something is so. But when I say, “why,” a middle is implied. Hence it is clear that opinion can proceed to immediates through a middle, whether opinion be of things that are contingent in their very nature, or of things accepted as contingent.

Then (89a16) he solves the aforesaid doubts. First, he solves the second one which is concerned with the identity of science and opinion. Secondly, he solves the first one which is concerned with the identity of the scientifically knowable and the opinable (89a23).

He says therefore first (89a16) that if someone proceeds through middles to immediates in such a way that the middles are not considered capable of being otherwise, but are considered to behave as definitions which are the middles through which demonstrations proceed, there will not be opinion but science. But if someone proceeds to the immediates through certain true middles which, nevertheless, are either not verified of the things of which they are said per se (as definitions which are predicated essentially or signify the species of a thing)-or he does not take them as being in these things per se, then he will have opinion and will not truly know the quia and the propter quid at once, even if he proceeds as far as the immediates: for then he will be forming an opinion through the immediates and will not know scientifically. But if he does not proceed through immediates but through mediates, then he will not be forming an opinion bearing on the propter quid but only the quia. For even scientific knowledge, when it is not propter quid and immediate, is not propter quid science but quia.

Then (89a23) he solves the first doubt which concerns the identity of the scientifically known and the object of opinion. Here it should be noted first of all that it is not unfitting that what is scientifically known by one person be a matter of opinion for another, because what one takes as impossible to be otherwise than stated and as scientifically known, another might take as possible to be otherwise and as a matter for opinion. But to say that the same man has opinion and scientific knowledge of the same thing at the same time, is not entirely true, but only in a way. For just as there can be true and false opinion about the same thing somehow but not absolutely, so also in regard to opinion and scientific knowledge. For if someone were to say that there could be a completely true and completely false opinion about the same thing, as some do say (for example, those who said that whatever seems to be true to a person is true, as it is stated in Metaphysics IV), then a person who would say such a thing would be forced to admit certain absurdities and the many other things pointed out in Metaphysics IV which follow from stating that the same thing is true and false. In particular this absurdity would follow, namely, that no opinion is false and that what he opined to be false he did not opine. Hence it cannot be said that there is true and false opinion of the same thing absolutely.

But because “same” is said in many ways, it happens that somehow there is true and false opinion about a same thing, and somehow not. For if “a same thing about which there are opinions” means the same real extramental subject, then there can be true and false opinion about a same thing. But if “same” refers to the proposition that is opined, then it is impossible. For example, to opine that the diagonal is commensurable with the side of a square is to opine with a false opinion; and it is absurd to say that this statement can be opined as a true opinion.

But if we take diagonal as the enunciable subject, then there can be true and false opinion of a same thing. For in regard to a diagonal, one opines truly that it is really incommensurable, and another opines falsely that it is commensurable. Thus, it is clear that although that concerning which there is science and opinion be the same in subject, it is not the same in notion. For the diagonal which is said to be commensurable and not to be commensurable is the same in subject, because the subject of each statement is the same. Nevertheless, it is clear that there is a mental distinction between diagonal insofar as it is taken as commensurable and insofar as it is taken as incommensurable. Consequently, there can be true and false opinion about the same subject, but not of the same subject according to notion.

The same is true of science and opinion. For it is science when someone knows this to be an animal in such a way that it cannot not be an animal; but it is opinion when someone knows it to be an animal in such a way that it might not be an animal. For example, if there is science concerning the fact that man is truly that which is an animal in such a way that it is impossible for it to be otherwise, there will be opinion concerning the fact that man is not truly that which is an animal, and as a result could be otherwise. For it is clear that the subject both known and opined is the same, namely, man; but in notion it is not the same. Thus, therefore, from the foregoing it is obvious that it is impossible to know scientifically and to opine the same thing at the same time, because a man would at one and the same time have a conjecture that could be otherwise and that could not be otherwise. But this could occur in the case of two men, namely, that one have science about a thing and another have opinion about the same thing, as has been stated. But this does not occur in the same person for the reason given.

Then (89b6) he compares science to certain habits that are concerned with truth. First, to those habits which deal with principles and conclusions. Secondly, to the habit which is specifically concerned with the middle (89b10).

