Commentary Aristotle Posterior Analytics
Lecture 22
Each science has its own deceptions and areas of ignorance b16. Since there are “geometrical” questions, does it follow that there are also distinctively “ungeometrical” questions?
b17. Further, in each special science—geometry for instance—what kind of error is it that may vitiate questions, and yet not exclude them from that science?
b19. Again, is the erroneous conclusion one constructed from premises opposite to the true premises, or is it formal fallacy though drawn from geometrical premises?
b21. Or, perhaps, the erroneous conclusion is due to the drawing of premises from another science; e.g., in a geometrical controversy a musical question is distinctively ungeometrical, b23. whereas the notion that parallels meet is in one sense geometrical, being ungeometrical in a different fashion: the reason being that ‘ungeometrical’, like ‘unrhythmical’, is equivocal, meaning in the one case not geometry at all, in the other bad geometry? It is this error, that is, error based on premises of this kind — ‘of’ the science but false — that is the contrary of science.
b27. In mathematics the formal fallacy is not so common, because it is the middle term in which the ambiguity lies, since the major is predicated of the whole of the middle and the middle of the whole of the minor (the predicate of course never has the prefix ‘all’); and in mathematics one can, so to speak, see these middle terms with an intellectual vision, while in dialectic the ambiguity may escape detection. e.g., ‘Is every circle a figure?’ A diagram shows that this is so, but the minor premise ‘Are epics circles?’ is shown by the diagram to be false.
b34. If a proof has an inductive minor premise, one should not bring an ‘objection’ against it. For since every premise must be applicable to a number of cases (otherwise it will not be true in every instance, which, since the syllogism proceeds from universals, it must be), then assuredly the same is true of an ‘objection’; since premises and ‘objections’ are so far the same that anything which can be validly advanced as an ‘objection’ must be such that it could take the form of a premise, either demonstrative or dialectical.
b40. On the other hand, arguments formally illogical do sometimes occur through taking as middles mere attributes of the major and minor terms. An instance of this is Caeneus’ proof that fire increases in geometrical proportion: ‘Fire,’ he argues, ‘increases rapidly, and so does geometrical proportion’. There is no syllogism so, but there is a syllogism if the most rapidly increasing proportion is geometrical and the most rapidly increasing proportion is attributable to fire in its motion.
a5. Sometimes, no doubt, it is impossible to reason from premises predicating mere attributes: but sometimes it is possible, though the possibility is overlooked.
a8. If false premises could never give true conclusions ‘resolution’ would be easy, for premises and conclusion would in that case inevitably reciprocate. I might then argue thus: let A be an existing fact; let the existence of A imply such and such facts actually known to me to exist, which we may call B. I can now, since they reciprocate, infer A from B.
a10. Reciprocation of premises and conclusion is more frequent in mathematics, because mathematics takes definitions, but never an accident, for its premises—a second characteristic distinguishing mathematical reasoning from dialectical disputations.
a13. A science expands not by the interposition of fresh middle terms, but by the apposition of fresh extreme terms. e.g., A is predicated of B, B of C, C of D, and so indefinitely. Or the expansion may be lateral: e.g., one major A, may be proved of two minors, C and E. Thus let A represent number — a number or number taken indeterminately; B determinate odd number; C any particular odd number. We can then predicate A of C. Next let D represent determinate even number, and E even number. Then A is predicable of E.
Since there are “geometrical” questions. 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, at: or, perhaps, the erroneous conclusion (77b21; [182]).
Accordingly, he poses three questions.
The first (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, at: further, in each special science (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, at: again, is the erroneous conclusion (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, that is, 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, that is, 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, at: or, perhaps, the erroneous conclusion (77b21) he solves these questions.
First, he solves the first.
Second, the second, at: whereas the notion (77b23; [183]).
Third, the third, at: in mathematics the formal fallacy (77b27; [184]).
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, at: whereas the notion (77b23) he solves the second question.
He says that a question about geometry, that is, 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, that is, equidistant lines coming together), would be geometric in one sense and non-geometric in another. For just as arrythmon, that is, 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, at: in mathematics (77b27) he solves the third question.
And he does two things.
First, he shows that in demonstrative sciences there is no paralogism in language.
Second, nor apart from language, at: if a proof has an inductive minor premise (77b34; [185]).
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, that is, 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, that is, 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,” that is, 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, that is, in argumentations, the ambiguity may go unnoticed, that is, 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, that is, 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, at: if a proof has an inductive minor premise (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, he does two things.
First, he shows how an objection should be presented in demonstrative matters.
Second, he shows that in demonstrative matters there cannot be paralogism according to fallacy outside the language, at: on the other hand, arguments (77b40; [186]).
He says therefore first (77b34) that one should not bring an objection against it, that is, against a paralogism, by citing an inductive, that is, 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, at: on the other hand, arguments (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.
Second, that deception does not take place in this manner in demonstrative matters, at: sometimes, no doubt (78a5; [188]).
He says therefore first (77b40) that illogical arguments do sometimes occur, that is, do not observe syllogistic form, because they admit both inherences, that is, 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, that is, 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, at: sometimes, no doubt (78a5) he shows that deception arising from this form of reasoning does not occur in demonstrative sciences.
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.
Second, at: if false premises could never give (78a7), he shows that it is possible to syllogize in the abovementioned manner without deception.
And he shows this in three ways.
The first 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, at: reciprocation of premises (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, that is, in demonstrative sciences, are for the most part convertible, because these sciences do not admit as a middle anything predicated accidentally, but only definitions which are principles of demonstration, as we have explained. And this is their point of difference from the statements in dialogues, that is, in dialectical syllogisms, in which accidents are frequently admitted.
Then, at: a science expands not by the interposition (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, that is, 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, that is, 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, that is, “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), that is, 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: a science expands (78a13) 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.