11 Barbara, major A, minor A.
12 24b 28.
13 Celarent, major E, minor A.
14 Major A, minor E.
15 Major E, minor E.
16 Darii.
17 24b 28.
18 Ferio.
19 24b 30.
20 The Aristotelian formula for the proposition, AB, in which B represents the subject and A the predicate (A belongs to B), has been retained throughout, because in most places this suits the context better than the modern formula in which A represents the subject and B the predicate.
21 Major I or O, minor A.
22 Major I or O, minor E.
23 Major A, minor O.
24 Major E, minor O.
25 a 2.
26 Major A, minor O.
27 i. e. the major premiss.
28 Major E, minor O.
29 II, OO, IO, OI.
30 Cesare.
31 25b 40.
32 Camestres.
33 l. 3.
34 Festino.
35 Baroco.
36 a 21.
37 l. 18.
38 Darapti.
39 Felapton.
40 See note 20.
41 Disamis.
42 Datisi.
43 Bocardo.
44 27b 20.
45 28a 30.
46 Ferison.
47 Fesapo, Fresison.
48 sc. in the first figure.
49 viz. by reduction per impossibile to Celarent and Barbara.
50 25b 21.
51 Post An. i. 8.
52 Cf. 25b 32.
53 Aristotle is thinking of the method of establishing a proposition A is B by inducing the opponent to agree that A is B if X is Y. All that remains then is to establish syllogistically that X is Y. That A is B thus follows from the agreement.
54 The diagram Aristotle has in mind appears to be the following:
Here A and B are the equal sides, E and F the angles at the base of the isosceles triangle. C and D are the angles formed by the base with the circumference. The angles formed by the equal sides with the base are loosely called AC, BD.
55 sc. but imperfect.
56 40b 30.
57 l. 6.
58 The reference is to the new premisses produced by conversion, when a syllogism in the second or third figure is being reduced to one in the first. Cf. 24b 24.
59 Post An. i. 19–22.
60 i. e. on the major and minor terms. Two affirmative premisses in the second figure give no conclusion.
61 44b 20.
62 We thus get a syllogism in Barbara.
63 Darapti.
64 Cesare.
65 Camestres.
66 By converting the major premiss of the Cesare syllogism or the minor premiss of the Camestres syllogism.
67 Felapton, by conversion.
68 i. e. the consequents of A and E.
69 27a 18–20,b 23–8.
70 i. e. if this false conclusion is replaced by its contradictory and this is treated as a premiss.
71 ii. 14.
72 Cf. 41a 39.
73 Topics, especially i. 14.
74 Aristotle is thinking of Plato’s establishment of definitions by means of division by dichotomy.
75 In cc. 1–30.
BOOK II
16 … To beg and assume the original question is a species of failure to demonstrate the problem proposed; but this happens in many ways. [64b] A man may not reason syllogistically at all, (30) or he may argue from premisses which are less known or equally unknown, or he may establish the antecedent by means of its consequents; for demonstration proceeds from what is more certain and is prior. Now begging the question is none of these: but since we get to know some things naturally through themselves, and other things by means of something else (the first principles through themselves, (35) what is subordinate to them through something else), whenever a man tries to prove what is not self-evident by means of itself, then he begs the original question. [65a] This may be done by assuming what is in question at once; it is also possible to make a transition to other things which would naturally be proved through the thesis proposed, (40) and demonstrate it through them, e. g. if A should be proved through B, and B through C, though it was natural that C should be proved through A: for it turns out that those who reason thus are proving A by means of itself. This is what those persons do who suppose that they are constructing parallel straight lines: for they fail to see that they are assuming facts which it is impossible to demonstrate unless the parallels exist. (5) So it turns out that those who reason thus merely say a particular thing is, if it is: in this way everything will be self-evident. But that is impossible. (10)
If then it is uncertain whether A belongs to C, and also whether A belongs to B, and if one should assume that A does belong to B, it is not yet clear whether he begs the original question, but it is evident that he is not demonstrating: for what is as uncertain as the question to be answered cannot be a principle of a demonstration. If however B is so related to C that they are identical, (15) or if they are plainly convertible, or the one belongs to the other, the original question is begged. For one might equally well prove that A belongs to B through those terms if they are convertible. But if they are not convertible, it is the fact that they are not that prevents such a demonstration, not the method of demonstrating. But if one were to make the conversion, then he would be doing what we have described and effecting a reciprocal proof with three propositions.
