A System of Logic, Ratiocinative and Inductive · John Stuart Mill
168 P. 271.
The objector (as has been acutely remarked by another correspondent) applies to the problem under consideration, a mode of calculation only suited to the reverse problem. Had the question been—If two of every three Bs are As and three out of every four Bs are Cs, how many Bs will be both As and Cs, his reasoning would have been correct. For the Bs that are both As and Cs must be fewer than either the Bs that are As or the Bs that are Cs, and to find their number we must abate either of these numbers in the ratio due to the other. But when the problem is to find, not how many Bs are both As and Cs, but how many things that are both As and Cs are Bs, it is evident that among these the proportion of Bs must be not less, but greater, than among things which are only A, or among things which are only B.
The true theory of the chances is best found by going back to the scientific grounds on which the proportions rest. The degree of frequency of a coincidence depends on, and is a measure of, the frequency, combined with the efficacy, of the causes in operation that are favorable to it. If out of every twelve As taken indiscriminately eight are Bs and four are not, it is implied that there are causes operating on A which tend to make it a B, and that these causes are sufficiently constant and sufficiently powerful to succeed in eight out of twelve cases, but fail in the remaining four. So if of twelve Cs, nine are Bs and three are not, there must be causes of the same tendency operating on C, which succeed in nine cases and fail in three. Now suppose twelve cases which are both As and Cs. The whole twelve are now under the operation of both sets of causes. One set is sufficient to prevail in eight of the twelve cases, the other in nine. The analysis of the cases shows that six of the twelve will be Bs through the operation of both sets of causes; two more in virtue of the causes operating on A; and three more through those operating on C, and that there will be only one case in which all the causes will be inoperative. The total number, therefore, which are Bs will be eleven in twelve, and the evaluation in the text is correct.
197 Supra, book i., chap. v. 198 Supra, book i., chap. v., § 1, and book ii., chap, v., § 5. 199 The axiom, “Equals subtracted from equals leave equal differences,” may be demonstrated from the two axioms in the text. If A = _a_ and B = _b_, A-B = _a-b_. For if not, let A-B = _a-b+c_. Then since B = _b_, adding equals to equals, A = _a+c_. But A = _a_. Therefore _a = a+c_, which is impossible. This proposition having been demonstrated, we may, by means of it, demonstrate the following: “If equals be added to unequals, the sums are unequal.” If A = _a_ and B not = _b_, A+B is not = _a+b_. For suppose it to be so. Then, since A = _a_ and A+B = _a+b_, subtracting equals from equals, B = _b_; which is contrary to the hypothesis. So again, it may be proved that two things, one of which is equal and the other unequal to a third thing, are unequal to one another. If A = _a_ and A not = B, neither is _a_ = B. For suppose it to be equal. Then since A = _a_ and _a_ = B, and since things equal to the same thing are equal to one another A = B; which is contrary to the hypothesis.
200 Geometers have usually preferred to define parallel lines by the property of being in the same plane and never meeting. This, however, has rendered it necessary for them to assume, as an additional axiom, some other property of parallel lines; and the unsatisfactory manner in which properties for that purpose have been selected by Euclid and others has always been deemed the opprobrium of elementary geometry. Even as a verbal definition, equidistance is a fitter property to characterize parallels by, since it is the attribute really involved in the signification of the name. If to be in the same plane and never to meet were all that is meant by being parallel, we should feel no incongruity in speaking of a curve as parallel to its asymptote. The meaning of parallel lines is, lines which pursue exactly the same direction, and which, therefore, neither draw nearer nor go farther from one another; a conception suggested at once by the contemplation of nature. That the lines will never meet is of course included in the more comprehensive proposition that they are everywhere equally distant. And that any straight lines which are in the same plane and not equidistant will certainly meet, may be demonstrated in the most rigorous manner from the fundamental property of straight lines assumed in the text, viz., that if they set out from the same point, they diverge more and more without limit.
