  In the two books of Analytica Priora, Aristotle has carried us through the full doctrine of the functions and varieties of the Syllogism; with an intimation that it might be applied to two purposes — Demonstration and Dialectic. We are now introduced to these two distinct applications of the Syllogism: first, in the Analytica Posteriora, to Demonstration; next, in the Topica, to Dialectic. We are indeed distinctly told that, as far as the forms and rules of Syllogism go, these are alike applicable to both;1 but the difference of matter and purpose in the two cases is so considerable as to require a distinct theory and precepts for the one and for the other. The contrast between Dialectic (along with Rhetoric) on the one hand and Science on the other is one deeply present to the mind of Aristotle. He seems to have proceeded upon the same fundamental antithesis as that which appears in the Platonic dialogues; but to have modified it both in meaning and in terminology, dismissing at the same time various hypotheses with which Plato had connected it. The antithesis that both thinkers have in view is Opinion or Common Sense versus Science or Special Teaching and Learning; those aptitudes, acquirements, sentiments, antipathies, &c., which a man imbibes and appropriates insensibly, partly by his own doing and suffering, partly by living amidst the drill and example of a given society — as distinguished from those accomplishments which he derives from a teacher already known to possess them, and in which both the time of his apprenticeship and the steps of his progress are alike assignable. Common Sense is the region of Opinion, in which there is diversity of authorities and contradiction of arguments without any settled truth; all affirmations being particular and relative, true at one time and place, false at another. Science, on the contrary, deals with imperishable Forms and universal truths, Like Plato, Aristotle distinguishes the region of Common Sense or Opinion from that of Science, and regards Universals as the objects of Science. But his Universals are very different from those of Plato: they are not self-existent realities, known by the mind from a long period of pre-existence, and called up by reminiscence out of the chaos of sensible impressions. To operate such revival is the great function that Plato assigns to Dialectic. But in the philosophy of Aristotle Dialectic is something very different. It is placed alongside of Rhetoric in the region of Opinion. Both the rhetor and the dialectician deal with all subjects, recognizing no limit; they attack or defend any or all conclusions, employing the process of ratiocination which Aristotle has treated under the name of Syllogism; they take up as premisses any one of the various opinions in circulation, for which some plausible authority may be cited; they follow out the consequences of one opinion in its bearing upon others, favourable or unfavourable, and thus become well furnished We shall see more fully how Aristotle deals with Dialectic, when we come to the Topica: here I put it forward briefly, in order that the reader may better understand, by contrast, its extreme antithesis, viz., Demonstrative Science and Necessary Truth as conceived by Aristotle. First, instead of two debaters, one of whom sets up a thesis which he professes to understand and undertakes to maintain, while the other puts questions upon it, — Demonstrative Science assumes a teacher who knows, and a learner conscious of ignorance but wishing to know. The teacher lays down premisses which the learner is bound to receive; or if they are put in the form of questions, the learner must answer them as the teacher expects, not according to his own knowledge. Secondly, instead of the unbounded miscellany of subjects treated in Dialectic, Demonstrative Science is confined to a few special subjects, in which alone appropriate premisses can be obtained, and definitions framed. Thirdly, instead Amidst all these diversities, Dialectic and Demonstrative Science have in common the process of Syllogism, including such assumptions as the rules of syllogizing postulate. In both, the conclusions are hypothetically true (i.e. granting the premisses to be so). But, in demonstrative syllogism, the conclusions are true universally, absolutely, and necessarily; deriving this character from their premisses, which Aristotle holds up as the cause, reason, or condition of the conclusion. What he means by Demonstrative Science, we may best conceive, by taking it as a small t?e??? or specially cultivated enclosure, subdivided into still smaller separate compartments — the extreme antithesis to the vast common land of Dialectic. Between the two lies a large region, neither essentially determinate like the one, nor essentially indeterminate like the other; an intermediate region in which are comprehended the subjects of the treatises forming the very miscellaneous EncyclopÆdia of Aristotle. These subjects do not admit of being handled with equal exactness; accordingly, he admonishes us that it is important to know how much exactness is attainable in each, and not to aspire to more.7 The passages above named in the Nikomachean Ethica are remarkable: ?????t? d’ ?? ??a???, e? ?at? t?? ?p??e????? ???? d?asaf??e??· t? ??? ?????? ??? ????? ?? ?pas? t??? ?????? ?p???t?t???, ?spe? ??d’ ?? t??? d?????????????. t?? ????e?a? ? ????? ?? ?pas?? ?p???te?? (???), ???’ ?? ???st??? ?at? t?? ?p??e????? ????, ?a? ?p? t?s??t?? ?f’ ?s?? ???e??? t? e??d?. Compare Metaphys. E. p. 1025, b. 13: ?p?de??????s?? ? ??a??a??te??? ? a?a??te???. The different degrees of exactness attainable in different departments of science, and the reasons upon which such difference depends are well explained in the sixth book of Mr. John Stuart Mill’s System of Logic, vol. II. chap. iii. pp. 422-425, 5th ed. Aristotle says that there can be no scientific theory or cognition about t? s?e???? which he defines to be that which belongs to a subject neither necessarily, nor constantly, nor usually, but only on occasion (Metaphys. E. p. 1026, b. 3, 26, 33; K. p. 1065, a. 1, meaning t? s?e???? ? ?a?’ a?t?, — Analyt. Post. I. 6, 75, a. 18; for he uses the term in two different senses — Metaph. ?. p. 1025, a. 31). In his view, there can be no science except about constant conjunctions; and we find the same doctrine in the following passage of Mr. Mill:— “Any facts are fitted, in themselves, to be a subject of science, which follow one another according to constant laws; although those laws may not have been discovered, nor even be discoverable by our existing resources. Take, for instance, the most familiar class of meteorological phenomena, those of rain and sunshine. Scientific inquiry has not yet succeeded in ascertaining the order of antecedence and consequence among these phenomena, so as to be able, at least in our regions of the earth, to predict them with certainty, or even with any high degree of probability. Yet no one doubts that the phenomena depend on laws.… Meteorology not only has in itself every requisite for being, but actually is, a science; though from the difficulty of observing the facts upon which the phenomena depend (a difficulty inherent in the peculiar nature of those phenomena), the science is extremely imperfect; and were it perfect, might probably be of little avail in practice, since the data requisite for applying its principles to particular instances would rarely be procurable. “A case may be conceived of an intermediate character between the perfection of science, and this its extreme imperfection. It may happen that the greater causes, those on which the principal part of the phenomena depends, are within the reach of observation and measurement; so that, if no other causes intervened, a complete explanation could be given, not only of the phenomenon in general, but of all the variations and modifications which it admits of. But inasmuch as other, perhaps many other, causes, separately insignificant in their effects, co-operate or conflict in many or in all cases with those greater causes, the effect, accordingly, presents more or less of aberration from what would be produced by the greater causes alone. Now if these minor causes are not so constantly accessible, or not accessible at all, to accurate observation, the principal mass of the effect may still, as before, be accounted for, and even predicted; but there will be variations and modifications which we shall not be competent to explain thoroughly, and our predictions will not be fulfilled accurately, but only approximately. “It is thus, for example, with the theory of the Tides.… And this is what is or ought to be meant by those who speak of sciences which are not exact sciences. Astronomy was once a science, without being an exact science. It could not become exact until not only the general course of the planetary motions, but the perturbations also, were accounted for and referred to their causes. It has become an exact science because its phenomena have been brought under laws comprehending the whole of the causes by which the phenomena are influenced, whether in a great or only in a trifling degree, whether in all or only in some cases, and assigning to each of those causes the share of effect that really belongs to it.… The science of human nature falls far short of the standard of exactness now realized in Astronomy; but there is no reason that it should not be as much a science as Tidology is, or as Astronomy was when its calculations had only mastered the main phenomena, but not the perturbations.” Aristotle, here at the beginning, seeks to clear up a difficulty which had been raised in the time of Plato as between knowledge and learning. How is it possible to learn at all? is a question started in the Menon.10 You either know a thing already, and, on this supposition, you do not want to learn it; or you do not know it, and in this case you cannot learn it, because, even when you have learnt, you cannot tell whether the matter learnt is what you were in search of. To this difficulty, the reply made in the Menon is, that you never do learn any thing really new. What you are said to learn, is nothing more than reminiscence of what had once been known in an anterior life, and forgotten at birth into the present life; what is supposed to be learnt is only the recall of that which you once knew, but had forgotten. Such is the Platonic doctrine of Reminiscence. Aristotle will not accept that doctrine as a solution; but he acknowledges the difficulty, and intimates that others had already tried to solve it without success. His own solution is that there are two grades of cognition: (1) the full, complete, absolute; (2) the partial, incomplete, qualified. What you already know by the first of these grades, you cannot be said to learn; but you may learn that which you know only by the second grade, and by such learning you bring your incomplete cognition up to completeness. Thus, you have learnt, and you know, the universal truth, that every triangle has its three angles equal to two right angles; but you do not yet know that A B C, D E F, G H I, &c., have their two angles equal to two right angles; for you have not yet seen any of these figures, and you do not know that they are triangles. The moment that you see A B C, or hear what Aristotle reports the solution given by others, but from which he himself dissented, of the Platonic puzzle. The respondent was asked, Do you know that every Dyad is even? — Yes. Some Dyad was then produced, which the respondent did not know to be a Dyad; accordingly he did not know it to be even. Now the critics alluded to by Aristotle said that the respondent made a wrong answer; instead of saying I know