Similarly if he should assume that B belongs to C, (20) this being as uncertain as the question whether A belongs to C, the question is not yet begged, but no demonstration is made. If however A and B are identical either because they are convertible or because A follows B, then the question is begged for the same reason as before. For we have explained the meaning of begging the question, (25) viz. proving that which is not self-evident by means of itself.
If then begging the question is proving what is not self-evident by means of itself, in other words failing to prove when the failure is due to the thesis to be proved and the premiss through which it is proved being equally uncertain, either because predicates which are identical belong to the same subject, or because the same predicate belongs to subjects which are identical, the question may be begged in the middle and third figures in both ways, (30) though, if the syllogism is affirmative, only in the third and first figures. If the syllogism is negative, the question is begged when identical predicates are denied of the same subject; and both premisses do not beg the question indifferently (in a similar way the question may be begged in the middle figure), because the terms in negative syllogisms are not convertible. In scientific demonstrations the question is begged when the terms are really related in the manner described, (35) in dialectical arguments when they are according to common opinion so related.
17 The objection that ‘this is not the reason why the result is false’, which we frequently make in argument, is made primarily in the case of a reductio ad impossibile, to rebut the proposition which was being proved by the reduction. [65b] (40) For unless a man has contradicted this proposition he will not say, ‘False cause’, but urge that something false has been assumed in the earlier parts of the argument; nor will he use the formula in the case of an ostensive proof; for here what one denies is not assumed as a premiss. Further when anything is refuted ostensively by the terms ABC, it cannot be objected that the syllogism does not depend on the assumption laid down. (5) For we use the expression ‘false cause’, when the syllogism is concluded in spite of the refutation of this position; but that is not possible in ostensive proofs: since if an assumption is refuted, a syllogism can no longer be drawn in reference to it. It is clear then that the expression ‘false cause’ can only be used in the case of a reductio ad impossibile, (10) and when the original hypothesis is so related to the impossible conclusion, that the conclusion results indifferently whether the hypothesis is made or not. The most obvious case of the irrelevance of an assumption to a conclu
sion which is false is when a syllogism drawn from middle terms to an impossible conclusion is independent of the hypothesis, as we have explained in the Topics.1 For to put that which is not the cause as the cause, (15) is just this: e. g. if a man, wishing to prove that the diagonal of the square is incommensurate with the side, should try to prove Zeno’s theorem that motion is impossible, and so establish a reductio ad impossibile: for Zeno’s false theorem has no connexion at all with the original assumption. (20) Another case is where the impossible conclusion is connected with the hypothesis, but does not result from it. This may happen whether one traces the connexion upwards or downwards, e. g. if it is laid down that A belongs to B, B to C, and C to D, (25) and it should be false that B belongs to D: for if we eliminated A and assumed all the same that B belongs to C and C to D, the false conclusion would not depend on the original hypothesis. Or again trace the connexion upwards; e. g. suppose that A belongs to B, E to A, (30) and F to E, it being false that F belongs to A. In this way too the impossible conclusion would result, though the original hypothesis were eliminated. But the impossible conclusion ought to be connected with the original terms: in this way it will depend on the hypothesis, e. g. when one traces the connexion downwards, (35) the impossible conclusion must be connected with that term which is predicate in the hypothesis: for if it is impossible that A should belong to D, the false conclusion will no longer result after A has been eliminated. If one traces the connexion upwards, the impossible conclusion must be connected with that term which is subject in the hypothesis: for if it is impossible that F should belong to B, the impossible conclusion will disappear if B is eliminated. (40) Similarly when the syllogisms are negative.