205 Not, however, as might at first sight appear, 999 times as much. A complete analysis of the cases shows that (always assuming the veracity of the witness to be ⁹⁄₁₀) in 10,000 drawings, the drawing of No. 79 will occur nine times, and be announced incorrectly once; the credibility, therefore, of the announcement of No. 79 is ⁹⁄₁₀; while the drawing of a white ball will occur nine times, and be announced incorrectly 999 times. The credibility, therefore, of the announcement of white is ⁹⁄₁₀₀₈, and the ratio of the two 1008:10; the one announcement being thus only about a hundred times more credible than the other, instead of 999 times. 206 Supra, book iii., chap. ii., § 3, 4, 5. 207 Mr. Bailey has given the best statement of this theory. “The general name,” he says, “raises up the image sometimes of one individual of the class formerly seen, sometimes of another, not unfrequently of many individuals in succession; and it sometimes suggests an image made up of elements from several different objects, by a latent process of which I am not conscious.” (Letters on the Philosophy of the Human Mind, 1st series, letter 22.) But Mr. Bailey must allow that we carry on inductions and ratiocinations respecting the class, by means of this idea or conception of some one individual in it. This is all I require. The name of a class calls up some idea, through which we can, to all intents and purposes, think of the class as such, and not solely of an individual member of it.
208 I have entered rather fully into this question in chap. xvii. of _An Examination of Sir William Hamilton’s Philosophy_, headed “The Doctrine of Concepts or General Notions,” which contains my last views on the subject. 209 Other examples of inappropriate conceptions are given by Dr. Whewell (_Phil. Ind. Sc._ ii., 185) as follows: “Aristotle and his followers endeavored in vain to account for the mechanical relation of forces in the lever, by applying the _inappropriate_ geometrical conceptions of the properties of the circle: they failed in explaining the _form_ of the luminous spot made by the sun shining through a hole, because they applied the _inappropriate_ conception of a circular _quality_ in the sun’s light: they speculated to no purpose about the elementary composition of bodies, because they assumed the _inappropriate_ conception of _likeness_ between the elements and the compound, instead of the genuine notion of elements merely _determining_ the qualities of the compound.” But in these cases there is more than an inappropriate conception; there is a false conception; one which has no prototype in nature, nothing corresponding to it in facts. This is evident in the last two examples, and is equally true in the first; the “properties of the circle” which were referred to, being purely fantastical. There is, therefore, an error beyond the wrong choice of a principle of generalization; there is a false assumption of matters of fact. The attempt is made to resolve certain laws of nature into a more general law, that law not being one which, though real, is inappropriate, but one wholly imaginary.
210 Professor Bain. 211 This sentence having been erroneously understood as if I had meant to assert that belief is nothing but an irresistible association, I think it necessary to observe that I express no theory respecting the ultimate analysis either of reasoning or of belief, two of the most obscure points in analytical psychology. I am speaking not of the powers themselves, but of the previous conditions necessary to enable those powers to exert themselves: of which conditions I am contending that language is not one, senses and association being sufficient without it. The irresistible association theory of belief, and the difficulties connected with the subject, have been discussed at length in the notes to the new edition of Mr. James Mill’s _Analysis of the Phenomena of the Human Mind_. 212 Mr. Bailey agrees with me in thinking that whenever “from something actually present to my senses, conjoined with past experience, I feel satisfied that something has happened, or will happen, or is happening, beyond the sphere of my personal observation,” I may with strict propriety be said to reason: and of course to reason inductively, for demonstrative reasoning is excluded by the circumstances of the case. (_The Theory of Reasoning_, 2d ed., p. 27.) _ 213 Novum Organum Renovatum_, pp. 35–37. _ 214 Novum Organum Renovatum_, pp. 39, 40.
