empirical generalization is far from being universally true; no doubt because it is a remote, and therefore easily defeated, result of general laws too deep for us at present to penetrate; but it will probably in time suggest processes of inquiry, leading to the discovery of those laws.
172 Or, according to Laplace’s theory, the sun and the sun’s rotation.
173 Supra, book iii., chap. v., § 7.
174 Supra, book iii., chap. x., § 2 175 In the preceding discussion, the _mean_ is spoken of as if it were exactly the same thing with the _average_. But the mean, for purposes of inductive inquiry, is not the average, or arithmetical mean, though in a familiar illustration of the theory the difference may be disregarded. If the deviations on one side of the average are much more numerous than those on the other (these last being fewer but greater), the effect due to the invariable cause, as distinct from the variable ones, will not coincide with the average, but will be either below or above the average, the deviation being toward the side on which the greatest number of the instances are found. This follows from a truth, ascertained both inductively and deductively, that small deviations from the true central point are greatly more frequent than large ones. The mathematical law is, “that the most probable determination of one or more invariable elements from observation is that in which the _sum of the squares_ of the individual aberrations,” or deviations, “_shall be the least possible_.” See this principle stated, and its grounds popularly explained, by Sir John Herschel, in his review of Quetelet on _ 176 Essai Philosophique sur les Probabilités_, fifth Paris edition, p.
7.
177 It even appears to me that the calculation of chances, where there are no data grounded either on special experience or on special inference, must, in an immense majority of cases, break down, from sheer impossibility of assigning any principle by which to be guided in setting out the list of possibilities. In the case of the colored balls we have no difficulty in making the enumeration, because we ourselves determine what the possibilities shall be. But suppose a case more analogous to those which occur in nature: instead of three colors, let there be in the box all possible colors, we being supposed ignorant of the comparative frequency with which different colors occur in nature, or in the productions of art. How is the list of cases to be made out? Is every distinct shade to count as a color? If so, is the test to be a common eye, or an educated eye—a painter’s, for instance? On the answer to these questions would depend whether the chances against some particular color would be estimated at ten, twenty, or perhaps five hundred to one. While if we knew from experience that the particular color occurs on an average a certain number of times in every hundred or thousand, we should not require to know any thing either of the frequency or of the number of the other possibilities.
179 “If this be not so, why do we feel so much more probability added by the first instance than by any single subsequent instance? Why, except that the first instance gives us its possibility (a cause _adequate_ to it), while every other only gives us the frequency of its conditions? If no reference to a cause be supposed, possibility would have no meaning; yet it is clear that, antecedent to its happening, we might have supposed the event impossible, _i.e._, have believed that there was no physical energy really existing in the world equal to producing it...After the first time of happening, which is, then, more important to the whole probability than any other single instance (because proving the possibility), the _number_ of times becomes important as an index to the intensity or extent of the cause, and its independence of any particular time. If we took the case of a tremendous leap, for instance, and wished to form an estimate of the probability of its succeeding a certain number of times; the first instance, by showing its possibility (before doubtful) is of the most importance; but every succeeding leap shows the power to be more perfectly under control, greater and more invariable, and so increases the probability; and no one would think of reasoning in this case straight from one instance to the next, without referring to the physical energy which each leap indicated. Is it not, then, clear that we do not ever” (let us rather say, that we do not in an advanced state of our knowledge) “conclude directly from the happening of an event to the probability of its happening again; but that we refer to the cause, regarding the past cases as an index to the cause, and the cause as our guide to the future?”—_Ibid._ 180 The writer last quoted says that the valuation of chances by comparing the number of cases in which the event occurs with the number in which it does not occur, “would generally be wholly erroneous,” and “is not the true theory of probability.” It is at least that which forms the foundation of insurance, and of all those calculations of chances in the business of life which experience so abundantly verifies. The reason which the reviewer gives for rejecting the theory is, that it “would regard an event as certain which had hitherto never failed; which is exceedingly far from the truth, even for a very large number of constant successes.” This is not a defect in a particular theory, but in any theory of chances.