[66a] It is clear then that when the impossibility is not related to the original terms, the false conclusion does not result on account of the assumption. Or perhaps even so it may sometimes be independent. For if it were laid down that A belongs not to B but to K, (5) and that K belongs to C and C to D, the impossible conclusion would still stand. Similarly if one takes the terms in an ascending series. Consequently since the impossibility results whether the first assumption is suppressed or not, it would appear to be independent of that assumption. Or perhaps we ought not to understand the statement that the false conclusion results independently of the assumption, in the sense that if something else were supposed the impossibility would result; but rather we mean that when the first assumption is eliminated, (10) the same impossibility results through the remaining premisses; since it is not perhaps absurd that the same false result should follow from several hypotheses, e. g. that parallels meet, both on the assumption that the interior angle is greater than the exterior and on the assumption that a triangle contains more than two right angles. (15)
18 A false argument depends on the first false statement in it. Every syllogism is made out of two or more premisses. If then the false conclusion is drawn from two premisses, one or both of them must be false: for (as was proved2) a false syllogism cannot be drawn from true premisses. (20) But if the premisses are more than two, e. g. if C is established through A and B, and these through D, E, F, and G, one of these higher propositions must be false, and on this the argument depends: for A and B are inferred by means of D, E, F, and G. Therefore the conclusion and the error results from one of them.
19 In order to avoid having a syllogism drawn against us, (25) we must take care, whenever an opponent asks us to admit the reason without the conclusions, not to grant him the same term twice over in his premisses, since we know that a syllogism cannot be drawn without a middle term, and that term which is stated more than once is the middle. How we ought to watch the middle in reference to each conclusion, is evident from our knowing what kind of thesis is proved in each figure. (30) This will not escape us since we know how we are maintaining the argument.
That which we urge men to beware of in their admissions, they ought in attack to try to conceal. This will be possible first, if, instead of drawing the conclusions of preliminary syllogisms, (35) they take the necessary premisses and leave the conclusions in the dark; secondly if instead of inviting assent to propositions which are closely connected they take as far as possible those that are not connected by middle terms. For example suppose that A is to be inferred to be true of F; B, C, D, and E being middle terms. One ought then to ask whether A belongs to B, and next whether D belongs to E, instead of asking whether B belongs to C; after that he may ask whether B belongs to C, (40) and so on. [66b] And if the syllogism is drawn through one middle term, he ought to begin with that: in this way he will most likely deceive his opponent.
20 Since we know when a syllogism can be formed and how its terms must be related, it is clear when refutation will be possible and when impossible. (5) A refutation is possible whether everything is conceded, or the answers alternate (one, I mean, being affirmative, the other negative). For as has been shown a syllogism is possible whether the terms are related in affirmative propositions or one proposition is affirmative, the other negative: consequently, if what is laid down is contrary to the conclusion, (10) a refutation must take place: for a refutation is a syllogism which establishes the contradictory. But if nothing is conceded, a refutation is impossible: for no syllogism is possible (as we saw3) when all the terms are negative: therefore no refutation is possible. For if a refutation were possible, a syllogism must be possible; although if a syllogism is possible it does not follow that a refutation is possible. (15) Similarly refutation is not possible if nothing is conceded universally: since the fields of refutation and syllogism are defined in the same way.
21 It sometimes happens that just as we are deceived in the arrangement of the terms,4 (20) so error may arise in our thought about them, e. g. if it is possible that the same predicate should belong to more than one subject immediately, but although knowing the one, a man may forget the other and think the opposite true. Suppose that A belongs to B and to C in virtue of their nature, and that B and C belong to all D in the same way. If then a man thinks that A belongs to all B, and B to D, but A to no C, and C to all D, (25) he will both know and not know the same thing5 in respect of the same thing.6 Again if a man were to make a mistake about the members of a single series; e. g. suppose A belongs to B, B to C, and C to D, but some one thinks that A belongs to all B, but to no C: he will both know that A belongs to D, (30) and think that it does not. Does he then maintain after this simply that what he knows, he does not think? For he knows in a way that A belongs to C through B, since the part is included in the whole; so that what he knows in a way, this he maintains he does not think at all: but that is impossible.