215 P. 217, 4to edition. 216 “E, ex, extra, extraneus, étranger, stranger.” Another etymological example sometimes cited is the derivation of the English _uncle_ from the Latin _avus_. It is scarcely possible for two words to bear fewer outward marks of relationship, yet there is but one step between them, _avus_, _avunculus_, _uncle_. So _pilgrim_, from _ager_: _per agrum_, _peragrinus_, _peregrinus_, _pellegrino_, _pilgrim_. Professor Bain gives some apt examples of these transitions of meaning. “The word ‘damp’ primarily signified moist, humid, wet. But the property is often accompanied with the feeling of cold or chilliness, and hence the idea of cold is strongly suggested by the word. This is not all. Proceeding upon the superadded meaning, we speak of damping a man’s ardor, a metaphor where the cooling is the only circumstance concerned; we go on still further to designate the iron slide that shuts off the draft of a stove, ‘the damper,’ the primary meaning being now entirely dropped. ‘Dry,’ in like manner, through signifying the absence of moisture, water, or liquidity, is applied to sulphuric acid containing water, although not thereby ceasing to be a moist, wet, or liquid substance.” So in the phrases, dry sherry, or Champagne. “‘Street,’ originally a paved way, with or without houses, has been extended to roads lined with houses, whether paved or unpaved. ‘Impertinent’ signified at first irrelevant, alien to the purpose in hand: through which it has come to mean, meddling, intrusive, unmannerly, insolent.” (_Logic_, ii., 173, 174.)
217 Pp. 226, 227. _ 218 Essays_, p. 214. _ 219 Essays_, p. 215. 220 Though no such evil consequences as take place in these instances are likely to arise from the modern freak of writing _sanatory_ instead of sanitary, it deserves notice as a charming specimen of pedantry ingrafted upon ignorance. Those who thus undertake to correct the spelling of the classical English writers, are not aware that the meaning of _sanatory_, if there were such a word in the language, would have reference not to the preservation of health, but to the cure of disease. _ 221 Historical Introduction_, vol. i., pp. 66–68. _ 222 History of Scientific Ideas_, ii., 110, 111. _ 223 History of Scientific Ideas_, ii., 111–113. _ 224 Nov. Org. Renov._, pp. 286, 287. _ 225 History of Scientific Ideas_, ii., 120–122. _ 226 Nov. Org. Renov._, p. 274. _ 227 Hist. Sc. Id._, i. 133. 228 Dr. Whewell, in his reply (_Philosophy of Discovery_, p. 270) says that he “stopped short of, or rather passed by, the doctrine of a series of organized beings,” because he “thought it bad and narrow philosophy.” If he did, it was evidently without understanding this form of the doctrine; for he proceeds to quote a passage from his “History,” in which the doctrine he condemns is designated as that of “a mere linear progression in nature, which would place each genus in contact only with the preceding and succeeding ones.” Now the series treated of in the text agrees with this linear progression in nothing whatever but in being a progression.
229 Supra, p. 137. _ 230 Vulgar Errors_, book v., chap. 21. _ 231 Pharmacologia_, Historical Introduction, p. 16. 232 The author of one of the Bridgewater Treatises has fallen, as it seems to me, into a similar fallacy when, after arguing in rather a curious way to prove that matter may exist without any of the known properties of matter, and may therefore be changeable, he concludes that it can not be eternal, because “eternal (passive) existence necessarily involves incapability of change.” I believe it would be difficult to point out any other connection between the facts of eternity and unchangeableness, than a strong association between the two ideas. Most of the _a priori_ arguments, both religious and anti-religious, on the origin of things, are fallacies drawn from the same source. 233 Supra, book ii., chap. v., § 6, and chap. vii., § 1, 2, 3, 4. See also _Examination of Sir William Hamilton’s Philosophy_, chap. vi. and elsewhere. 234 It seems that this doctrine was, before the time I have mentioned, disputed by some thinkers. Dr. Ward mentions Scotus, Vasquez, Biel, Francis Lugo, and Valentia. 235 I quote this passage from Playfair’s celebrated _Dissertation on the Progress of Mathematical and Physical Science_. 236 This statement I must now correct, as too unqualified. The maxim in question was maintained with full conviction by no less an authority than Sir William Hamilton. See my _Examination_, chap. xxiv.
