It is remarkable that Mill's review of Whately, the outcome of these discussions, says very little about Induction. At that stage Mill's chief concern seems to have been to uphold the usefulness of Deductive Logic, and he even goes so far as to scoff at its eighteenth century detractors and their ambition to supersede it with a system of Induction. The most striking feature of the article is the brilliant defence of the Syllogism as an analysis of arguments to which I have already referred. He does not deny that an Inductive Logic might be useful as a supplement, but apparently he had not then formed the design of supplying such a supplement. When, however, that design seriously entered his mind, consequent upon the felt need of a method for social investigations, it was Whately's conception of Induction that he fell back upon. Historically viewed, his System of Logic was an attempt to connect the practical conditions of proof set forth in Herschel's discourse with the theoretic view of Induction propounded in Whately's. The tag by which he sought to attach the new material to the old system was the Inductive Enthymeme of the Schoolmen as interpreted by Whately.

Whately's interpretation--or misinterpretation--of this Enthymeme, and the conception of Induction underlying it, since it became Mill's ruling conception of Induction, and virtually the formative principle of his system, deserves particular attention.

"This, that and the other horned animal, ox, sheep, goat, ruminate; _therefore_, all horned animals ruminate."

The traditional view of this Enthymeme I have given in my chapter on Formal Induction (p. 238). It is that a Minor Premiss is suppressed: "This, that and the other constitute the whole class". This is the form of the Minor in Aristotle's Inductive Syllogism.

But, Whately argued, how do we know that this, that and the other--the individuals we have examined--constitute the whole class? Do we not assume that what belongs to the individuals examined belongs to the whole class? This tacit assumption, he contended, is really at the bottom of the Enthymeme, and its proper completion is to take this as the Major Premiss, with the enumeration of individuals as the Minor.

Thus:--

What belongs to the individuals examined belongs to the whole class.

The property of the ruminating belongs to the individuals examined, ox, sheep, goat, etc.

_Therefore_, it belongs to all.

In answer to this, Hamilton repeated the traditional view, treating Whately's view merely as an instance of the prevailing ignorance of the history of Logic. He pointed out besides that Whately's Major was the postulate of a different kind of inference from that contemplated in Aristotle's Inductive Syllogism, Material as distinguished from Formal inference. This is undeniable if we take this syllogism purely as an argumentative syllogism. The "all" of the conclusion simply covers the individuals enumerated and admitted to be "all" in the Minor Premiss. If a disputant admits the cases produced to be all and can produce none to the contrary, he is bound to admit the conclusion.

Now the inference contemplated by Whately was not inference from an admission to what it implies, but inference from a series of observations to all of a like kind, observed and unobserved.

It is not worth while discussing what historical justification Whately had for his view of Induction. It is at least arguable that the word had come to mean, if it did not mean with Aristotle himself, more than a mere summation of particulars in a general statement. Even Aristotle's respondent in the concession of his Minor admitted that the individuals enumerated constituted all in the truly general sense, not merely all observed but all beyond the range of observation. The point, however, is insignificant. What really signifies is that while Hamilton, after drawing the line between Formal Induction and Material, fell back and entrenched himself within that line, Mill caught up Whately's conception of Induction, pushed forward, and made it the basis of his System of Logic.

In Mill's definition, the mere summation of particulars, _Inductio per enumerationem simplicem ubi non reperitur instantia contradictoria_, is Induction improperly so called. The only process worthy of the name is Material Induction, inference to the unobserved. Here only is there an advance from the known to the unknown, a veritable "inductive hazard".

Starting then with this conception of inference to the unobserved as the only true inference, and with an empirical law--a generality extended from observed cases to unobserved--as the type of such inference, Mill saw his way to connecting a new Logic with the old. We must examine this junction carefully, and the brilliant and plausible arguments by which he supported it; we shall find that, biased by this desire to connect the new with the old, he gave a misleading dialectic setting to his propositions, and, in effect, confused the principles of Argumentative conclusion on the one hand and of Scientific Observation and Inference on the other. The conception of Inference which he adopted from Whately was too narrow on both sides for the uses to which he put it. Be it understood that in the central methods both of Syllogistic and of Science, Mill was substantially in accord with tradition; it is in his mode of junction, and the light thereby thrown upon the ends and aims of both, that he is most open to criticism.

As regards the relation between Deduction and Induction, Mill's chief proposition was the brilliant paradox that all inference is at bottom Inductive, that Deduction is only a partial and accidental stage in a process the whole of which may be called Induction. An opinion was abroad--fostered by the apparently exclusive devotion of Logic to Deduction--that all inference is essentially Deductive. Not so, answered Mill, meeting this extreme with another: all inference is essentially Inductive. He arrives at this through the conception that Induction is a generalisation from observed particulars, while Deduction is merely the extension of the generalisation to a new case, a new particular. The example that he used will make his meaning plain.