He says therefore first (89b6) that in regard to those forms of knowledge that are other than opinion, the question of how they are distinguished into reason and understanding and science and art and prudence and wisdom, pertains somehow to the considerations of first philosophy or even of natural philosophy and somehow to the considerations of moral philosophy which is called ethics.

To understand this it should be noted that in Ethics VI, Aristotle lays down five things which are always concerned with the truth, namely, art, science, wisdom, prudence and understanding, adding two that are concerned with the true and false, namely, suspicion and opinion. Now the first five are concerned solely with the true, because they imply rectitude of reason. But three of them, namely, wisdom, science and understanding imply reason’s rectitude in regard to necessary truths: science in regard to conclusions, understanding in regard to principles, and wisdom in regard to the highest causes, which are the divine causes. But the other two, namely, art and prudence, imply the rectitude of reason in regard to contingent things: prudence in regard to things to be done, i.e., in regard to acts which remain within the one acting, such as to love, hate, choose, and similar acts that pertain to moral activity, of which prudence is the director. Art, however, implies the rectitude of reason in regard to things to be made, i.e., things that are done to external matter; for example, cutting and other activities in which art directs.

To this list he adds “reason,” which pertains to bringing, principles to their conclusions. Now to treat of wisdom, namely, what it is and how it works, and to treat of science and understanding and art, pertain somehow to first philosophy. Prudence, on the other hand, pertains to moral consideration; understanding and reason, insofar as they name certain powers, pertain to natural consideration, as is clear in On the Soul.

Then (89b10) he makes mention of the habit which is specifically concerned with the middle. And he says that quick wit is a certain accurate and facile conjecturing of the middle (the reason) why something occurs; and this when one does not have much time to inspect or deliberate, as when a person, seeing that the moon has all its brightness whenever it is turned toward the sun, immediately understands why this is, namely, because it is illumined by the sun. Similarly, in the case of human acts: if a person sees a poor man arguing with a rich man, he knows that the rich man took something from him, and the argument is about its return. Or if he should see persons formerly enemies now made friends, he knows that the reason is because both have a common enemy.

Secondly, he shows in two ways that to know the middle is to know the “why”: first, with the following reason, namely, because such a quick witted person, seeing all the middle causes, knows all the ultimate ones into which they are all finally analyzed, and through which the “why” is known. Secondly, he manifests this with an example presented in a syllogism. Thus, let us suppose the moon to be C, i.e., the minor extreme, and “to be bright because of its opposition to the sun,” to be A. the major extreme; but “to be illumined by the sun,” to be B, i.e., the middle. For C is B. i.e., “The moon has light from the sun,” and A is in B, because “that which has light from the sun is bright when turned toward the sun.” Thus it is proved that A is in C through B.

Hence it is clear that quick wit is the faculty of quickly apprehending the middle, and is present in persons as a natural gift or is acquired by training. But he presented various examples of quick wit to show that in all the aforesaid habits, namely, prudence, wisdom, and so on, there can be quick wit.

BOOK II

Lecture 1
(89b21-90a35)
EACH OF THE FOUR QUESTIONS WHICH PERTAIN TO SCIENCE IS ONE WAY OR ANOTHER A QUESTION OF THE MIDDLE

b2l. The kinds of question— b26. Thus, when our question— b28. On the other hand,— b32. but for some objects— b38. Now when we ask— a2. (By distinguishing— a5. We conclude that in— a24. Cases in which the

After determining about the demonstrative syllogism in the first book, the Philosopher intends in this [second] book to treat concerning its principles. But there are two principles of the demonstrative syllogism, namely, its middle and the first indemonstrable propositions. Therefore this book is divided into two parts. In the first he determines concerning the knowledge of the middle in demonstrations. In the second concerning the knowledge of the first propositions (99b18) [L. 20]. For since it has been established in the first book that every doctrine and every discipline takes its start from pre-existing knowledge, and since in demonstrations the knowledge of the conclusion is acquired through some middle and through the first indemonstrable propositions, we are left with the task of investigating how these come to be known.

