Commentary Aristotle Posterior Analytics
Lecture 29
Syllogism of ignorance in regard to mediate propositions b17. In the case of attributes not atomically connected with or disconnected from their subjects, (a) (i) as long as the false conclusion is inferred through the ‘appropriate’ middle, only the major and not both premises can be false. By ‘appropriate middle’ I mean the middle term through which the contradictory—that is, the true—conclusion is inferrible. Thus, let A be attributable to B through a middle term C: then, since to produce a conclusion the premise C-B must be taken affirmatively, it is clear that this premise must always be true, for its quality is not changed. But the major A-C is false, for it is by a change in the quality of A-C that the conclusion becomes its contradictory—that is, true.
b27. Similarly (ii) if the middle is taken from another series of predication; e.g., suppose D to be not only contained within A as a part within its whole but also predicable of all B. Then the premise D-B must remain unchanged, but the quality of A-D must be changed; so that D-B is always true, A-D always false. Such error is practically identical with that which is inferred through the ‘appropriate’ middle.
b32. On the other hand, (b) if the conclusion is not inferred through the ‘appropriate’ middle—(i) when the middle is subordinate to A but is predicable of no B, both premises must be false, because if there is to be a conclusion both must be posited as asserting the contrary of what is actually the fact, and so posited both become false: e.g., suppose that actually all D is A but no B is D; then if these premises are changed in quality, a conclusion will follow and both of the new premises will be false. When, however, (ii) the middle D is not subordinate to A, A-D will be true, D-B false—A-D true because A was not subordinate to D, D-B false because if it had been true, the conclusion too would have been true; but it is ex hypothesi false.
a5. When the erroneous inference is in the second figure, both premises cannot be entirely false; since if B is subordinate to A, there can be no middle predicable of all of one extreme and of none of the other, as was stated before. One premise, however, may be false, and it may be either of them. Thus, if C is actually an attribute of both A and B, but is assumed to be an attribute of A only and not of B, C-A will be true, C-B false: or again if C be assumed to be attributable to B but to no A, C-B will be true, C-A false. We have stated when and through what kinds of premises error will result in cases where the erroneous conclusion is negative.
a16. If the conclusion is affirmative, (a) (i) it may be inferred through the ‘appropriate’ middle term. In this case both premises cannot be false since, as we said before, C-B must remain unchanged if there is to be a conclusion, and consequently A-C, the quality of which is changed, will always be false.
a20. This is equally true if (ii) the middle is taken from another series of predication, as was stated to be the case also with regard to negative error; for D-B must remain unchanged, while the quality of A-D must be converted, and the type of error is the same as before.
a25. (b) The middle may be inappropriate. Then (i) if D is subordinate to A, A-D will be true, but D-B false; since A may quite well be predicable of several terms no one of which can be subordinated to another. If, however, (ii) D is not subordinate to A, obviously A-D, since it is affirmed, will always be false, while D-B may be either true or false; for A may very well be an attribute of no D, whereas all B is D, e.g., no science is animal, all music is science. Equally well A may be an attribute of no D, and D of no B. It emerges, then, that if the middle term is not subordinate to the major, not only both premises but either singly may be false.
a35. Thus we have made it clear how many varieties of erroneous inference are liable to happen and through what kinds of premises they occur, in the case both of immediate and of demonstrable truths.
In the case of attributes not atomically. 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.
Second, how a false affirmative is concluded which is opposed to a true negative, at: if the conclusion is affirmative (81a16).
Concerning the first he does two things.
First, he shows this in the first figure.
Second, in the second, at: when the erroneous inference (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.
Second, 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, at: similarly, if the middle (80b27).
Third, he shows how such a syllogism is constructed through an extraneous middle, at: on the other hand, if the conclusion (80b32).
He says therefore first (80b17) that when a syllogism is constructed which concludes something false in propositions which are not individual, that is, 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, that is, 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, at: similarly, if the middle is taken (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, at: on the other hand, if the conclusion (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, AD, 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, that is, 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, at: when the erroneous inference (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 or 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, at: if the conclusion is affirmative (81a16) he shows how an affirmative syllogism of deception is formulated in mediate propositions.
First, when it is formulated through a proper middle.
Second, when it is formulated through a middle similar to a proper middle, at: this is equally true if the middle (81a20).
Third, when it is formulated through an extraneous middle, at: the middle may be inappropriate (81a25).
He says therefore first (81a16), 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, at: this is equally true if the middle (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, at: the middle may be inappropriate (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. Therefore, 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.