Take a common Syllogism:--

All men are mortal.

Socrates is a man.

Socrates is mortal.

"The proposition," Mill says, "that Socrates is mortal is evidently an inference. It is got at as a conclusion from something else. But do we in reality conclude it from the proposition, All men are mortal?" He answers that this cannot be, because if it is not true that Socrates is mortal it cannot be true that all men are mortal. It is clear that our belief in the mortality of Socrates must rest on the same ground as our belief in the mortality of men in general. He goes on to ask whence we derive our knowledge of the general truth, and answers: "Of course from observation. Now all which man can observe are individual cases.... A general truth is but an aggregate of particular truths.

But a general proposition is not merely a compendious form for recording a number of particular facts.... It is also a process of inference. From instances which we have observed we feel warranted in concluding that what we have found true in those instances, holds in all similar ones, past, present, and future. We then record all that we have observed together with what we infer from our observations, in one concise expression." A general proposition is thus at once a summary of particular facts and a memorandum of our right to infer from them. And when we make a deduction we are, as it were, interpreting this memorandum. But it is upon the particular facts that the inference really rests, and Mill contends that we might if we chose infer to the particular conclusion at once without going through the form of a general inference. Thus Mills seeks to make good his point that all inference is essentially Inductive, and that it is only for convenience that the word Induction has been confined to the general induction, while the word Deduction is applied to the process of interpreting our memorandum.

Clear and consecutive as this argument is, it is fundamentally confusing. It confuses the nature of Syllogistic conclusion or Deduction, and at the same time gives a partial and incomplete account of the ground of Material inference.

The root of the first confusion lies in raising the question of the ground of material inference in connexion with the Syllogism. As regards the usefulness of the Syllogism, this is an IGNORATIO ELENCHI.

That the Major and the conclusion rest upon the same ground as matters of belief is indisputable: but it is irrelevant. In so far as "Socrates is mortal" is an inference from facts, it is not the conclusion of a Syllogism. This is implicitly and with unconscious inconsistency recognised by Mill when he represents Deduction as the interpretation of a memorandum. To represent Deduction as the interpretation of a memorandum--a very happy way of putting it and quite in accordance with Roger Bacon's view--is really inconsistent with regarding Deduction as an occasional step in the process of Induction. If Deduction is the interpretation of a memorandum, it is no part of the process of inference from facts. The conditions of correct interpretation as laid down in Syllogism are one thing, and the methods of correct inference from the facts, the methods of science that he was in search of, are another.

Let us emphasise this view of Deduction as the interpretation of a memorandum. It corresponds exactly with the view that I have taken in discussing the utility of the Syllogism. Suppose we want to know whether a particular conclusion is consistent with our memorandum, what have we to look to? We have to put our memorandum into such a form that it is at once apparent whether or not it covers our particular case. The Syllogism aspires to be such a form. That is the end and aim of it. It does not enable us to judge whether the memorandum is a legitimate memorandum or not. It only makes clear that if the memorandum is legitimate, so is the conclusion. How to make clear and consistent memoranda of our beliefs in words is a sufficiently complete description of the main purpose of Deductive Logic.

Instead, then, of trying to present Deduction and Induction as parts of the same process, which he was led to do by his desire to connect the new and the old, Mill ought rather, in consistency as well as in the interests of clear system, to have drawn a line of separation between the two as having really different ends, the conditions of correct conclusion from accepted generalities on the one hand, and the conditions of correct inference from facts on the other. Whether the first should be called inference at all is a question of naming that ought to have been considered by itself. We may refuse to call it inference, but we only confuse ourselves and others if we do not acknowledge that in so doing we are breaking with traditional usage.

Perhaps the best way in the interests of clearness is to compromise with tradition by calling the one Formal Inference and the other Material Inference.

It is with the latter that the Physical Sciences are mainly concerned, and it was the conditions and methods of its correct performance that Mill desired to systematise in his Inductive Logic. We have next to see how his statement of the grounds of Material Inference was affected by his connexion of Deduction and Induction. Here also we shall find a reason for a clearer separation between the two departments of Logic.

