Uniformities of co-existence, then, not only when they are consequences of laws of succession, but also when they are ultimate truths, must be ranked, for the purposes of logic, among empirical laws, and are amen been formed among simple substances, (and the attempt has been often made,) have, with the progress of experience, either faded into inanity, or been proved to be erroneous; and each Kind of simple substance remains with its own collection of pro-able in every respect to the same perties apart from the rest, saving a rules with those unresolved uniforcertain parallelism with a few other mities which are known to be depenKinds, the most similar to itself. In dent on causation. organised beings, indeed, there are abundance of propositions ascertained to be universally true of superior genera, to many of which the discovery hereafter of any exceptions must be regarded as extremely improbable. But these, as already observed, are, we have every reason to believe, properties dependent on causation. * * 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 generalisations 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 CHAPTER XXIII. OF APPROXIMATE GENERALISATIONS, §1. IN our inquiries into the nature of the inductive process, we must 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. * 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 dis covery 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. not confine our notice to such genera- | universal proposition, Every B is C, lisations from experience as profess to arrive at the conclusion that Most A be universally true. There is a class are C. But when a second proposiof inductive truths avowedly not uni- tion of the approximate kind is introversal, in which it is not pretended duced, or even when there is but that the predicate is always true of one, if that one be the major premise, the subject, but the value of which, nothing can in general be positively as generalisations, is nevertheless extremely great. An important portion of the field of inductive knowledge does not consist of universal truths, but of approximations to such truths; and when a conclusion is said to rest on probable evidence, the premises it is drawn from are usually generalisations of this sort. concluded. When the major is Most B are D, then, even if the minor be Every A is B, we cannot infer that most A are D, or with any certainty that even some A are D. Though the majority of the class B have the attribute signified by D, the whole of the sub-class A may belong to the minority.* As every certain inference respect- Though so little use can be made, ing a particular case implies that in science, of approximate generalisathere is ground for a general propositions, except as a stage on the road to tion, of the form, Every A is B; so something better, for practical guiddoes every probable inference suppose that there is ground for a proposition of the form, Most A are B; and the degree of probability of the inference in an average case will depend on the proportion between the number of instances existing in nature which accord with the generalisation, and the number of those which conflict with it. § 2. Propositions in the form, Most A are B, are of a very different degree of importance in science, and in the practice of life. To the scientific inquirer they are valuable chiefly as materials for, and steps towards, universal truths. The discovery of these is the proper end of science : its work is not done if it stops at the proposition that a majority of A are B, without circumscribing that majority by some cominon character, fitted to distinguish them from the minority. Independently of the inferior precision of such imperfect generalisations, and the inferior assurance with which they can be applied to individual cases, it is plain that, compared with exact generalisations, they are almost useless as means of discovering ulterior truths by way of deduction. We may, it is true, by combining the proposition Most A are B, with an ance they are often all we have to rely on. Even when science has really determined the universal laws of any phenomenon, not only are those laws generally too much encumbered with conditions to be adapted for everyday use, but the cases which present themselves in life are too complicated, and our decisions require to be taken too rapidly, to admit of waiting till the existence of a phenomenon can be proved by what have been scientifically ascertained to be universal marks of it. To be indecisive and reluctant to act, because we have not evidence of a perfectly conclusive character to act on, is a defect sometimes incident to scientific minds, but which, wherever it exists, renders them unfit for practical emergencies. If we would succeed in action, we must judge by indications which, though they do not generally mislead us, sometimes do; and must make up, as far as possible, for the * 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 generalisations, when the precise degree of their approximation to universality is unknown or undefined. incomplete conclusiveness of any one indication, by obtaining others to corroborate it. The principles of induction applicable to approximate generalisation are therefore a not less important subject of inquiry than the rules for the investigation of universal truths, and might reasonably be expected to detain us almost as long, were it not that these principles are mere corollaries from those which have