Did Aristotle make a mistake in the Prior Analytics, Part 5?

Question

Did Aristotle make a mistake in Part 5 of the prior analytics?

“If the middle term is related universally to one of the extremes, a particular negative syllogism must result whenever the middle term is related universally to the major whether positively or negatively, and particularly to the minor and in a manner opposite to that of the universal statement: by 'an opposite manner' I mean, if the universal statement is negative, the particular is affirmative: if the universal is affirmative, the particular is negative. For if M belongs to no N, but to some O, it is necessary that N does not belong to some O. For since the negative statement is convertible, N will belong to no M: but M was admitted to belong to some O: therefore N will not belong to some O: for the result is reached by means of the first figure. Again if M belongs to all N, but not to some O, it is necessary that N does not belong to some O: for if N belongs to all O, and M is predicated also of all N, M must belong to all O: but we assumed that M does not belong to some O. And if M belongs to all N but not to all O, we shall conclude that N does not belong to all O: the proof is the same as the above. But if M is predicated of all O, but not of all N, there will be no syllogism. Take the terms animal, substance, raven; animal, white, raven. Nor will there be a conclusion when M is predicated of no O, but of some N. Terms to illustrate a positive relation between the extremes are animal, substance, unit: a negative relation, animal, substance, science.”

He says there’s a conclusion for “all N is M, some O is not M,” namely some O is not N. But then he says there’s no conclusion for “all O is M, some N is not M,” which I’d assume to be some N is not O

Answer 1

Revised edition

The 2nd figure is defined as follows: the Middle is predicated in both premises and 1st premise is universal.

The valid syllogism in this figure are: EAE, AEE, EIO and AOO.

Example of AOO (Baroco):

if M belongs to all N, but not to some O, it is necessary that N does not belong to some O.

We use Alexander of Aphrodisias' commentary [see On Aristotle Prior Analytics 1.14-22, page 155].

We can compare Alexander's comment to Aristotle counterexample method in An.Prior, 26a9-12:

Nor can a deduction be formed when neither the first term belongs to any of the middle, nor the middle to any of the last [i.e. "A si predicated of no B" and "B is predicated of no C" is invalid.]

In this case, the two instances provided show that the truth of the premises is compatible both with an A and an E conclusion. But they entails respectively I and O [see The Square of Opposition], and thus the EE form is not valid.

What about the two passages discussed above ?

The first is a counterexample to an OA form:

if M is predicated of all O, but not of all N, there will be no syllogism. Take the terms animal, substance, raven; animal, white, raven.

I think that the key-point is: we cannot exchange the two premises.

If so, the conclusion must express a relation of predication between the major (O) and the minor (N): "raven is predicated (not predicated) of all (some) substance (animal)"

The counterexamples use for M,N,O: animal, substance, raven; animal, white, raven.

With the first three terms we have that the 1st premise is: "animal (M) is predicated of every raven (O)", and the 2nd one is: "animal (M) is not predicated of every substance (N)".

But "substance (N) is predicated of every raven (O)" which means that "raven is predicated of some substance".

The conclusion is I, which implies that the premises cannot validly conclude with E (the contradictory of I).

But, at the same time, "raven is not predicated of some substance" is false, and thus the premises cannot validly conclude with O.

With the next three terms we have that the 1st premise is: "animal (M) is predicated of every raven (O)", and the 2nd one is: "animal (M) is not predicated of every white (N)".

But "white (N) is predicated of no raven (O)", i.e. "raven is predicated of no white", which is an E proposition.

This implies that the premises cannot validly conclude with I (the contradictory of I). But E implies O, and thus also A is excluded.

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Conclusion:

if we agree that we cannot exchange major and minor premises, and thus OA is different from AO, there is no OA valid syllogism in the 2nd figure.