In his antagonism to a supposed doctrine that all reasoning is from general to particular, Mill maintained _simpliciter_ that all reasoning is from particulars to particulars. Now this is true only _secundum quid_, and although in the course of his argument Mill introduced the necessary qualifications, the unqualified thesis was confusing. It is perfectly true that we may infer--we can hardly be said to reason--from observed particulars to unobserved. We may even infer, and infer correctly, from a single case. The village matron, called in to prescribe for a neighbour's sick child, infers that what cured her own child will cure the neighbour's, and prescribes accordingly. And she may be right. But it is also true that she may be wrong, and that no fallacy is more common than reasoning from particulars to particulars without the requisite precautions. This is the moral of one of the fables of Camerarius. Two donkeys were travelling in the same caravan, the one laden with salt, the other with hay. The one laden with salt stumbled in crossing a stream, his panniers dipped in the stream, the salt melted, and his burden was lightened. When they came to another stream, the donkey that was laden with hay dipped his panniers in the water, expecting a similar result. Mill's illustrations of correct inference from particulars to particulars were really irrelevant. What we are concerned with in considering the grounds of Inference, is the condition of correct inference, and no inference to an unobserved case is sound unless it is of a like kind with the observed case or cases on which it is founded, that is to say, unless we are entitled to make a general proposition. We need not go through the form of making a general proposition, but if a general proposition for all particulars of a certain description is not legitimate, no more is the particular inference. Mill, of course, did not deny this, he was only betrayed by the turn of his polemic into an unqualified form of statement that seemed to ignore it.

But this was not the worst defect of Mill's attempt at a junction of old and new through Whately's conception of Induction. A more serious defect was due to the insufficiency of this conception to represent all the modes of scientific inference. When a certain attribute has been found in a certain connexion in this, that, and the other, to the extent of all observed instances, we infer that it will be found in all, that the connexion that has obtained within the range of our actual experience has obtained beyond that range and will obtain in the future. Call this an observed uniformity of nature: we hold ourselves justified in expecting that the observed uniformities of nature will continue. Such an observed uniformity--that All animals have a nervous system, that All animals die, that Quinine cures ague--is also called an Empirical Law.

But while we are justified in extending an empirical law beyond the limits within which it has been observed to hold good, it is a mistake to suppose that the main work of science is the collection of empirical laws, and that the only scientific inference is the inference from the observed prevalence of an empirical law to its continuance. With science the collection of empirical laws is only a preliminary: "the goal of science," in Herschel's phrase, "is explanation". In giving such prominence to empirical laws in his theory, Mill confined Induction to a narrower scope than science ascribes to it. Science aims at reaching "the causes of things": it tries to penetrate behind observed uniformities to the explanation of them. In fact, as long as a science consists only of observed uniformities, as long as it is in the empirical stage, it is a science only by courtesy. Astronomy was in this stage before the discovery of the Law of Gravitation. Medicine is merely empirical as long as its practice rests upon such generalisations as that Quinine cures ague, without knowing why. It is true that this explanation may consist only in the discovery of a higher or deeper uniformity, a more recondite law of connexion: the point is that these deeper laws are not always open to observation, and that the method of reaching them is not merely observing and recording.

In the body of his Inductive Logic, Mill gave a sufficient account of the Method of Explanation as practised in scientific inquiry. It was only his mode of approaching the subject that was confusing, and made it appear as if the proper work of science were merely extending observed generalities, as when we conclude that all men will die because all men have died, or that all horned animals ruminate because all hitherto observed have had this attribute. A minor source of confusion incident to the same controversy was his refusing the title of Induction proper to a mere summary of particulars. He seemed thereby to cast a slight upon the mere summation of particulars. And yet, according to his theory, it was those particulars that were the basis of the Induction properly so called. That all men will die is an inference from the observation summed up in the proposition that all men have died. If we refuse the name of Induction to the general proposition of fact, what are we to call it? The truth is that the reason why the word Induction is applied indifferently to the general proposition of fact and the general proposition applicable to all time is that, once we are sure of the facts, the transition to the inference is so simple an affair that it has not been found necessary in practice to distinguish them by different names.

Our criticism of Mill would itself mislead if it were taken to mean that the methods of science which he formulated are not the methods of science or that his system of those methods is substantially incomplete. His Inductive Logic as a system of scientific method was a great achievement in organisation, a veritable _Novum Organum_ of knowledge. What kept him substantially right was that the methods which he systematised were taken from the practice of men of science.

Our criticism amounts only to this, that in correlating the new system with the old he went upon a wrong track. For more than two centuries Deduction had been opposed to Induction, the _ars disserendi_ to the _ars inveniendi_. In trying to reconcile them and bring them under one roof, Mill drew the bonds too tight. In stating the terms of the union between the two partners, he did not separate their spheres of work with sufficient distinctness.

Mill's theory of Deduction and Induction and the voluminous criticism to which in its turn it has been subjected have undoubtedly been of great service in clearing up the foundations of reasoning. But the moral of it is that if we are to make the methods of Science a part of Logic, and to name this department Induction, it is better to discard altogether the questions of General and Particular which are pertinent to Syllogism, and to recognise the new department simply as being concerned with a different kind of inference, inference from facts to what lies beyond them, inference from the observed to the unobserved.