No principle of evaluation can provide for such a case as that which the reviewer supposes. If an event has never once failed, in a number of trials sufficient to eliminate chance, it really has all the certainty which can be given by an empirical law; it _is_ certain during the continuance of the same collocation of causes which existed during the observations. If it ever fails, it is in consequence of some change in that collocation. Now, no theory of chances will enable us to infer the future probability of an event from the past, if the causes in operation, capable of influencing the event, have intermediately undergone a change.
181 Pp. 18, 19. The theorem is not stated by Laplace in the exact terms in which I have stated it; but the identity of import of the two modes of expression is easily demonstrable.
182 For a fuller treatment of the many interesting questions raised by the theory of probabilities, I may now refer to a recent work by Mr.
Venn, Fellow of Caius College, Cambridge, “The Logic of Chance;” one of the most thoughtful and philosophical treatises on any subject connected with Logic and Evidence which have been produced, to my knowledge, for many years. Some criticisms contained in it have been very useful to me in revising the corresponding chapters of the present work. In several of Mr. Venn’s opinions, however, I do not agree. What these are will be obvious to any reader of Mr. Venn’s work who is also a reader of this.
in Priestley’s curtailed edition.
184 I am happy to be able to quote the following excellent passage from Mr. Baden Powell’s _Essay on the Inductive Philosophy_, in confirmation, both in regard to history and to doctrine, of the statement made in the text. Speaking of the “conviction of the universal and permanent uniformity of nature,” Mr. Powell says (pp.
98–100): “We may remark that this idea, in its proper extent, is by no means one of popular acceptance or natural growth. Just so far as the daily experience of every one goes, so far indeed he comes to embrace a certain persuasion of this kind, but merely to this limited extent, that what is going on around him at present, in his own narrow sphere of observation, will go on in like manner in future. The peasant believes that the sun which rose to-day will rise again to-morrow; that the seed put into the ground will be followed in due time by the harvest this year as it was last year, and the like; but has no notion of such inferences in subjects beyond his immediate observation. And it should be observed that each class of persons, in admitting this belief within the limited range of his own experience, though he doubt or deny it in every thing beyond, is, in fact, bearing unconscious testimony to its universal truth. Nor, again, is it only among the _most_ ignorant that this limitation is put upon the truth. There is a very general propensity to believe that every thing beyond common experience, or especially ascertained laws of nature, is left to the dominion of chance or fate or arbitrary intervention; and even to object to any attempted explanation by physical causes, if conjecturally thrown out for an apparently unaccountable phenomenon.
“The precise doctrine of the _generalization_ of this idea of the uniformity of nature, so far from being obvious, natural, or intuitive, is utterly beyond the attainment of the many. In all the extent of its universality it is characteristic of the philosopher.
It is clearly the result of philosophic cultivation and training, and by no means the spontaneous offspring of any primary principle naturally inherent in the mind, as some seem to believe. It is no mere vague persuasion taken up without examination, as a common prepossession to which we are always accustomed; on the contrary, all common prejudices and associations are against it. It is pre-eminently _an acquired idea_. It is not attained without deep study and reflection. The best informed philosopher is the man who most firmly believes it, even in opposition to received notions; its acceptance depends on the extent and profoundness of his inductive studies.”
185 Supra, book iii., chap. iii., § 1 186 It deserves remark, that these early generalizations did not, like scientific inductions, presuppose causation. What they did presuppose, was _uniformity_ in physical facts. But the observers were as ready to presume uniformity in the co-existence of facts as in the sequences. On the other hand, they never thought of assuming that this uniformity was a principle pervading all nature: their generalizations did not imply that there was uniformity in every thing, but only that as much uniformity as existed within their observation, existed also beyond it. The induction, fire burns, does not require for its validity that all nature should observe uniform laws, but only that there should be uniformity in one particular class of natural phenomena; the effects of fire on the senses and on combustible substances. And uniformity to this extent was not assumed, anterior to the experience, but proved by the experience.
The same observed instances which proved the narrower truth, proved as much of the wider one as corresponded to it. It is from losing sight of this fact, and considering the law of causation in its full extent as necessarily presupposed in the very earliest generalizations, that persons have been led into the belief that the law of causation is known _a priori_, and is not itself a conclusion from experience.
