Analysis of Mr. Mill's System of Logic Part 5

An a.s.sumed cause, on the other hand, cannot be accepted as true simply _because_ it explains the phenomena (since two conflicting hypotheses often do this even originally, or, as Dr. Whewell himself allows, may at any rate by modifications be made to do it); nor _because_ it moreover leads to the prediction of other results which turn out true (since this shows only what was indeed apparent already from its agreement with the old facts, viz. that the phenomena are governed by laws partially identical with the laws of other causes); nor _because_ we cannot imagine any other hypothesis which will account for the facts (since there may be causes unknown to our present experience which will equally account for them). The utility of such a.s.sumptions _of causes_ depends on their being, in their own nature, _capable_ (as Descartes' Vortices were not, though possibly the Luminiferous Ether may be) of being, at some time or other, proved directly by independent evidence to be the causes. And this was, perhaps, all that Newton meant by his _verae causae_, which alone, he said, may be a.s.signed as causes of phenomena.

a.s.sumptions of causes, which fulfil this condition, are, in science, even indispensable, with a view both to experimental inquiry, and still more to the application of the Deductive Method. They may be accepted, not indeed, as Dr. Whewell thinks they may be, as proof, but as suggesting a line of experiment and observation which may result in proof. And this is actually the method used by practical men for eliciting the truth from involved statements. They first extemporise, from a few of the particulars, a rude theory of the mode in which the event happened; and then keep altering it to square with the rest of the facts, which they review one by one.

The attempting, as in Geology, to conjecture, in conformity with known laws, in what former collocations of known agents (though _not_ known to have been formerly present) individual existing facts may have originated, is not Hypothesis but Induction; for then we do not _suppose_ causes, but legitimately infer from known effects to unknown causes. Of this nature was Laplace's theory, whether weak or not, as to the origin of the earth and planets.

CHAPTER XV.

PROGRESSIVE EFFECTS, AND CONTINUED ACTION OF CAUSES.

Sometimes a complex effect results, not (as has been supposed in the last four chapters) from several, but from _one_ law. The following is the way.

Some effects are instantaneous (e.g. some sensations), and are prolonged only by the prolongation of the causes; others are in their own nature permanent. In some cases of the latter cla.s.s, the original is also the proximate cause (e.g. Exposure to moist air is both the original and the proximate cause of iron rust). But in others of the same cla.s.s, the permanency of the effect is only the permanency of a series of changes. Thus, e.g. in cases of Motion, the original force is only the _remote_ cause of any link (after the very first) in the series; and the motion immediately preceding it, being itself a compound of the original force and any r.e.t.a.r.ding agent, is its _proximate_ cause.

When the original cause is permanent as well as the effect (e.g. Suppose a continuance of the iron's exposure to moist air), we get a progressive series of effects arising from the cause's acc.u.mulating influence; and the sum of these effects amounts exactly to what a number of successively introduced similar causes would have produced. Such cases fall under the head of Composition of Causes, with this peculiarity, that, as the causes (to regard them as plural) do not come into play all at once, the effect at each instant is the sum of the effects only of the then acting causes, and the result will appear as an ascending series. Each addition in such case takes place according to a fixed law (equal quant.i.ties in equal times); and therefore it can be computed deductively. Even when, as is sometimes the case, a cause is at once permanent and progressive (as, e.g. the sun, by its position becoming more vertical, increases the heat in summer) so that the quant.i.ties added are unequal, the effect is still progressive, resulting from its cause's continuance and progressiveness combined.

In _all_ cases whatever of progressive effects, the succession not merely between the cause and the effect, but also between the first and latter stages of the effect, is uniform. Hence, from the invariable sequence of two terms (e.g. Spring and Summer) in a series going through any continued and uniform process of variation, we do not presume that one is the cause and the others the effect, but rather that the whole series is an effect.

CHAPTER XVI.

EMPIRICAL LAWS.

Empirical laws are derivative laws, of which the derivation is not known. They are observed uniformities, which we compare with the result of any deduction to verify it; but of which the _why_, and also the limits, are unrevealed, through their being, though resolvable, not yet resolved into the simpler laws. They depend usually, not solely on the ultimate laws into which they are resolvable; but on those, together with an ultimate fact, viz. the mode of coexistence of some of the component elements of the universe. Hence their untrustworthiness for scientific purposes; for, till they have been resolved (and then a derivative law ceases to be empirical), we cannot know whether they result from the different effects of one cause, or from effects of different causes; that is, whether they depend on laws, or on laws and a collocation. And if they thus depend on a collocation, they can be received as true only within the limits of time and s.p.a.ce, and also circ.u.mstance, in which they have been observed, since the mode of the collocation of the permanent causes is not reducible to a law, there being no principle known to us as governing the distribution and relative proportions of the primaeval natural agents.