been already treated of. $3. There are two sorts of cases in which we are forced to guide ourselves by generalisations of the imperfect form, Most A are B. The first is, when we have no others; when we have not been able to carry our investigation of the laws of the phenomena any farther; as in the following propositions: Most darkeyed persons have dark hair; Most springs contain mineral substances; Most stratified formations contain fossils. The importance of this class of generalisations is not very great; for though it frequently happens that we see no reason why that which is true of most individuals of a class is not true of the remainder, nor are able to bring the former under any general description which can distinguish them from the latter, yet if we are willing to be satisfied with propositions of a less degree of generality, and to break down the class A into sub-classes, we may generally obtain a collection of propositions exactly true. We do not know why most wood is lighter than water, nor can we point out any general property which discriminates wood that is lighter than water from that which is heavier. But we know exactly what species are the one and what the other. And if we meet with a specimen not conformable to any known species, (the only case in which our previous knowledge affords no other guidance than the approximate generalisation,) we can generally make a specific experiment, which is a surer resource. It often happens, however, that the proposition, Most A are B, is not the ultimatum of our scientific attainments, though the knowledge we possess beyond it cannot conveniently be brought to bear upon the particular instance. We may know well enough what circumstances distinguish the portion of A which has the attribute B from the portion which has it not, but may have no means, or may not have time to examine whether those characteristic circumstances exist or not in the individual case. This is the situation we are generally in when the inquiry is of the kind called moral, that is, of the kind which has in view to predict human actions. To enable us to affirm anything universally concerning the actions of classes of human beings, the classification must be grounded on the circumstances of their mental culture and habits, which in an individual case are seldom exactly known; and classes grounded on these distinctions would never precisely accord with those into which mankind are divided for social purposes. All propositions which can be framed respecting the actions of human beings as ordinarily classified, or as classified according to any kind of outward indications, are merely approximate. We can only say, Most persons of a particular age, profession, country, or rank in society have such and such qualities; or, Most persons when placed in certain circumstances act in such and such a way. Not that we do not often know well enough on what causes the qualities depend, or what sort of persons they are who act in that particular way; but we have seldom the means of knowing whether any individual person has been under the influence of those causes, or is a person of that particular sort. We could replace the approximate generalisations by propositions universally true; but these would hardly ever be capable of being applied to practice. We should be sure of our majors, but we should not be able to get minors to fit: we are forced, therefore, to draw | perhaps affect it. The proposition, our conclusions from coarser and more fallible indications. § 4. Proceeding now to consider what is to be regarded as sufficient evidence of an approximate generalisation, we can have no difficulty in at once recognising that when admissible at all, it is admissible only as an empirical law. Propositions of the form, Every A is B, are not necessarily laws of causation, or ultimate uniformities of co-existence; propositions like Most A are B, cannot be so. Propositions hitherto found true in every observed instance may yet be no necessary consequence of laws of causation or of ultimate uniformities, and unless they are so, may, for aught we know, be false beyond the limits of actual observation: still more evidently must this be the case with propositions which are only true in a mere majority of the observed instances. There is some difference, however, in the degree of certainty of the proposition, Most A are B, according as that approximate generalisation comprises the whole of our knowledge of the subject or not. Suppose, first, that the former is the case. We know only that most A are B, not why they are so, nor in what respect those which are, differ from those which are not. How then did we learn that most A are B? Precisely in the manner in which we should have learnt, had such happened to be the fact, that all A are B. We collected a number of instances sufficient to eliminate chance, and having done so, compared the number of instances in the affirmative with the number in the negative. The result, like other unresolved derivative laws, can be relied on solely within the limits not only of place and time, but also of circumstance, under which its truth has been actually observed; for as we are supposed to be ignorant of the causes which make the proposition true, we cannot tell in what manner any new circumstance might Most judges are inaccessible to bribes, would probably be found true of Englishmen, Frenchmen, Germans, North Americans, and so forth; but if on this evidence alone we extended the