187 Book ii., chap. iii.
188 One of the most rising thinkers of the new generation in France, M.
Taine (who has given, in the _Revue des Deux Mondes_, the most masterly analysis, at least in one point of view, ever made of the present work), though he rejects, on this and similar points of psychology, the intuition theory in its ordinary form, nevertheless assigns to the law of causation, and to some other of the most universal laws, that certainty beyond the bounds of human experience, which I have not been able to accord to them. He does this on the faith of our faculty of abstraction, in which he seems to recognize an independent source of evidence, not indeed disclosing truths not contained in our experience, but affording an assurance which experience can not give, of the universality of those which it does contain. By abstraction M. Taine seems to think that we are able, not merely to analyze that part of nature which we see, and exhibit apart the elements which pervade it, but to distinguish such of them as are elements of the system of nature considered as a whole, not incidents belonging to our limited terrestrial experience. I am not sure that I fully enter into M.
Taine’s meaning; but I confess I do not see how any mere abstract conception, elicited by our minds from our experience, can be evidence of an objective fact in universal Nature, beyond what the experience itself bears witness of; or how, in the process of interpreting in general language the testimony of experience, the limitations of the testimony itself can be cast off.
Dr. Ward, in an able article in the _Dublin Review_ for October, 1871, contends that the uniformity of nature can not be proved from experience, but from “transcendental considerations” only, and that, consequently, all physical science would be deprived of its basis, if such transcendental proof were impossible.
When physical science is said to depend on the assumption that the course of nature is invariable, all that is meant is that the conclusions of physical science are not known as _absolute_ truths: the truth of them is _conditional_ on the uniformity of the course of nature; and all that the most conclusive observations and experiments can prove, is that the result arrived at will be true if, and as long as, the present laws of nature are valid. But this is all the assurance we require for the guidance of our conduct. Dr.
Ward himself does not think that his transcendental proofs make it practically greater; for he believes, as a Catholic, that the course of nature not only has been, but frequently and even daily is, suspended by supernatural intervention.
But though this conditional conclusiveness of the evidence of experience, which is sufficient for the purposes of life, is all that I was necessarily concerned to prove, I have given reasons for thinking that the uniformity, as itself a part of experience, is sufficiently proved to justify undoubting reliance on it. This Dr.
Ward contests, for the following reasons: First (p. 315), supposing it true that there has hitherto been no well authenticated case of a breach in the uniformity of nature; “the number of natural agents constantly at work is incalculably large; and the observed cases of uniformity in their action must be immeasurably fewer than one thousandth of the whole. Scientific men, we assume for the moment, have discovered that in a certain proportion of instances—immeasurably fewer than one thousandth of the whole—a certain fact has prevailed; the fact of uniformity; and they have not found a single instance in which that fact does _not_ prevail. Are they justified, we ask, in inferring from these premises that the fact is universal? Surely the question answers itself. Let us make a very grotesque supposition, in which, however, the conclusion would really be tried according to the arguments adduced. In some desert of Africa there is an enormous connected edifice, surrounding some vast space, in which dwell certain reasonable beings, who are unable to leave the inclosure. In this edifice are more than a thousand chambers, which some years ago were entirely locked up, and the keys no one knew where. By constant diligence twenty-five keys have been found, out of the whole number; and the corresponding chambers, situated promiscuously throughout the edifice, have been opened. Each chamber, when examined, is found to be in the precise shape of a dodecahedron. Are the inhabitants justified on that account in holding with certitude that the remaining 975 chambers are built on the same plan?”
Not with perfect certitude, but (if the chambers to which the keys have been found are really “situated promiscuously”) with so high a degree of probability that they would be justified in acting upon the presumption until an exception appeared.
Dr. Ward’s argument, however, does not touch mine as it stands in the text. My argument is grounded on the fact that the uniformity of the course of nature as a whole, is constituted by the uniform sequences of special effects from special natural agencies; that the number of these natural agencies in the part of the universe known to us is not incalculable, nor even extremely great; that we have now reason to think that at least the far greater number of them, if not separately, at least in some of the combinations into which they enter, have been made sufficiently amenable to observation, to have enabled us actually to ascertain some of their fixed laws; and that this amount of experience justifies the same degree of assurance that the course of nature is uniform throughout, which we previously had of the uniformity of sequence among the phenomena best known to us. This view of the subject, if correct, destroys the force of Dr.