That this is the general aim and proper work of Science is evident from its history. Get at the secrets of Nature by the study of Nature, penetrate to what is unknown and unexperienced by help of what is known and has been experienced, was the cry of the early reformers of Science. Thus only, in Roger Bacon's phrase, could certainty--assured, well grounded, rational belief--be reached. This doctrine, like every other, can be understood only by what it was intended to deny. The way of reaching certainty that Roger Bacon repudiated was argument, discussion, dialectic. This "concludes a question but does not make us feel certain, or acquiesce in the contemplation of truth that is not also found in Experience". Argument is not necessarily useless; the proposition combated is only that by it alone--by discussion that does not go beyond accepted theories or conceptions--rational belief about the unknown cannot be reached. The proposition affirmed is that to this end the conclusions of argument must be tested by experience.

Observation of facts then is a cardinal part of the method of Science.

The facts on which our inferences are based, by which our conclusions are tested, must be accurate. But in thus laying emphasis on the necessity of accurate observation, we must beware of rushing to the opposite extreme, and supposing that observation alone is enough.

Observation, the accurate use of the senses (by which we must understand inner as well as outer sense), is not the whole work of Science. We may stare at facts every minute of our waking day without being a whit the wiser unless we exert our intellects to build upon them or under them. To make our examination fruitful, we must have conceptions, theories, speculations, to bring to the test. The comparison of these with the facts is the inductive verification of them. Science has to exercise its ingenuity both in making hypotheses and in contriving occasions for testing them by observation. These contrived occasions are its artificial experiments, which have come to be called experiments simply by contrast with conclusive observations for which Nature herself furnishes the occasion. The observations of Science are not passive observations. The word experiment simply means trial, and every experiment, natural or artificial, is the trial of a hypothesis. In the language of Leonardo da Vinci, "Theory is the general, Experiments are the soldiers".

Observation and Inference go hand in hand in the work of Science, but with a view to a methodical exposition of its methods, we may divide them broadly into Methods of Observation and Methods of Inference.

There are errors specially incident to Observation, and errors specially incident to Inference. How to observe correctly and how to make correct inferences from our observations are the two objects of our study in Inductive Logic: we study the examples of Science because they have been successful in accomplishing those objects.

That all inference to the unobserved is founded on facts, on the data of experience, need not be postulated. It is enough to say that Inductive Logic is concerned with inference in so far as it is founded on the data of experience. But inasmuch as all the data of experience are not of equal value as bases of inference, it is well to begin with an analysis of them, if we wish to take a comprehensive survey of the various modes of inference and the conditions of their validity.

[Footnote 1: Hamilton's _Reid_, p. 712.]

[Footnote 2: The _Novum Organum_ was never completed. Of the nine heads of special aids to the intellect in the final interpretation he completed only the first, the list of Prerogative Instances.]

[Footnote 3: _Sylva Sylvarum_, Century I, 24.]

[Footnote 4: _Sylva Sylvarum_, Century I, 5.]

CHAPTER I.

THE DATA OF EXPERIENCE AS GROUNDS OF INFERENCE OR RATIONAL BELIEF.

If we examine any of the facts or particulars on which an inference to the unobserved is founded, we shall find that they are not isolated individuals or attributes, separate objects of perception or thought, but relations among things and their qualities, constituents, or ingredients.

Take the "particular" from which Mill's village matron inferred, the fact on which she based her expectation of a cure for her neighbour's child. It is a relation between things. We have the first child's ailment, the administration of the drug, and the recovery, a series of events in sequence. This observed sequence is the fact from which she is said to infer, the datum of experience. She expects this sequence to be repeated in the case of her neighbour's child.

Similarly we shall find that, in all cases where we infer, the facts are complex, are not mere isolated things, but relations among things--using the word thing in its widest sense--relations which we expect to find repeated, or believe to have occurred before, or to be occurring now beyond the range of our observation. These relations, which we may call coincidences or conjunctions, are the data of experience from which we start in our beliefs or inferences about the unexperienced.

The problem of Inductive Logic being to determine when or on what conditions such beliefs are rational, we may begin by distinguishing the data of coincidence or conjunction accordingly. There are certain coincidences that we expect to find repeated beyond the occasions on which we have observed them, and others that we do not expect to find repeated. If it is a sound basis of inference that we are in search of, it is evidently to these first, the coincidences that we are assured of finding again, that we must direct our study. Let us see whether they can be specified.

(1) If there is no causal connexion between A and B, using these as symbols for the members of a coincidence--the objects that are presented together--we do not expect the coincidence to be repeated.

If A and B are connected as cause and effect, we expect the effect to recur in company with the cause. We expect that when the cause reappears in similar circumstances, the effect also will reappear.