But the first part is divided into two parts. In the first he inquires what is a middle in demonstrations. In the second part he inquires how that middle is made known to us (90a36) [L. 2]. But because the middle in demonstrations is employed in order to make known something about which there might have been doubt or question, therefore in regard to the first he does two things. First, he lays down the number of questions. Secondly, from these questions he pursues his investigation by showing how the questions pertain to the middle of demonstrations (89b38). Regarding the first he does three things. First, he enumerates the questions. Secondly, he explains complex questions (89b26). Thirdly, simple questions (89b32).

He says therefore first (89b21) that the number of questions is equal to the number of things that are scientifically known. The reason for this is that science is knowledge acquired through demonstration. But things which we previously did not know are those of which we must seek knowledge by demonstration: for it is in regard to things which we do not know that we form questions. Hence it follows that the things we inquire about are equal in number to the things we know through science. But there are four things that we ask, namely, quia [i.e., is it a fact that], propter quid [i.e., why, or what is the cause or reason], si est [if it is, i.e., whether it is], quid est [what is it]. To these four can be reduced whatever is scientifically inquirable or knowable.

However in Topics I he divides questions or problems into four kinds in a different way, but all of them are included under one of the questions listed here, namely, the one called quia. For in the Topics he is concerned only with questions to be disputed dialectically.

Then (89b26) he clarifies the questions he laid down; and first of all the complex ones. To understand this it should be noted that science bears only on the true, and the true is not signified except by an enunciation; therefore, only the enunciation can be scientifically knowable and so inquirable. But, as it is stated in On Interpretation II, the enunciation is formed in two ways: in one way from a name and a verb without an appositive, as when it is stated that man is; in another way when some third item is set adjacent, as when it is stated that man is white.

Therefore the questions we form can be reduced either to the first type of enunciation so that we get, as it were, a simple question; or to the second type, and then the question will be, as it were, complex or put in number, because, namely, the question concerns the putting together of two items.

According to this latter type a twofold question can be formed: one of them is whether this is true which is stated. This question he expounds first, saying that when we ask concerning some thing whether that thing is this or that, so that in effect we are somehow putting it in number, namely, by taking two things one of which is predicate and the other subject, as when we ask if the sun is failing because of an eclipse or not, and is man an animal or not, then we are said to ask quia: not in the sense that the word “quia” functions as a question mark, but because we are asking in order to find out quia [i.e., that] it is so. An indication of this is that when we have discovered it through demonstration, we cease our questioning; and if we had known it at the very beginning, we would not have asked whether it is so. But inquiry does not cease until that is obtained which was asked. And so, since the question in which we ask whether this is this ceases once we have certified that it is so, it is clear what a question of this kind asks.

Then (89b28) he clarifies the next question which also puts in number, ‘I saying that when we know that it is so, we ask propter quid [i.e., why] it is so. For example, when we know that the sun is failing through an eclipse and that the earth is moved during an earthquake, we ask why the sun is failing or why the earth is being moved. Therefore we ask it in this way, namely, by putting in number.

Then (89b32) he clarifies the other two questions which do not put in number but are simple. And he says that we ask certain things in a manner different from the aforesaid questions, namely, by not putting in number; as when we ask whether or not there be centaurs, for in this case the question we ask concerning the centaur is simply whether it exists and not whether the centaur be this, say white, or not. And just as when we knew that this is this, we then asked why, so once we know of something simply that it is, we ask what it is, for example, what is God or what is man. These then and so many are the things we ask; and when we have found the answer, we are said to know scientifically.

Then (89b38) he shows how the aforesaid questions are related to the middle. Concerning this he does three things. First, he states what he intends. Secondly, he explains what he had said (90a2). Thirdly, he proves his proposition (90a5).