every Dyad is even, he ought to have said, Every Dyad which I know to be a Dyad is even. Aristotle pronounces that this criticism is incorrect. The respondent knows the conclusion which had previously been demonstrated to him; and that conclusion was, Every triangle has its three angles equal to two right angles; it was not, Every thing which I know to be a triangle has its three angles equal to two right angles. This last proposition had never been demonstrated, nor even stated: ??de?a ??? p??tas?? ?a??eta? t??a?t?, ?t? ?? s? ??da? ??????, ? ? s? ??da? e?????a??, ???? ?at? pa?t?? (b. 3-5). This discussion, in the commencement of the Analytica Posteriora (combined with Analyt. Priora, II. xxi.), is interesting, because it shows that even then the difficulties were felt, about the major proposition of the Syllogism, which Mr. John Stuart Mill has so ably cleared up, for the first time, in his System of Logic. See Book II. ch. iii. of that work, especially as it stands in the sixth edition, with the note there added, pp. 232-233. You affirm, in the major proposition of the Syllogism, that every triangle has its three angles equal to two right angles; does not this include the triangle A, B, C, and is it not therefore a petitio principii? Or, if it be not so, does it not assert more than you know? The Sophists (upon whom both Plato and Aristotle are always severe, but who were valuable contributors to the theory of Logic by fastening upon the weak points) attacked it on this ground, and raised against it the puzzle described by Aristotle (in this chapter), afterwards known as the Sophism entitled ? ???e?a?????? (see Themistius Paraphras. I. i.; also ‘Plato and the Other Companions of Sokrates,’ Vol. III. ch. xxxviii. p. 489). The critics whom Aristotle here cites and disapproves, virtually admitted the pertinence of this puzzle by modifying their assertion, and by cutting it down to “Everything which we know to be a triangle has its three angles equal to two right angles.” Aristotle finds fault with this modification, which, however, is one way of abating the excess of absolute and peremptory pretension contained in the major, and of intimating the want of a minor to be added for interpreting and supplementing the major; while Aristotle himself arrives at the same result by admitting that the knowledge corresponding to the major proposition is not yet absolute, but incomplete and qualified; and that it is only made absolute when supplemented by a minor. The very same point, substantially, is raised in the discussion between Mr. John Stuart Mill and an opponent, in the note above referred to. “A writer in the ‘British Quarterly Review’ endeavours to show that there is no petitio principii in the Syllogism, by denying that the proposition All men are mortal, asserts or assumes that Socrates is mortal. In support of this denial, he argues that we may, and in fact do, admit the general proposition without having particularly examined the case of Socrates, and even without knowing whether the individual so named is a man or something else. But this of course was never denied. That we can and do draw inferences concerning cases specifically unknown to us, is the datum from which all who discuss this subject must set out. The question is, in what terms the evidence or ground on which we draw these conclusions may best be designated — whether it is most correct to say that the unknown case is proved by known cases, or that it is proved by a general proposition including both sets of cases, the known and the unknown? I contend for the former mode of expression. I hold it an abuse of language to say, that the proof that Socrates is mortal, is that all men are mortal. Turn it in what way we will, this seems to me asserting that a thing is the proof of itself. Whoever pronounces the words, All men are mortal, has affirmed that Socrates is mortal, though he may never have heard of Socrates; for since Socrates, whether known to be a man or not, really is a man, he is included in the words, All men, and in every assertion of which they are the subject.… The reviewer acknowledges that the maxim (Dictum de Omni et Nullo) as commonly expressed — ‘Whatever is true of a class is true of everything included in the class,’ is a mere identical proposition, since the class is nothing but the things included in it. But he thinks this defect would be cured by wording the maxim thus: ‘Whatever is true of a class is true of everything which can be shown to be a member of the class:’ as if a thing could be shown to be a member of the class without being one.” The qualified manner in which the maxim is here enunciated by the reviewer (what can be shown to be a member of the class) corresponds with the qualification introduced by those critics whom Aristotle impugns (????s? ??? ?? f?s???te? e?d??a? p?sa? d??da ??t?a? ??sa?, ???’ ?? ?sas?? ?t? d???); and the reply of Mr. Mill would have suited for these critics as well as for the reviewer. The puzzle started in the Platonic Menon is, at bottom, founded on the same view as that of Mr. Mill, when he states that the major proposition of the Syllogism includes beforehand the conclusion. “The general principle, (says Mr. Mill, p. 205), instead of being given as evidence of the particular case, cannot itself be taken for true without exception, until every shadow of doubt which could affect any case comprised in it is dispelled by evidence aliunde; and then what remains for the syllogism to prove? From a general principle we cannot infer any particulars but those which the principle itself assumes as known.” To enunciate this in the language of the Platonic Menon, we learn nothing by or through the evidence of the Syllogism, except a part of what we have already professed ourselves to know by asserting the major premiss. So also Zabarella (In lib. I. Post. Anal. Comm., p. 340, Op. ed. Venet. 