Uniformities cannot be proved by the Method of Agreement alone to be laws of causation; they must be tested by the Method of Difference, or explained deductively. But laws of causation themselves are either ultimate or derivative. Signs, previous to actual proof by _resolution_ of them, of their being derivative, are, either that we can _surmise_ the existence of a link between the known antecedent and the consequent, as e.g. in the laws of chemical action; or, that the antecedent is some very complex fact, the effects of which are probably (since most complex cases fall under the Composition of Causes) compounded of the effects of its different elements. But the laws which, though laws of causation, are thus presumably derivative laws only, need, equally with the uniformities which are not known to be laws of causation at all, to be explained by deduction (which they then in turn verify), and are less _certain_ than when they have been resolved into the ultimate laws.

Consequently they come under the definition of Empirical Laws, equally with uniformities not known to be laws of causation. However, the latter are far more _uncertain_; for as, till they are resolved, we cannot tell on how many collocations, as well as laws, they may not depend, we must not rely on them beyond the exact limits in which the observations were made. Therefore, the name _Empirical Laws_ will generally be confined here to these.

CHAPTER XVII.

CHANCE, AND ITS ELIMINATION.

Empirical laws are certain only in those limits within which they have been _observed_ to be true. But, even within those limits, the connection of two phenomena may, as the same effect may be produced by several different causes, be due to Chance; that is, it may, though being, as all facts must be, the result of some law, be a coincidence whence, simply because we do not know all the circ.u.mstances, _we_ have no ground to infer an uniformity. When neither Deduction, nor the Method of Difference, can be applied, the only way of inferring that coincidences are not casual, is by observing the frequency of their occurrence, not their absolute frequency, but whether they occur _more_ often than chance would (that is, more often than the positive frequency of the phenomena would) account for. If, in such cases, we could ascend to the causes of the two phenomena, we should find at some stage some cause or causes common to both. Till we can do this, the fact of the connection between them is only an empirical law; but still it is a law.

Sometimes an effect is the result partly of chance, and partly of law: viz. when the total effect is the result partly of the effects of casual conjunctions of causes, and partly of the effects of some constant cause which they blend with and modify. This is a case of Composition of Causes. The object being to find _how much_ of the result is attributable to a given constant cause, the only resource, when the variable causes cannot be wholly excluded from the experiment, is to ascertain what is the effect of all of _them_ taken together, and then to eliminate this, which is the casual part of the effect, in reckoning up the results. If the results of frequent experiments, in which the constant cause is kept invariable, oscillate round one point, that average or middle point is due to the constant cause, and the variable remainder to chance; that is, to causes the coexistence of which with the constant cause was merely casual. The test of the sufficiency of such an induction is, whether or not an increase in the number of experiments materially alters the average.

We can thus discover not merely _how much_ of the effect, but even whether _any_ part of it whatever is due to a constant cause, when this latter is so uninfluential as otherwise to escape notice (e.g. the loading of dice). This case of the Elimination of Chance is called _The discovery of a residual phenomenon by eliminating the effects of chance._

The mathematical doctrine of chances, or Theory of Probabilities, considers what deviation from the average chance by itself can possibly occasion in some number of instances smaller than is required for a fair average.

CHAPTER XVIII.

THE CALCULATION OF CHANCES.

In order to calculate chances, we must know that of several events one, and no more, must happen, and also not know, or have any reason to suspect, which of them that one will be. Thus, with the simple knowledge that the issue must be one of a certain number of possibilities, we _may_ conclude that one supposition is most probable _to us_. For this purpose it is not _necessary_ that specific experience or reason should have also proved the occurrence of each of the several events to be, as a fact, equally frequent. For, the probability of an event is not a quality of the event (since every event is in itself certain), but is merely a name for the degree of ground _we_ have, with our present evidence, for expecting it. Thus, if we know that a box contains red, white, and black b.a.l.l.s, though we do not know in what proportions they are mingled, we have numerically appreciable grounds for considering the probability to be two to one against any one colour. Our judgment may indeed be said in this case to rest on the experience we have of the laws governing the frequency of occurrence of the different cases; but such experience is universal and axiomatic, and not specific experience about a particular event. Except, however, in games of chance, the purpose of which requires ignorance, such specific experience can generally be, and should be gained. And a slight improvement in the data profits more than the most elaborate application of the calculus of probabilities to the bare original data, e.g. to such data, when we are calculating the credibility of a witness, as the proportion, even if it could be verified, between the number of true and of erroneous statements a man, _qua_ man, may be supposed to make during his life.

Before applying the Doctrine of Chance, therefore, we should lay a foundation for an evaluation of the chances by gaining positive knowledge of the facts. Hence, though not a _necessary_, yet a most usual condition for calculating the probability of a fact is, that we should possess a _specific_ knowledge of the proportion which the cases in which facts of the particular sort occur bear to the cases in which they do not occur.

Inferences drawn correctly according to the Doctrine of Chances depend ultimately on causation. This is clearest, when, as sometimes, the probability of an event is deduced from the frequency of the occurrence of the causes. When its probability is calculated by merely counting and comparing the number of cases in which it has occurred with those in which it has not, the law, being arrived at by the Method of Agreement, is only empirical. But even when, as indeed generally, the numerical data are obtained in the latter way (since usually we can judge of the frequency of the causes only through the medium of the empirical law, which is based on the frequency of the effects), still then, too, the inference really depends on causation alone. Thus, an actuary infers from his tables that, of any hundred living persons under like conditions, five will reach a given age, not simply because that proportion have reached it in times past, but because that fact shows the existence there of a particular proportion between the _causes_ which shorten and the causes which prolong life to the given extent.