257 “Thus Fourcroy,” says Dr. Paris, “explained the operation of mercury by its specific gravity, and the advocates of this doctrine favored the general introduction of the preparations of iron, especially in scirrhus of the spleen or liver, upon the same hypothetical principle; for, say they, whatever is most forcible in removing the obstruction must be the most proper instrument of cure: such is steel, which, besides the attenuating power with which it is furnished, has still a greater force in this case from the gravity of its particles, which, being seven times specifically heavier than any vegetable, acts in proportion with a stronger impulse, and therefore is a more powerful deobstruent. This may be taken as a specimen of the style in which these mechanical physicians reasoned and practiced.”—_Pharmacologia_, pp. 38, 39. _ 258 Pharmacologia_, pp. 39, 40. 259 I quote from Dr. Whewell’s _Hist. Ind. Sc._, 3d ed., i., 129. _ 260 Hist. Ind. Sc._, i., 52. _ 261 Nov. Org._, Aph. 60. 262 “An advocate,” says Mr. De Morgan (_Formal Logic_, p. 270), “is sometimes guilty of the argument _à dicto secundum quid ad dictum simpliciter_: it is his business to do for his client all that his client might _honestly_ do for himself. Is not the word in italics frequently omitted? _Might_ any man honestly try to do for himself all that counsel frequently try to do for him? We are often reminded of the two men who stole the leg of mutton; one could swear he had not got it, the other that he had not taken it. The counsel is doing his duty by his client, the client has left the matter to his counsel. Between the unexecuted intention of the client, and the unintended execution of the counsel, there may be a wrong done, and, if we are to believe the usual maxims, no wrong-doer.”
The same writer justly remarks (p. 251) that there is a converse fallacy, _à dicto simpliciter ad dictum secundum quid_, called by the scholastic logicians _fallacia accidentis_; and another which may be called _à dicto secundum quid ad dictum secundum alterum quid_ (p. 265). For apt instances of both, I must refer the reader to Mr. De Morgan’s able chapter on Fallacies. 263 An example of this fallacy is the popular error that _strong_ drink must be a cause of _strength_. There is here fallacy within fallacy; for granting that the words “strong” and “strength” were not (as they are) applied in a totally different sense to fermented liquors and to the human body, there would still be involved the error of supposing that an effect must be like its cause; that the conditions of a phenomenon are likely to resemble the phenomenon itself; which we have already treated of as an _a priori_ fallacy of the first rank. As well might it be supposed that a strong poison will make the person who takes it strong. 264 In his later editions, Archbishop Whately confines the name of Petitio Principii “to those cases in which one of the premises either is manifestly the same in sense with the conclusion, or is actually proved from it, or is such as the persons you are addressing are not likely to know, or to admit, except as an inference from the conclusion; as, _e.g._, if any one should infer the authenticity of a certain history, from its recording such and such facts, the reality of which rests on the evidence of that history.”
265 No longer even a probable hypothesis, since the establishment of the atomic theory; it being now certain that the integral particles of different substances gravitate unequally. It is true that these particles, though real _minima_ for the purposes of chemical combination, may not be the ultimate particles of the substance; and this doubt alone renders the hypothesis admissible, even as an hypothesis. _ 266 Hist. Ind. Sc._, i., 34. 267 “And coxcombs vanquish Berkeley with a grin.” 268 Some arguments and explanations, supplementary to those in the text, will be found in _An Examination of Sir William Hamilton’s Philosophy_, chap. xxvi. _ 269 Supra_, p. 424. 270 When this chapter was written, Professor Bain had not yet published even the first part (“The Senses and the Intellect”) of his profound Treatise on the Mind. In this the laws of association have been more comprehensively stated and more largely exemplified than by any previous writer; and the work, having been completed by the publication of “The Emotions and the Will,” may now be referred to as incomparably the most complete analytical exposition of the mental phenomena, on the basis of a legitimate Induction, which has yet been produced. More recently still, Mr. Bain has joined with me in appending to a new edition of the “Analysis,” notes intended to bring up the analytic science of Mind to its latest improvements.