In the former case, (35) where the middle term does not belong to the same series, it is not possible to think both the premisses with reference to each of the two middle terms: e. g. that A belongs to all B, but to no C, and both B and C belong to all D. For it turns out that the first premiss of the one syllogism is either wholly or partially contrary to the first premiss of the other. For if he thinks that A belongs to everything to which B belongs, (40) and he knows that B belongs to D, then he knows that A belongs to D. [67a] Consequently if again he thinks that A belongs to nothing to which C belongs, he thinks that A does not belong to some of that to which B belongs; but if he thinks that A belongs to everything to which B belongs, and again thinks that A does not belong to some of that to which B belongs, (5) these beliefs are wholly or partially contrary. In this way then it is not possible to think; but nothing prevents a man thinking one premiss of each syllogism or both premisses of one of the two syllogisms: e. g. A belongs to all B, and B to D, and again A belongs to no C. An error of this kind is similar to the error into which we fall concerning particulars: e. g. if A belongs to all B, and B to all C, (10) A will belong to all C. If then a man knows that A belongs to everything to which B belongs, he knows that A belongs to C. But nothing prevents his being ignorant that C exists; e. g. let A stand for two right angles, B for triangle, C for a particular diagram of a triangle. A man might think that
C did not exist, though he knew that every triangle contains two right angles; consequently he will know and not know the same thing at the same time. (15) For the expression ‘to know that every triangle has its angles equal to two right angles’ is ambiguous, meaning to have the knowledge either of the universal or of the particulars. Thus then he knows that C contains two right angles with a knowledge of the universal, but not with a knowledge of the particulars; consequently his knowledge will not be contrary to his ignorance. (20) The argument in the Meno7 that learning is recollection may be criticized in a similar way. For it never happens that a man starts with a foreknowledge of the particular, but along with the process of being led to see the general principle he receives a knowledge of the particulars, by an act (as it were) of recognition. For we know some things directly; e. g. that the angles are equal to two right angles, if we know that the figure is a triangle. (25) Similarly in all other cases.
By a knowledge of the universal then we see the particulars, but we do not know them by the kind of knowledge which is proper to them; consequently it is possible that we may make mistakes about them, but not that we should have the knowledge and error that are contrary to one another: rather we have the knowledge of the universal but make a mistake in apprehending the particular. (30) Similarly in the cases stated above.8 The error in respect of the middle term is not contrary to the knowledge obtained through the syllogism, nor is the thought in respect of one middle term contrary to that in respect of the other. Nothing prevents a man who knows both that A belongs to the whole of B, and that B again belongs to C, thinking that A does not belong to C, e. g. (35) knowing that every mule is sterile and that this is a mule, and thinking that this animal is with foal: for he does not know that A belongs to C, unless he considers the two propositions together. So it is evident that if he knows the one and does not know the other, he will fall into error. And this is the relation of knowledge of the universal to knowledge of the particular. For we know no sensible thing, once it has passed beyond the range of our senses, even if we happen to have perceived it, except by means of the universal and the possession of the knowledge which is proper to the particular, but without the actual exercise of that knowledge. [67b] For to know is used in three senses: it may mean either to have knowledge of the universal or to have knowledge proper to the matter in hand or to exercise such knowledge: consequently three kinds of error also are possible. (5) Nothing then prevents a man both knowing and being mistaken about the same thing, provided that his knowledge and his error are not contrary. And this happens also to the man whose knowledge is limited to each of the premisses and who has not previously considered the particular question. For when he thinks that the mule is with foal he has not the knowledge in the sense of its actual exercise, (10) nor on the other hand has his thought caused an error contrary to his knowledge: for the error contrary to the knowledge of the universal would be a syllogism.
The Basic Works of Aristotle (Modern Library Classics) Page 15