In regard to the first it should be noted that two of the aforesaid questions put in number and two do not. From the first member of each of these groups, he forms another grouping composed of the question that it is and the question if it is. And he says that when we ask that this is this, or when we simply ask concerning something, if it is, we are not asking anything else than whether or not a middle is to be found of that which we ask; and this is something not conveyed by the form of the question. For when I ask whether the sun is eclipsed or whether man exists, it is not obvious from the form of the question that I am asking whether there is some middle by which it might be demonstrated that the sun is eclipsed or that man exists; but if the sun is eclipsed or man does exist, the consequence is that some middle can be found to demonstrate the things which are inquired. For no one forms a question concerning immediate things which, although they are true, do not have a middle, since things of this sort, being evident, do not fall under a question. Thus, therefore, one who asks whether this is this or whether this absolutely is, as a consequence is asking whether there is a middle of this sort. For in the question if it is or that it is, one is asking whether that which is a middle exists, because that which is the middle is the reason of that concerning which one asks whether this is this or simply whether it is, as will be explained below. Nevertheless the question is not being asked under the aspect of middle.

Now it happens that having found the answer to what is asked by these two questions, one knows either that it is or if it is: one of which consists in knowing the existence absolutely; but the other in part, as when we know that man is white, because to be white does not signify the existence of man in his entirety but signifies him to be something. This is why when a man is becoming white, we do not say that he is coming to be absolutely, but in a qualified sense. But when it is asserted that man is, his existence is signified absolutely, so that when a man comes to be, he is said to become absolutely.

Therefore, when we know that it is and ask why it is, or when we know if it is and ask what it is, we are asking what is the middle. And as in the other cases, so here, this is gathered not from the form of the question but by way of concomitance. For one who seeks the cause why the sun is eclipsed is not seeking it as a middle which demonstrates, but he is seeking that which is a middle, because it is by way of consequence that once he has it, he can demonstrate. And the same applies to the question what is it.

Then (904) he manifests what he had said, namely, that that it is and if it is differ as in part and absolutely differ. For when we inquire whether the moon is waning or waxing, it is a question in part, since in a question of this type we are asking if the moon is something, namely, is it waning or has it waxed or not. But when we ask whether the moon exists or whether it is night, the question bears on existence absolutely.

Then (90a5) he proves his point, namely, that the aforesaid questions pertain to the middle. First, he proves it with a reason. Secondly, with a sign (90a24).

He concludes therefore first (90a5) from the explanation given above that in all the aforesaid questions one is either asking whether there is a middle, namely, in the question that it is and in the question if it is, or what the middle is, namely, in the question why and in the question what is it.

And he proves that the question why inquires what the middle is. For it is obvious that a cause is the middle in a demonstration which enables one to know scientifically, because to know scientifically is to know the cause of a thing. But it is precisely the cause that is being sought in all the above questions. That this is so he manifests first in regard to the question that. For when it is asked whether the moon is waning, then according to the manner explained above, what is being asked is whether or not something is the cause of this waning. Then he shows this for the question why. For once we know that something is the cause of the moon’s waning, we ask what the cause is; and this is to inquire why.

The same applies to the other two questions, as he shows in the following way. For he says that whether we assert not that something is this or that (for example, when I say that man is white or is a grammarian), but that the substance itself exists absolutely; or do not assert that some thing exists absolutely, but that some thing is something by putting in number (whether that something be a thing predicated per se or a thing predicated per accidens); no matter in which of these ways we take the thing to be, its cause is the middle for demonstrating it.

Then he explains what he means by a substance to exist absolutely when we inquire concerning the moon or the earth or a triangle or any other subject whether it is and then take some middle to demonstrate this. I say that a thing is something when we inquire concerning eclipse in regard to the moon, or equality or inequality in regard to triangle, or whether it is in the middle of the universe or not in regard to the earth. And he asserts that as far as the present point is concerned it makes no difference which way a thing is taken to be, because in all these cases what it is is the same as why.

He manifests this first of all in regard to the waning, of the moon. For if one asks what is the eclipse of the moon, the answer is that it is the absence of light in the moon because of the earth’s being set between it and the sun. And this same answer is given when it is inquired why the moon is eclipsed. For we say that the moon is eclipsed because there is a lack of light due to the earth’s opposition.

