Commentary Aristotle Posterior Analytics
Lecture 28
How by syllogizing in the first or second figure a false negative is concluded contrary to an immediate affirmative a8. On the other hand, an error of non-attribution may occur either in the first or in the second figure. Let us therefore first explain the various forms it takes in the first figure and the character of the premises in each case.
a11. (c) It may occur when both premises are false; e.g., supposing A atomically connected with both C and B, if it be then assumed that no C is A and all B is C, both premises are false.
a14. (d) It is also possible when one is false. This may be either premise indifferently. A-C may be true, C-B false — A-C true because A is not an attribute of all things, C-B false because C, which never has the attribute A, cannot be an attribute of B; for if C-B were true, the premise A-C would no longer be true, and besides if both premises were true, the conclusion would be true.
a20. Or again, C-B may be true and A-C false; e.g., if both C and A contain B as genera, one of them must be subordinate to the other, so that if the premise takes the form No C is A, it will be false. This makes it clear that whether either or both premises are false, the conclusion will equally be false.
a27. In the second figure the premises cannot both be wholly false; for if all B is A, no middle term can be with truth universally affirmed of one extreme and universally denied of the other: but premises in which the middle is affirmed of one extreme and denied of the other are the necessary condition if one is to get a valid inference at all. Therefore, if, taken in this way, they are wholly false, their contraries conversely should be wholly true. But this is impossible. On the other hand, there is nothing to prevent both premises being partially false; e.g., if actually some A is C and some B is C, then if it is premised that all A is C and no B is C, both premises are false, yet partially, not wholly, false. The same is true if the major is made negative instead of the minor.
a38. Or one premise may be wholly false, and it may be either of them. Thus, supposing that actually an attribute of all A must also be an attribute of all B, then if C is yet taken to be a universal attribute of all but universally non-attributable to B, C-A will be true but C-B false. Again, actually that which is an attribute of no B will not be an attribute of all A either; for if it be an attribute of all A, it will also be an attribute of all B, which is contrary to supposition; but if C be nevertheless assumed to be a universal attribute of A, but an attribute of no B, then the premise C-B is true but the major is false.
b6. The case is similar if the major is made the negative premise. For in fact what is an attribute of no A will not be an attribute of any B either; and if it be yet assumed that C is universally non-attributable to A, but a universal attribute of B, the premise C-A is true but the minor wholly false. Again, in fact it is false to assume that that which is an attribute of all B is an attribute of no A, for if it be an attribute of all B, it must be an attribute of some A. If then C is nevertheless assumed to be an attribute of all B but of no A, C-B will be true but C-A false.
b14. It is thus clear that in the case of atomic propositions erroneous inference will be possible not only when both premises are false but also when only one is false.
On the other hand, an error of non-attribution. 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.
Second, in the second, at: in the second figure the premises (80a27).
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 and 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.
Second, at: thus, let A be atomically disconnected (80a11), he establishes his proposition.
First, he shows how such a proposition is formed from two false premises in the first figure.
Second, how it is formed from one false and one true premise, at: it is also possible when one is false (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, that is, 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, at: it is also possible when one is false (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.
Second, how it might be the reverse, at: or again, C-B may be true (80a20).
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, that is, “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, at: or again, C-B may be true (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', that is, C. It is necessary, therefore, according to the aforesaid, that as between these two, namely, A and C, one should be under the other, that is, 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, at: in the second figure the premises (80a27) he shows how a negative syllogism of ignorance is formed in the second figure.
First, when both are false.
Second, when one or the other is false, at: or one premise may be wholly false (80a38).
He says therefore first (80a27) 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, that is, if we should say, “No animal is male” and “Every man is male.”
Then, at: or one premise may be wholly false (80a38) he shows how it happens when one is false.
First, in the second mode of the second figure.
Second, in the first mode, at: the case is similar if the major (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 CB, 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, at: the case is similar if the major (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 proposition 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.