Ward’s first argument.
His second argument is, that many or most persons, both scientific and unscientific, believe that there _are_ well authenticated cases of breach in the uniformity of nature, namely, miracles. Neither does this consideration touch what I have said in the text. I admit no other uniformity in the events of nature than the law of Causation; and (as I have explained in the chapter of this volume which treats of the Grounds of Disbelief) a miracle is no exception to that law. In every case of alleged miracle, a _new antecedent_ is affirmed to exist; a _counteracting cause_, namely, the volition of a supernatural being. To all, therefore, to whom beings with superhuman power over nature are a _vera causa_, a miracle is a _case_ of the Law of Universal Causation, not a deviation from it.
Dr. Ward’s last, and as he says, strongest argument, is the familiar one of Reid, Stewart, and their followers—that whatever knowledge experience gives us of the past and present, it gives us none of the future. I confess that I see no force whatever in this argument.
Wherein does a future fact differ from a present or a past fact, except in their merely momentary relation to the human beings at present in existence? The answer made by Priestley, in his _Examination of Reid_, seems to me sufficient, viz., that though we have had no experience of what _is_ future, we have had abundant experience of what _was_ future. The “leap in the dark” (as Professor Bain calls it) from the past to the future, is exactly as much in the dark and no more, as the leap from a past which we have personally observed, to a past which we have not. I agree with Mr.
Bain in the opinion that the resemblance of what we have not experienced to what we have, is, by a law of our nature, presumed through the mere energy of the idea, before experience has proved it. This _psychological_ truth, however, is not, as Dr. Ward when criticising Mr. Bain appears to think, inconsistent with the _logical_ truth that experience does prove it. The proof comes after the presumption, and consists in its invariable _verification_ by experience when the experience arrives. The fact which while it was future could not be observed, having as yet no existence, is always, when it becomes present and _can_ be observed, found conformable to the past.
p. 257) that the uniformity of the course of nature is a different thing from the law of causation; and while he allows that the former is only proved by a long continuance of experience, and that it is not inconceivable nor necessarily incredible that there may be worlds in which it does not prevail, he considers the law of causation to be known intuitively. There is, however, no other uniformity in the events of nature than that which arises from the law of causation: so long therefore as there remained any doubt that the course of nature was uniform throughout, at least when not modified by the intervention of a new (supernatural) cause, a doubt was necessarily implied, not indeed of the reality of causation, but of its universality. If the uniformity of the course of nature has any exceptions—if any events succeed one another without fixed laws—to that extent the law of causation fails; there are events which do not depend on causes.
189 Book i., chap. vii.
190 In some cases, a Kind is sufficiently identified by some one remarkable property: but most commonly several are required; each property considered singly, being a joint property of that and of other Kinds. The color and brightness of the diamond are common to it with the paste from which false diamonds are made; its octohedral form is common to it with alum, and magnetic iron ore; but the color and brightness and the form together, identify its Kind: that is, are a mark to us that it is combustible; that when burned it produces carbonic acid; that it can not be cut with any known substance; together with many other ascertained properties, and the fact that there exist an indefinite number still unascertained.
191 This doctrine of course assumes that the allotropic forms of what is chemically the same substance are so many different Kinds; and such, in the sense in which the word Kind is used in this treatise, they really are.
192 Professor Bain (Logic, ii., 13) mentions two empirical laws, which he considers to be, with the exception of the law connecting Gravity with Resistance to motion, “the two most widely operating laws as yet discovered whereby two distinct properties are conjoined throughout substances generally.” The first is, “a law connecting Atomic Weight and Specific Heat by an inverse proportion. For equal weights of the simple bodies, the atomic weight multiplied by a number expressing the specific heat, gives a nearly uniform product.
The products, for all the elements, are near the constant number 6.”
The other is a law which obtains “between the specific gravity of substances in the gaseous state, and the atomic weights. The relationship of the two numbers is in some instances equality; in other instances the one is a multiple of the other.”