1617): “DuÆ illÆ dictiones (primis et immediatis) unam tantum significant conditionem ordine secundam, non duas; idem namque est, principia esse medio carentia, ac esse prima.” In Aristotle’s time two doctrines had been advanced, in opposition to the preceding theory: (1) Some denied the necessity of any indemonstrable principia, and affirmed the possibility of, demonstrating backwards ad infinitum; (2) Others agreed in denying the necessity of any indemonstrable principia, but contended that demonstration in a circle is valid and legitimate — e.g. that A may be demonstrated by means of B, and B by means of A. Against both these doctrines Aristotle enters his protest. The first of them — the supposition of an interminable regress — he pronounces to be obviously absurd: the second he declares tantamount to proving a thing by itself; the circular demonstration, besides, having been shown to be impossible, except in the First figure, with propositions in which the predicate reciprocates or is co-extensive with the subject — a very small proportion among propositions generally used in demonstrating.15 ?? ??a ?e??e?a ?p? t?? ?p??? ?p?st?t?? ?a?’ a?t? ??t?? ?? ???p???e?? t??? ?at??????????? ? ???p???es?a? d?’ a?t? t? ?st? ?a? ?? ??????? (b. 16, seq.). Line must be included in the definition of the opposites straight or curve. Also it is essential to every line that it is either straight or curve. Number must be included in the definition of the opposites odd or even; and to be either odd or even is essentially predicable of every number. You cannot understand what is meant by straight or curve unless you have the notion of a line. The example given by Aristotle of causal conjunction (the death of an animal under the sacrificial knife) shows that he had in his mind the perfection of Inductive Observation, including full application of the Method of Difference. About the precise signification of ?a????? in Aristotle, see a valuable note of Bonitz (ad Metaphys. Z. iii.) p. 299; also Waitz (ad Aristot. De Interpr. c. vii.) I. p. 334. Aristotle gives it here, b. 26: ?a????? d? ???? ? ?? ?at? pa?t?? te ?p???? ?a? ?a?’ a?t? ?a? ? a?t?. Compare Themistius, Paraphr. p. 19, Spengel. ?? ?a?’ a?t? is described by Aristotle confusedly. ?? ?a?????, is that which is predicable of the subject as a whole or summum genus: t? ?at? pa?t??, that which is predicable of every individual, either of the summum genus or of any inferior species contained therein. Cf. Analyt. Post. I. xxiv. p. 85, b. 24: ? ??? ?a?’ a?t? ?p???e? t?, t??t? a?t? a?t? a?t??? — the subject is itself the cause or fundamentum of the properties per se. See the explanation and references in Kampe, Die Erkenntniss-theorie des Aristoteles, ch. v. pp. 160-165. Aristotle remarks that there is great liability to error about these Universalia Prima. We sometimes demonstrate a predicate to be true, universally and per se, of a lower species, without being aware that it might also be demonstrated to be true, universally and per se, of the higher genus to which that species belongs; perhaps, indeed, that higher genus may not yet have obtained a current name. That proportions hold by permutation, was demonstrated severally for numbers, lines, solids, and intervals of time; but this belongs to each of them, not from any separate property of each, but from what is common to all: that, however, which is common to all had received no name, so that it was not known that one demonstration might comprise all the four.18 In like manner, a man may know that an equilateral and an isosceles triangle have their three angles equal to two right angles, and also that a scalene triangle has its three angles equal to two right angles; yet he may not know (except sophistically and by accident19) that a triangle in genere has its three angles equal to two right angles, though there be no other triangles except equilateral, isosceles, and scalene. He does not know that this may be demonstrated of every triangle quatenus triangle. The only way to obtain a In every demonstration the principia or premisses must be not only true, but necessarily true; the conclusion also will then be necessarily true, by reason of the premisses, and this constitutes Demonstration. Wherever the premisses are necessarily true, the conclusion will be necessarily true; but you cannot say, vice versÂ, that wherever the conclusion is necessarily true, the syllogistic premisses from which it follows must always be necessarily true. They may be true without being necessarily true, or they may even be false: if, then, the conclusion be necessarily true, it is not so by reason of these premisses; and the syllogistic proof is in this case no demonstration. Your syllogism may have true premisses and may lead to a conclusion which is true by reason of them; but still you have not demonstrated, since neither premisses nor conclusion are necessarily true.21 When an opponent contests your demonstration, he succeeds if he can disprove the necessity of your conclusion; if he can show any single case in which it either is or may be false.22 It is not enough to proceed upon a premiss which is either probable or simply true: it may be true, yet not appropriate to the case: you must take your departure from the first or highest universal of the genus about which you attempt to demonstrate.23 Again, unless you can state the why of your conclusion; that is to say, unless the middle term, by reason of which the conclusion is necessarily true, be itself necessarily true, — you have not demonstrated it, nor do you know it absolutely. Your On the point last mentioned, M. BarthÉlemy St. Hilaire observes in his note, p. 41: “Dans les questions de dialectique, la conclusion est nÉcessaire en ce sens, qu’elle suit nÉcessairement des prÉmisses; elle n’est pas du tout nÉcessaire en ce sens, que la chose qu’elle