CHAPTER XIX.

THE EXTENSION OF DERIVATIVE LAWS TO ADJACENT CASES.

Derivative laws are inferior to ultimate laws, both in the extent of the propositions, and in their degree of certainty within that extent. In particular, the uniformities of coexistence and sequence which obtain between effects depending on different primaeval causes, vary along with any variation in the collocation of these causes. Even when the derivative uniformity is between different effects of the same cause, it cannot be trusted to, since one or more of the effects may be producible by another cause also. The effects, even, of derivative laws of _causation_ (resulting, i.e. the laws, from the combination of several causes) are not independent of collocations; for, though laws of causation, whether ultimate or derivative, are themselves universal, being fulfilled even when counteracted, the peculiar probability of the latter kind of laws of causation being counteracted (as compared with ultimate laws, which are liable to frustration only from one set of counteracting causes) is fatal to the universality of the derivative uniformities made up of the sequences or coexistences of their effects; and, therefore, such derivative uniformities as the latter are to be relied on only when the collocations are known not to have changed.

Derivative laws, not causative, may certainly be extended beyond the limits of observation, but only to cases _adjacent_ in time. Thus, we may not predict that the sun will rise this day 20,000 years, but we can predict that it will rise to-morrow, on the ground that it has risen every day for the last 5,000 years. The latter prediction is lawful, _because_, while we know the causes on which its rising depends, we know, also, that there has existed hitherto no perceptible cause to counteract them; and that it is opposed to experience that a cause imperceptible for so long should start into immensity in a day. If the uniformity is empirical only, that is, if we do not know the causes, and if we infer that they remain uncounteracted from their effects alone, we still can extend the law to adjacent cases, but only to cases still more closely adjacent in time; since we can know neither whether changes in these unknown causes may not have occurred, nor whether there may not exist now an adverse cause capable after a time of counteracting them.

An empirical law cannot generally be extended, in reference to _Place_, even to adjacent cases (since there is no uniformity in the collocations of primaeval causes). Such an extension is lawful only if the new cases are _presumably_ within the influence of the same individual causes, even though unknown. When, however, the causes are known, and the conjunction of the effects is deducible from laws of the causes, the derivative uniformity may be extended over a wider s.p.a.ce, and with less abatement for the chance of counteracting causes.

CHAPTER XX.

a.n.a.lOGY.

One of the many meanings of _a.n.a.logy_ is, Resemblance of Relations. The value of an a.n.a.logical argument in this sense depends on the showing that, on the common circ.u.mstance which is the _fundamentum relationis_, the rest of the circ.u.mstances of the case depend. But, generally, _to argue from a.n.a.logy_ signifies to infer from resemblance in some points (not necessarily in _relations_) resemblance in others. Induction does the same: but a.n.a.logy differs from induction in not requiring the previous proof, by comparison of instances, of the invariable conjunction between the known and the unknown properties; though it requires that the latter should not have been ascertained to be _unconnected_ with the common properties.

If a fair proportion of the properties of the two cases are known, every resemblance affords ground for expecting an indefinite number of other resemblances, among which the property in question may perhaps be found.

On the other hand, every dissimilarity will lead us to expect that the two cases differ in an indefinite number of properties, including, perhaps, the one in question. These dissimilarities may even be such as would, in regard to one of the two cases, imply the absence of that property; and then every resemblance, as showing that the two cases have a similar nature, is even a reason for presuming against the presence of that property. Hence, the value of an a.n.a.logical argument depends on the extent of ascertained resemblance as compared, first, with the amount of ascertained difference, and next, with the extent of the unexplored region of unascertained properties.

The conclusions of a.n.a.logy are not of direct use, unless when the case to which we reason is a case _adjacent_, not, as before, in time or place, but in _circ.u.mstances_. Even then a complete induction should be sought after. But the great value of a.n.a.logy, even when faint, in science, is that it may suggest observations and experiments, with a view to establis.h.i.+ng positive scientific truths, for which, however, the hypotheses based on a.n.a.logies must never be mistaken.

CHAPTER XXI.

THE EVIDENCE OF THE LAW OF UNIVERSAL CAUSATION.

The validity of all the four inductive methods depends on our a.s.suming that there is a cause for every event. The belief in this, i.e. in the law of universal causation, some affirm, is an instinct which needs no warrant other than all men's disposition to believe it; and they argue that to demand evidence of it is to appeal to the intellect from the intellect. But, though there is no appeal from the faculties all together, there may be from one to another: and, as belief is not proof (for it may be generated by a.s.sociation of ideas as well as by evidence), a case of belief does require to be proved by an appeal to something else, viz. to the faculties of sense and consciousness.