Then he manifests the same idea with another example. For if one asks what is a chord, the answer is given that it is a numerical ratio according to high and low notes. Again, if one asks why a high note and a low note are concordant, the answer is given that it is because the high note and the low note have a numerical ratio. And so the question what is it and the question why reduce to the same thing subjectively, although they differ in formality. Hence because the question why leads us to inquire what the middle is, as has been shown, what is left is that when one asks what is it, the middle is likewise inquired.

Then he shows the same thing in regard to the question that. For, as has been said, concordance is a numerical ratio between high and low notes. Men, therefore, one asks whether a high and low note concord, he is inquiring whether there is some numerical ratio of the high and low note; and this is the middle for demonstrating that a high and a low note concord. Consequently in the question that, one inquires whether there is a middle. But once we have found that there is a numerical ratio of the high and low note, we then ask what that ratio is: and this is to ask what or why.

Here Aristotle seems to say that the definition of a proper attribute is the middle in demonstration. However it must be remarked that the definition of the proper attribute cannot be completed without the definition of the subject. For it is obvious that the principles which the definition of the subject contains are the principles of the proper attribute. Hence a demonstration will not reach the first cause unless one takes as the middle of demonstration the definition of the subject. And so the proper attribute must be concluded of the subject by means of the definition of the proper attribute; furthermore, the definition of the proper attribute must be concluded of the subject through the definition of the subject. This is why it was laid down at the very beginning that one must know beforehand the what is it not only of the proper attribute but of the subject also; which would not be required unless the definition of the proper attribute were concluded of the subject through the definition of the subject.

This is clear from the following example: If we wish to demonstrate of triangle that it has three angles equal to two right angles, we first take as middle the fact that it is a figure having an exterior angle equal to its two opposite interior angles-which is, as it were, the definition of a proper attribute. But this in turn must be demonstrated by the definition of the subject, so that we would say: “Every closed figure of three straight lines has an exterior angle equal to its two opposite interior angles; but the triangle is such a figure. Therefore...” And the same is true if we were to demonstrate that a high note concords with a low note: for we would state the definition of the attribute which in this case consists in their having a numerical ratio; but then to demonstrate this attribute we would have to take the definition of high note and of low note. For a low note is one which is apt to act on the sense for a long time, whereas a high note is one which does so for a short time: but between the long and the short there is a numerical ratio. Therefore, there is a numerical ratio between a high note and a low note. And if the high note and the low note were defined some other way, it would make no difference. For in any case something pertaining to quantity would have to appear in their definition, so that it would be necessary to conclude that there is a numerical ratio between them.

Then (90a44) he manifests his point by means of a sensible sign. And he says that those cases in which the middle is perceptible by sense clearly show that every question is a question concerning a middle, because, namely, when the middle is known through the senses, no room is left for a question. For we ask one of the aforesaid questions in matters pertaining to sense, when the middle does not appear: thus, we ask whether or not there is an eclipse of the moon because we do not sensibly perceive the middle which is the cause making the moon to be eclipsed. But if we were to situate ourselves in a place above the moon, we would see how the moon became eclipsed by entering the earth’s shadow. Then we would no longer ask if it is or why it is, but both would at once be obvious to us.

But because someone could object that sense-perception bears on singulars—whereas it is universals that are being asked about, just as it is universals that are scientifically known—and consequently, it does not seem that the matter under question can be made known through sense; therefore, as though in answer to this objection he adds that it is precisely because we do sense the particular (namely, that this body of the moon enters this shadow of the earth), that we happen at once to know the universal. For our sense would observe the fact that the light of the sun is now blocked by the earth’s opposition; and through this it would be clear to us that the moon is now eclipsed. And because we would conjecture that the eclipse of the moon always occurs in this way, the sense knowledge of the singular would immediately become a universal in our science.

And so from this example he concludes that knowing the quod quid [quod quid refers to any or all of the items that constitute the essential nature of a thing.] and the why are the same. For from the fact that we observe the earth situated between the sun and moon, we would know scientifically both what an eclipse of the moon is and why the moon is eclipsed. And one of these, namely, the knowledge what it is, is reduced to the science by which we know that something simply is, and not that something is in some thing. But the why is reduced to our knowledge of things that are in some thing, as when we say that three angles are equal to or greater than or less than two right angles.