exprime soit nÉcessaire. Ainsi il faut distinguer la nÉcessitÉ de la forme et la nÉcessitÉ de la matiÈre: ou comme disent les scholastiques, necessitas illationis et necessitas materiÆ. La dialectique se contente de la premiÈre, mais la demonstration a essentiellement besoin des deux.” In every demonstration three things may be distinguished: (1) The demonstrated conclusion, or Attribute essential to a certain genus; (2) The Genus, of which the attributes per se are the matter of demonstration; (3) The Axioms, out of which, or through which, the demonstration is obtained. These Axioms may be and are common to several genera: but the demonstration cannot be transferred from one genus to another; both the extremes as well as the middle term must belong to the same genus. An arithmetical demonstration cannot be transferred to magnitudes and their properties, except in so far as magnitudes are numbers, which is partially true of some among them. The demonstrations in arithmetic may indeed be transferred to harmonics, because harmonics is subordinate to arithmetic; and, for the like reason, demonstrations in geometry may be transferred to mechanics and optics. But we cannot introduce into geometry any property of lines, which does not belong to them qu lines; such, for example, as that a straight line is the most beautiful of all lines, or is the contrary of a circular line; for these predicates belong to it, not qu line, but qu member of a different or more extensive genus.26 There can be no For complete demonstration, it is not sufficient that the premisses be true, immediate, and undemonstrable; they must, furthermore, be essential and appropriate to the class in hand. Unless they be such, you cannot be said to know the conclusion absolutely; you know it only by accident. You can only know a conclusion when demonstrated from its own appropriate premisses; and you know it best when it is demonstrated from its highest premisses. It is sometimes difficult to determine whether we really know or not; for we fancy that we know, when we demonstrate from true and universal principia, without being aware whether they are, or are not, the principia appropriate to the case.28 But these principia must always be assumed without demonstration — the class whose essential constituent properties are in question, the universal Axioms, and the Definition or meaning of the attributes to be demonstrated. If these definitions and axioms are not always formally enunciated, it is because we tacitly presume them to be already known and admitted by the learner.29 He may indeed always refuse to grant them in express words, but they are such that he cannot help granting them by internal assent in his mind, to which every syllogism must address itself. When you assume a premiss without demonstrating it, though it be really demonstrable, this, if the learner is favourable and willing to grant it, is an assumption or Hypothesis, valid relatively to him alone, but not valid absolutely: if he is reluctant or adverse, it is a Postulate, which Themistius, Paraphr. p. 37: ?spe? ??d’ ?? ?e??t?a? ??????ta? ta?? ??aa?? ?p?? ?? d?a?????ta? ?a? de??????s??, ???’ ?? ????s?? ?? t? ????, ?? e?s? s???a a? ??af?e?a?. A similar doctrine is asserted, Analyt. Prior. I. xli. p. 49, b. 35, and still more clearly in De Memoria et Reminiscentia, p. 450, a. 2-12. The process of Demonstration neither requires, nor countenances, the Platonic theory of Ideas — universal substances beyond and apart from particulars. But it does require that we should admit universal predications; that is, one and the same predicate truly applicable in the same sense to many different particulars. Unless this be so, there can be no universal major premiss, nor appropriate middle term, nor valid demonstrative syllogism.33 The Maxim or Axiom of Contradiction, in its most general enunciation, is never formally enunciated by any special science; but each of them assumes the Maxim so far as applicable to its own purpose, whenever the Reductio ad Absurdum is introduced.34 It is in this and the other common principles or Axioms that all the sciences find their point of contact and communion; and that Dialectic also comes into communion with all of them, as also the science (First Philosophy) that scrutinizes the validity or demonstrability of the Axioms.35 The dialectician is not confined The text is here very obscure. He proceeds to distinguish Geometry especially (also other sciences, though less emphatically) from t? ?? t??? d?a?????? (I. xii. p. 78, a. 12). Julius Pacius, ad Analyt. Post. I. viii. (he divides the chapters differently), p. 417, says:— “Differentia interrogationis dialecticÆ et demonstrativÆ hÆc est. Dialecticus ita interrogat, ut optionem det adversario, utrum malit affirmare an negare. Demonstrator vero interrogat ut rem evidentiorem faciat; id est, ut doceat ex principiis auditori notis.” “L’interrogation syllogistique se confondant avec la proposition, il s’ensuit que l’interrogation doit Être, comme la proposition, propre À la science dont il s’agit.” (BarthÉlemy St Hilaire, note, p. 70). Interrogation here has a different meaning from that which it bears in Dialectic. Knowledge of Fact and knowledge of the Cause must be distinguished, and even within the same Science.41 In some syllogisms the conclusion only brings out t? ?t? — the reality of certain facts; in others, it ends in t? d??t? — the affirmation of a cause, or of the Why. The syllogism of the Why is, where the middle term is not merely the cause, but the proximate cause, of the conclusion. Often, however, the effect is more notorious, so that we employ it as middle term, and conclude from it to its reciprocating cause; in which case our syllogism is only of the ?t?; and so it is also when we employ as middle term a cause not proximate but remote, concluding from that to the effect.42 