Neither of these generalizations has the smallest appearance of being an ultimate law. They point unmistakably to higher laws. Since the heat necessary to raise to a given temperature the same weight of different substances (called their specific heat) is inversely as their atomic weight, that is, directly as the number of atoms in a given weight of the substance, it follows that a single atom of every substance requires the same amount of heat to raise it to a given temperature; a most interesting and important law, but a law of causation. The other law mentioned by Mr. Bain points to the conclusion, that in the gaseous state all substances contain, in the same space, the same number of atoms; which, as the gaseous state suspends all cohesive force, might naturally be expected, though it could not have been positively assumed. This law may also be a result of the mode of action of causes, namely, of molecular motions. The cases in which one of the numbers is not identical with the other, but a multiple of it, may be explained on the nowise unlikely supposition, that in our present estimate of the atomic weights of some substances, we mistake two, or three, atoms for one, or one for several.
193 Dr. M’Cosh (p. 324 of his book) considers the laws of the chemical composition of bodies as not coming under the principle of Causation; and thinks it an omission in this work not to have provided special canons for their investigation and proof. But every case of chemical composition is, as I have explained, a case of causation. When it is said that water is composed of hydrogen and oxygen, the affirmation is that hydrogen and oxygen, by the action on one another which they exert under certain conditions, _generate_ the properties of water. The Canons of Induction, therefore, as laid down in this treatise, are applicable to the case. Such special adaptations as the Inductive methods may require in their application to chemistry, or any other science, are a proper subject for any one who treats of the logic of the special sciences, as Professor Bain has done in the latter part of his work; but they do not appertain to General Logic.
Dr. M’Cosh also complains (p. 325) that I have given no canons for those sciences in which “the end sought is not the discovery of Causes or of Composition, but of Classes; that is, Natural Classes.”
Such canons could be no other than the principles and rules of Natural Classification, which I certainly thought that I had expounded at considerable length. But this is far from the only instance in which Dr. M’Cosh does not appear to be aware of the contents of the books he is criticising.
194 Mr. De Morgan, in his _Formal Logic_, makes the just remark, that from two such premises as Most A are B, and Most A are C, we may infer with certainty that some B are C. But this is the utmost limit of the conclusions which can be drawn from two approximate generalizations, when the precise degree of their approximation to universality is unknown or undefined.
_ 195 Rationale of Judicial Evidence_, vol. iii., p. 224.
196 The evaluation of the chances in this statement has been objected to by a mathematical friend. The correct mode, in his opinion, of setting out the possibilities is as follows. If the thing (let us call it T) which is both an A and a C, is a B, something is true which is only true twice in every thrice, and something else which is only true thrice in every four times. The first fact being true eight times in twelve, and the second being true six times in every eight, and consequently six times in those eight; both facts will be true only six times in twelve. On the other hand, if T, although it is both an A and a C, is not a B, something is true which is only true once in every thrice, and something else which is only true once in every four times. The former being true four times out of twelve, and the latter once in every four, and therefore once in those four; both are only true in one case out of twelve. So that T is a B six times in twelve, and T is not a B, only once: making the comparative probabilities, not eleven to one, as I had previously made them, but six to one.
In the last edition I accepted this reasoning as conclusive. More attentive consideration, however, has convinced me that it contains a fallacy.
The objector argues, that the fact of A’s being a B is true eight times in twelve, and the fact of C’s being a B six times in eight, and consequently six times in those eight; both facts, therefore, are true only six times in every twelve. That is, he concludes that because among As taken indiscriminately only eight out of twelve are Bs and the remaining four are not, it must equally hold that four out of twelve are not Bs when the twelve are taken from the select portion of As which are also Cs. And by this assumption he arrives at the strange result, that there are fewer Bs among things which are both As and Cs than there are among either As or Cs taken indiscriminately; so that a thing which has both chances of being a B, is less likely to be so than if it had only the one chance or only the other.
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.
199 The axiom, “Equals subtracted from equals leave equal differences,” may be demonstrated from the two axioms in the text. If A = _a_ and 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 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.
202 See the two remarkable notes (A) and (F), appended to his _Inquiry into the Relation of Cause and Effect_.
203 Supra, p. 413.
204 A writer to whom I have several times referred, gives as the definition of an impossibility, that which there exists in the world no cause adequate to produce. This definition does not take in such