assertion to Orientals, we should step beyond the limits, not only of place but of circumstance, within which the fact had been observed, and should let in possibilities of the absence of the determining causes, or the presence of counteracting ones, which might be fatal to the approximate generalisation. In the case where the approximate proposition is not the ultimatum of our scientific knowledge, but only the most available form of it for practical guidance; where we know, not only that most A have the attribute B, but also the causes of B, or some properties by which the portion of A which has that attribute is distinguished from the portion which has it not; we are rather more favourably situated than in the preceding case. For we have now a double mode of ascertaining whether it be true that most A are B; the direct mode, as before, and an indirect one, that of examining whether the proposition admits of being deduced from the known cause, or from any known criterion, of B. Let the question, for example, be whether most Scotchmen can read? We may not have ob served or received the testimony of others respecting a sufficient number and variety of Scotchinen to ascertain this fact; but when we consider that the cause of being able to read is the having been taught it, another mode of determining the question presents itself, namely, by inquiring whether most Scotchmen have been sent to schools where reading is effectually taught. Of these two modes, sometimes one and sometimes the other is the more available. In some cases, the frequency of the effect is the more accessible to that extensive and varied observation which is indispensable to the establishment of an empirical law; at other times, the frequency of the causes, or of some collateral indications. It commonly happens that neither is susceptible of so satisfactory an induction as could be desired, and that the grounds on which the conclusion is received are compounded of both. Thus a person may believe that most Scotchmen can read, because, so far as his information extends, most Scotchmen have been sent to school, and most Scotch schools teach reading effectually; and also because most of the Scotchmen whom he has known or heard of could read; though neither of these two sets of observations may by itself fulfil the necessary conditions of extent and variety. know directly, and the other two points indirectly, by means of marks; as, for example, from his conduct on some former occasion, or from his reputation, which, though a very uncertain mark, affords an approximate generalisation, (as, for instance, Most persons who are believed to be honest by those with whom they have had frequent dealings are really so,) which approaches nearer to an universal truth than the approximate general proposition with which we set out, viz., Most persons on most occasions speak truth. As it seems unnecessary to dwell further on the question of the evidence of approximate generalisations, we shall proceed to a not less imAlthough the approximate gene-portant topic, that of the cautions to ralisation may in most cases be indis- be observed in arguing from these pensable for our guidance, even when incompletely universal propositions to we know the cause, or some certain particular cases. mark, of the attribute predicated; it needs hardly be observed that we may always replace the uncertain indication by a certain one, in any case in which we can actually recognise the existence of the cause or mark. For example, an assertion is made by a witness, and the question is whether to believe it. If we do not look to any of the individual circumstances of the case, we have nothing to direct us but the approximate generalisation that truth is more common than falsehood, or, in other words, that most persons, on most occasions, speak truth. But if we consider in what circumstances the cases where truth is spoken differ from those in which it is not, we find, for instance, the following the witness's being an honest person or not; his being an accurate observer or not; his having an interest to serve in the matter or not. Now, not only may we be able to obtain other approximate generalisations respecting the degree of frequency of these various possibilities, but we may know which of them is positively realised in the individual That the witness has or has not an interest to serve, we perhaps case. : $5. So far as regards the direct application of an approximate generalisation to an individual instance, this question presents no difficulty. If the proposition, Most A are B, has been established, by a sufficient induction, as an empirical law, we may conclude that any particular A is B with a probability proportioned to the preponderance of the number of affirmative instances over the number of exceptions. If it has been found practicable to attain numerical precision in the data, a corresponding degree of precision may be given to the evaluation of the chances of error in the conclusion. If it can be established as an empirical law that nine out of every ten A are B, there will be one chance in ten of error in assuming that any A not individually known to us is a B; but this, of course, holds only within the limits of time, place, and circumstance embraced in the observations, and therefore cannot be counted on for any sub-class or variety of A (or for A in any set of external circumstances) which were not included in the average. It must be added that we can guide ourselves |