Sometimes “Cum enim vera demonstratio, id est t?? d??t?, fiat per causam proximam, consequens est, ut demonstratio vel per effectum proximum, vel per causam remotam, sit demonstratio t?? ?t?” (Julius Pacius, Comm. p. 422). M. BarthÉlemy St. Hilaire observes (Note, p. 82):— “La cause ÉloignÉe non immÉdiate, donne un syllogisme dans la seconde figure. — Il est vrai qu’Aristote n’appelle cause que la cause immÉdiate; et que la cause ÉloignÉe n’est pas pour lui une vÉritable cause.” See in Schol. p. 188, a. 19, the explanation given by Alexander of the syllogism t?? d??t?. As there are some affirmative propositions that are indivisible, i.e., having affirmative predicates which belong to a subject at once, directly, immediately, indivisibly, — so there are also some indivisible negative propositions, i.e., with predicates that belong negatively to a subject at once, directly, &c. In all such there is no intermediate step to justify either the affirmation of the predicate, or the negation of the predicate, respecting the given subject. This will be the case where neither the predicate nor the subject is contained in any higher genus.45 For the like reason the Third figure is not mentioned here, but only the First and Second: because in the Third figure no universal conclusion can be proved (Julius Pacius, p. 431). Let us next assume the affirmative proposition, All B is A, to be true, but mediate and deducible through the middle term C. If you conclude the contrary of this (No B is A) through the same middle term C, in the First figure, your error cannot arise from falsity in the minor premiss, because your minor (by the laws of the figure) must be affirmative; your error must arise from a false major, because a negative major is not inconsistent with the laws of the First figure. On the other hand, if you conclude the contrary in the First figure through a different Such will be the case when the deducible proposition assumed to be true is affirmative, and when therefore the contrary conclusion which you profess to have proved is negative. But if the deducible proposition assumed to be true is negative, and if consequently the contrary conclusion must be affirmative, — then, if you try to prove this contrary through the same middle term, your premisses cannot both be false, but your major premiss must always be false.52 If, however, you try to prove the contrary through a different and inappropriate middle term, you cannot convert the minor premiss to its contrary (because the minor premiss must continue affirmative, in order that you may arrive at any conclusion at all), but the major can be so converted. Should the major premiss thus converted be true, the minor will be false; should the major premiss thus converted be false, the minor may be either true or false. Either one of the premisses, or both the premisses, may thus be false.53 Errors of simple ignorance (not concluded from false syllogism) may proceed from defect or failure of sensible perception, in one or other of its branches. For without sensation there can be no induction; and it is from induction only that the premisses for demonstration by syllogism are obtained. We cannot arrive at universal propositions, even in what are called abstract sciences, except through induction of particulars; nor can we demonstrate except from universals. Induction and Demonstration are the only two ways of learning; and the particulars composing our inductions can only be known through sense.54 Aristotle next proceeds to show (what in previous passages he In Dialectical Syllogism it is enough if the premisses be admitted or reputed as propositions immediately true, whether they are so in reality or not; but in Scientific or Demonstrative Syllogism they must be so in reality: the demonstration is not complete unless it can be traced up to premisses that are thus immediately or directly true (without any intervening middle term).60 That there are and must be such primary or immediate premisses, Aristotle now undertakes to prove, by some dialectical reasons, and other analytical or scientific reasons.61 He himself Aristotle, however, does not always adhere closely to the distinction. Thus, if we compare the logical or dialectical reasons given, p. 82, b. 37, seq., with the analytical, announced as beginning p. 84, a. 8, seq., we find the same main topic dwelt upon in both, namely, that to admit an infinite series excludes the possibility of Definition. Both Alexander and Ammonius agree in announcing this as the capital topic on which the proof turned; but Alexander inferred from hence that the argument was purely dialectical (??????? ?p??e???a), while Ammonius regarded it as a reason thoroughly convincing and evident: ? ??t?? f???s?f?? (Ammonius) ??e?e ? d?? t??t? ???e?? ?????? t? ?p??e???ata· ??a???? ??? ?t? e?s?? ???s??, e? ? ??ata????a? e?sa????e? (Schol. p. 227, a. 40, seq., Brand.). It is plain from Aristotle’s own words64 that he intended these four chapters (xix.-xxii.) as a confirmation of what he had already asserted in chapter iii. of the present treatise, and as farther refutation of the two distinct classes of opponents there indicated: (1) those who said that everything was demonstrable, demonstration in a circle being admissible; (2) those who said that nothing was demonstrable, inasmuch as the train of predication Mr. John Stuart Mill, in his System of Logic, takes for granted that there must be immediate, indemonstrable truths, to serve as a basis for deduction; “that there cannot be a chain of proof suspended from nothing;” that there must be ultimate laws of nature, though we cannot be sure that the laws now known to us are ultimate. On the other hand, we read in the recent work of an acute contemporary philosopher, Professor Delboeuf (Essai de Logique Scientifique, LiÈge, 1865, Pref. pp. v, vii, viii, pp. 46, 47:) — “Il est des