Many striking applications of the laws of association to the explanation of complex mental phenomena are also to be found in Mr. Herbert Spencer’s “Principles of Psychology.” 271 In the case of the moral sentiments the place of direct experiment is to a considerable extent supplied by historical experience, and we are able to trace with a tolerable approach to certainty the particular associations by which those sentiments are engendered. This has been attempted, so far as respects the sentiment of justice, in a little work by the present author, entitled _Utilitarianism_. 272 The most favorable cases for making such approximate generalizations are what may be termed collective instances; where we are fortunately enabled to see the whole class respecting which we are inquiring in action at once, and, from the qualities displayed by the collective body, are able to judge what must be the qualities of the majority of the individuals composing it. Thus the character of a nation is shown in its acts as a nation; not so much in the acts of its government, for those are much influenced by other causes; but in the current popular maxims, and other marks of the general direction of public opinion; in the character of the persons or writings that are held in permanent esteem or admiration; in laws and institutions, so far as they are the work of the nation itself, or are acknowledged and supported by it; and so forth. But even here there is a large margin of doubt and uncertainty. These things are liable to be influenced by many circumstances; they are partially determined by the distinctive qualities of that nation or body of persons, but partly also by external causes which would influence any other body of persons in the same manner. In order, therefore, to make the experiment really complete, we ought to be able to try it without variation upon other nations: to try how Englishmen would act or feel if placed in the same circumstances in which we have supposed Frenchmen to be placed; to apply, in short, the Method of Differences as well as that of Agreement. Now these experiments we can not try, nor even approximate to.
273 “To which,” says Dr. Whewell, “we may add, that it is certain, from the history of the subject, that in that case the hypothesis would never have been framed at all.” Dr. Whewell (_Philosophy of Discovery_, pp. 277–282) defends Bacon’s rule against the preceding strictures. But his defense consists only in asserting and exemplifying a proposition which I had myself stated, viz., that though the largest generalizations may be the earliest made, they are not at first seen in their entire generality, but acquire it by degrees, as they are found to explain one class after another of phenomena. The laws of motion, for example, were not known to extend to the celestial regions, until the motions of the celestial bodies had been deduced from them. This, however, does not in any way affect the fact, that the middle principles of astronomy, the central force, for example, and the law of the inverse square, could not have been discovered, if the laws of motion, which are so much more universal, had not been known first. On Bacon’s system of step-by-step generalization, it would be impossible in any science to ascend higher than the empirical laws; a remark which Dr. Whewell’s own Inductive Tables, referred to by him in support of his argument, amply bear out. _ 274 Supra_, page 317 to the end of the chapter.
Dr. Whewell equally misunderstands M. Comte’s doctrine respecting the second or metaphysical stage of speculation. M. Comte did not mean that “discussions concerning ideas” are limited to an early stage of inquiry, and cease when science enters into the positive stage. (_Philosophy of Discovery_, pp. 226 et seq.) In all M. Comte’s speculations as much stress is laid on the process of clearing up our conceptions as on the ascertainment of facts. When M. Comte speaks of the metaphysical stage of speculation, he means the stage in which men speak of “Nature” and other abstractions as if they were active forces, producing effects; when Nature is said to do this, or forbid that; when Nature’s horror of a vacuum, Nature’s non-admission of a break, Nature’s _vis medicatrix_, were offered as explanations of phenomena; when the qualities of things were mistaken for real entities dwelling in the things; when the phenomena of living bodies were thought to be accounted for by being referred to a “vital force;” when, in short, the abstract names of phenomena were mistaken for the causes of their existence. In this sense of the word it can not be reasonably denied that the metaphysical explanation of phenomena, equally with the theological, gives way before the advance of real science. That the final, or positive stage, as conceived by M. Comte, has been equally misunderstood, and that, notwithstanding some expressions open to just criticism, M. Comte never dreamed of denying the legitimacy of inquiry into all causes which are accessible to human investigation, I have pointed out in a former place.
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