Finally, he summarizes and concludes the main point, namely, that it is clear from the foregoing that in all questions there is question of the middle.

Lecture 2
(90a36-91a3)
WHETHER THE DEFINITION WHICH SIGNIFIES THE QUOD QUID OF A THING CAN BE DEMONSTRATED

a36. Let us now state— b1. beginning what we— b3. It might, I mean,— b7. And again, not even— b14. Induction too will— b16. Again, if to define— b19. What then? Can— b23. Moreover, the basic— b28. But if the definable— b33. Moreover, every demonstration— b38. Again, to prove essential

After showing that every question is in some sense a question of the middle, which is the quod quid and the propter quid, the Philosopher now begins to make manifest how the middle becomes known to us. And it is divided into two parts. In the first part he shows how the quod quid and propter quid are related to demonstration. In the second part he shows how the quod quid and propter quid should be investigated (96a22) [L. 13]. The first part is divided into two parts. In the first he manifests how the quid est is related to demonstration. In the second he manifests how the propter quid which signifies a cause is related to demonstration (94a20) [L. 9]. In regard to the first he does two things. First, he states his proposal. Secondly, he pursues his proposal (90b1).

He says therefore first (90a36) that since every question for whose solution a demonstration is adduced is a question of the middle which is quid and propter quid, our first task is to declare how the quod quid is made known to us, namely, is it through demonstration or division or some other way. Furthermore, we must point out how to reduce to the quod quid those items of a thing that are apparent. And because a definition is a statement which signifies the quod quid, it is necessary also to know what a definition is and why certain things are definable. Consequently we shall proceed in these matters in the following way. First, by raising difficulties. Secondly, by establishing the truth (93a1) [L. 7]. Then (90b1) he pursues his proposal in the order stated. First, he proceeds dialectically by raising difficulties. Secondly, by determining the truth (93a1) [L. 7]. In regard to the first he does two things. First, he proceeds by disputing about the definition which signifies quod quid. Secondly, about the quod quid which is signified by the definition (91a12) [L. 31. Concerning the first he does three things. First, he inquires disputatively whether there is definition of all things of which there is demonstration. Secondly, whether conversely there is demonstration of all things of which there is definition (91b19). Thirdly, whether there can be definition and demonstration of the same thing (90b27). In regard to the first he does two things. First, he states his intention. Secondly, he pursues his proposal (90b1).

He says therefore first (90b1) that when a number of items are planned for discussion, one should take his start from the item which is most fruitful in settling the “had,” i.e., the subsequent, problems. This would mean that in our case we should begin with the fact that someone might wonder whether the same thing and according to the same aspect could be known through definition and demonstration.

Then (90b4) he proves that there is not definition of all things of which there is demonstration. This he does in four ways, the first of which is the following: a definition is indicative of the quod quid; but anything which pertains to the quod quid is predicated both affirmatively and universally: therefore, a definition contains or signifies only those things which are predicated affirmatively and universally. But not all syllogisms demonstrate affirmative universal conclusions: in fact some are negative, as for example, all those in the second figure; and some are particular, as for example, all those in the third figure. Therefore, there is not definition of all things of which there is demonstration.

Secondly (90b7), he shows the same thing when he says that there cannot even be definition of all things which are concluded through affirmative syllogisms-which occurs only in the first figure, as when it is demonstratively syllogized that every triangle has three angles equal to two right angles. Now the reason for this statement, namely, that there cannot be definition of all things thus syllogized, is that to know something demonstratively is nothing else than to have a demonstration. From this it follows that if the science of all these things is had only through demonstration, there is not definition of them. For things which have a definition are made known through their definition. Otherwise it would follow that a person who does not have a demonstration of these things would have scientific knowledge of them, for the simple reason that nothing hinders a person who knows the definition from not having at the same time the demonstration-although the definition is a principle of the demonstration. For not everyone who knows the principles knows how to deduce the conclusion by demonstrating.

Thirdly (9014), he shows the same thing through an induction from which a conviction can arise sufficient for admitting the above-mentioned conclusion. He says, therefore, that demonstration is concerned with things which are per se i