points sur lesquels je crains de ne m’Être pas expliquÉ assez nettement, entre autres la question du fondement de la certitude. Je suis de ceux qui repoussent de toutes leurs forces l’axiome si spÉcieux qu’on ne peut tout dÉmontrer; cette proposition aurait, À mes yeux, plus besoin que toute autre d’une dÉmonstration. Cette dÉmonstration ne sera en partie donnÉe que quand on aura une bonne fois ÉnumÉrÉ toutes les propositions indÉmontrables; et quand on aura bien dÉfini le caractÈre auquel on les reconnait. Nulle part on ne trouve ni une semblable ÉnumÉration, ni une semblable dÉfinition. On reste À cet Égard dans une position vague, et par cela mÊme facile À dÉfendre.” It would seem, by these words, that M. Delboeuf stands in the most direct opposition to Aristotle, who teaches us that the ???a? or principia from which demonstration starts cannot be themselves demonstrated. But when we compare other passages of M. Delboeuf’s work, we find that, in rejecting all undemonstrable propositions, what he really means is to reject all self-evident universal truths, “C’est donc une vÉritable illusion d’admettre des vÉritÉs Évidentes par elles-mÊmes. Il n’y a pas de proposition fausse que nous ne soyons disposÉs d’admettre comme axiome, quand rien ne nous a encore autorisÉs À la repousser” (p. ix.). This is quite true in my opinion; but the immediate indemonstrable truths for which Aristotle contends as ???a? of demonstration, are not announced by him as self-evident, they are declared to be results of sense and induction, to be raised from observation of particulars multiplied, compared, and permanently formularized under the intellectual habitus called NoÛs. By Demonstration Aristotle means deduction in its most perfect form, beginning from these ???a? which are inductively known but not demonstrable (i. e. not knowable deductively). And in this view the very able and instructive treatise of M. Delboeuf mainly coincides, assigning even greater preponderance to the inductive process, and approximating in this respect to the important improvements in logical theory advanced by Mr. John Stuart Mill. Among the universal propositions which are not derived from Induction, but which serve as ???a? for Deduction and Demonstration, we may reckon the religious, ethical, Æsthetical, social, political, &c., beliefs received in each different community, and impressed upon all newcomers born into it by the force of precept, example, authority. Here the major premiss is felt by each individual as carrying an authority of its own, stamped and enforced by the sanction of society, and by the disgrace or other penalties in store for those who disobey it. It is ready to be interpreted and diversified by suitable minor premisses in all inferential applications. But these ???a? for deduction, differing widely at different times and places, though generated in the same manner and enforced by the same sanction, would belong more properly to the class which Aristotle terms t? ??d??a. We have thus recognized that there exist immediate (ultimate or primary) propositions, wherein the conjunction between predicate and subject is such that no intermediate term can be assigned between them. When A is predicated both of B and C, this may perhaps be in consequence of some common property possessed by B and C, and such common property will form a middle term. For example, equality of angles to two right angles belongs both to an isosceles and to a scalene triangle, and it belongs to them by reason of their common property — triangular figure; which last is thus the middle term. But this need not be always the case.66 It is possible that the two propositions — A predicated of B, A predicated of C — may both of them be immediate propositions; and that there may be no community of nature between B and C. Whenever a middle term can be found, demonstration is possible; but where no middle term can be found, demonstration is impossible. The proposition, whether affirmative or negative, is then an immediate or indivisible one. Such propositions, and the terms of which they are composed, are the ultimate elements or principia of Demonstration. Predicate and subject are brought constantly into closer and closer conjunction, until at last they become one and indivisible.67 Here we reach the unit or element Having thus, in the long preceding reasoning, sought to prove that all demonstration must take its departure from primary undemonstrable principia — from some premisses, affirmative and negative, which are directly true in themselves, and not demonstrable through any middle term or intervening propositions, Aristotle now passes to a different enquiry. We have some demonstrations in which the conclusion is Particular, others in which it is Universal: again, some Affirmative, some Negative, Which of the two, in each of these alternatives, is the best? We have also demonstrations Direct or Ostensive, and demonstrations Indirect or by way of Reductio ad Absurdum. Which of these two is the best? Both questions appear to have been subjected to debate by contemporary philosophers.70 Aristotle discusses these points dialectically (as indeed he points out in the Topica that the comparison of two things generally, as to better and worse, falls under the varieties of dialectical enquiry71), first stating and next refuting the arguments on the weaker side. Some persons may think (he says) that demonstration of the Particular is better than demonstration of the Universal: first, because it conducts to fuller cognition of that which the thing is in itself, and not merely that which it is quatenus member of a class; secondly, because demonstrations of the Universal are apt to generate an illusory belief, that the Universal is a distinct reality apart from and independent of all its particulars (i.e., that figure in general has a real existence apart from all particular figures, and number in general apart from all particular numbers, &c.), while demonstrations of the Particular do not lead to any such illusion.72 Such are the several reasons enumerated by Aristotle in refutation of the previous opinion stated in favour of the Particular. Evidently he does not account them all of equal value: he intimates that some are purely dialectical (??????); and he insists most upon the two following:— 1. He that knows the Universal knows in a certain sense the Particular; if he knows that every triangle has its three angles equal to two right angles, he knows potentially that the isosceles has its three angles equal to the same, though he may not know as yet that the isosceles is a triangle. But he that knows the Particular does not in any way know the Universal, either actually or potentially.75 2. The Universal is apprehended by Intellect or NoÛs, the highest of all cognitive powers; the Particular terminates in sensation. Here, I presume, he means, that, in demonstration of the Particular, the conclusion teaches you nothing more than you might have learnt from a direct observation of sense; whereas in that of the Universal the conclusion teaches you more than you could have learnt from direct sensation, and comes into correlation with the highest form of our intellectual nature.76 Next, Aristotle compares the Affirmative with the Negative demonstration, and shows that the Affirmative is the better. Of two demonstrations (he lays it down) that one which proceeds upon a smaller number of postulates, assumptions, or propositions, is better than the other; for, to say nothing of other reasons, it conducts you more speedily to knowledge than the other, and that is an advantage. Now, both in the affirmative and in the negative syllogism, you must have three terms and two propositions; but in the affirmative you assume only that something is; while in the negative you assume both that something is, and that something is not. Here is a double assumption instead of a single; therefore the negative is the worse or Philoponus says (Schol. pp. 234-235, Brand.) that the Commentators all censured Aristotle for the manner in which he here laid out the Syllogism d?’ ?d???t??. I do not, however, find any such censure in Themistius. Philoponus defends Aristotle from the censure. If we next compare one Science with another, the prior and more accurate of the two is, (1) That which combines at once the ?t? and the d??t?; (2) That which is abstracted from material conditions, as compared with that which is immersed therein — for example, arithmetic is more accurate than harmonics; (3) The more simple as compared with the more complex: thus, arithmetic is more accurate than geometry, a monad or unit is a substance without position, whereas a point (more concrete) is a substance with position.81 One and the same science is that which belongs to one and the same generic subject-matter. The premisses of a demonstration must be included in the same genus with the conclusion; and where the ultimate premisses are heterogeneous, the cognition derived from them must be considered as not one but a compound of several.82 You may find two or more distinct middle terms for demonstrating the same conclusion; sometimes out of the same logical series or table, sometimes out of different tables.83 Philoponus illustrates this (Schol. p. 235, b. 41, Br.): ???? t? Te?d?s??? sfa????? ?????ste?? ?st?? ?p?st?? t?? t?? ??t?????? pe?? ????????? sfa??a?. &c. Euclid, in the 20th Proposition of his first Book, demonstrates that any two sides of a triangle are together greater than the third side. According to Proklus, the Epikureans derided the demonstration of such a point as absurd; and it seems that some contemporaries of Aristotle argued in a similar way, judging by the phrase ?spe? fas? t??e?. By t? p??ta, he means the ???a? of Demonstration, which are treated especially in II. xix. See Biese, Die Philos. des Aristoteles, p. 277. The text of this and the succeeding words seems open to doubt, as well as that of Themistius (p. 63). Waitz in his note (p. 374) explains the meaning clearly:— “non ita quidem ut ipsa sensuum perceptio scientiam afferat; sed ita ut quod in singulis accidere videamus, idem etiam in omnibus accidere coniicientes universe intelligamus.” Aristotle next proceeds to refute, at some length, the supposition, that the principia of all syllogisms are the same. We see at once that this cannot be so, because some syllogisms are true, others false. But, besides, though there are indeed a few Axioms essential to the process of demonstration, and the same in all syllogisms, yet these are not sufficient of themselves for demonstration. There must farther be other premisses or matters of evidence — propositions immediately true (or established by prior demonstrations) belonging to each branch of Science specially, as distinguished from the others. Our demonstration relates to these special matters or premisses, though it is accomplished out of or by means of the common Axioms.90 Science or scientific Cognition differs from true Opinion, and the cognitum from the opinatum, herein, that Science is of the Universal, and through necessary premisses which cannot be otherwise; while Opinion relates to matters true, yet which at the same time may possibly be false. The belief in a proposition which is immediate (i. e., undemonstrable) yet not necessary, is Opinion; it is not Science, nor is it NoÛs or Intellect — the principium of Science or scientific Cognition. Such beliefs are With some remarks upon Sagacity, or the power of divining a middle term in a time too short for reflection (as when the friendship of two men is on the instant referred to the fact of their having a common enemy), the present book is brought to a close.93 |
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