Chance, Love, and Logic
International Library of Psychology
Philosophy and Scientific Method
Chance, Love, and Logic
International Library of Psychology
Philosophy and Scientific Method
General Editor: C. K. Ogden, m.a.
VOLUMES ALREADY PUBLISHED
PHILOSOPHICAL STUDIES, by G. E. Moore, Litt.D.
THE MISUSE OF MIND, by Karin Stephen. Prefatory note by Henri Bergson
CONFLICT AND DREAM, by W. H. R. Rivers, F.R.S.
PSYCHOLOGY AND POLITICS, by W. H. R. Rivers, F.R.S.
TRACTATUS LOGICO-PHILOSOPHICUS, by L. Wittgenstein. Introduction by Bertrand Russell
THE MEASUREMENT OF EMOTION, by W. Whately Smith. Introduction by William Brown
PSYCHOLOGICAL TYPES, by C. G. Jung, M.D., LL.D.
SCIENTIFIC THOUGHT, by C. D. Broad, Litt.D.
THE MEANING OF MEANING, by C. K. Ogden and I. A. Richards
CHARACTER AND THE UNCONSCIOUS, by J. H. van der Hoop
IN PREPARATION
SCIENTIFIC METHOD, by A. D. Ritchie
THE ANALYSIS OF MATTER, by Bertrand Russell, F.R.S.
PSYCHOLOGY AND ETHNOLOGY, by W. H. R. Rivers, F.R.S.
INDIVIDUAL PSYCHOLOGY, by Alfred Adler
MATHEMATICS FOR PHILOSOPHERS, by G. H. Hardy, F.R.S.
THE PSYCHOLOGY OF MYTHS, by G. Elliot Smith, F.R.S.
THE PHILOSOPHY OF THE UNCONSCIOUS by E. von Hartmann. Introduction by Prof. G. Elliot Smith
THE THEORY OF MEDICAL DIAGNOSIS, by F. G. Crookshank, M.D., F.R.C.P.
THE ELEMENTS OF PSYCHOTHERAPY, by William Brown, M.D., D.Sc.
EMOTION AND INSANITY, by S. Thalbitzer. Introduction by Prof. H. Höffding
THE LAWS OF FEELING, by F. Paulhan
THE PSYCHOLOGY OF MUSIC, by Edward J. Dent
COLOUR-HARMONY, by James Wood
THE DEVELOPMENT OF CHINESE THOUGHT, by Liang Che Chiao
THE HISTORY OF MATERIALISM, by F. A. Lange
PSYCHE, by E. Rohde
THE PRIMITIVE MIND, by P. Radin, Ph.D.
THE PSYCHOLOGY OF PRIMITIVE PEOPLES, by B. Malinowski, Ph.D., D.Sc.
THE STATISTICAL METHOD IN ECONOMICS, by P. Sargant Florence
THE PRINCIPLES OF CRITICISM, by I. A. Richards
THE PHILOSOPHY OF ‘AS IF’ by H. Vaihinger
CHANCE, LOVE, AND LOGIC
PHILOSOPHICAL ESSAYS
BY THE LATE
CHARLES S. PEIRCE
THE FOUNDER OF PRAGMATISM
EDITED WITH AN INTRODUCTION
BY
MORRIS R. COHEN
WITH A SUPPLEMENTARY ESSAY ON
THE PRAGMATISM OF PEIRCE
BY
JOHN DEWEY
NEW YORK
HARCOURT, BRACE & COMPANY, INC.
LONDON: KEGAN PAUL TRENCH, TRUBNER & CO., LTD.
1923
COPYRIGHT, 1923, BY
HARCOURT, BRACE AND COMPANY, INC.
PRINTED IN THE U.S.A.
PREFACE
In the essays gathered together in this volume we have the most developed and coherent available account of the philosophy of Charles S. Peirce, whom James, Royce, Dewey, and leading thinkers in England, France, Germany and Italy have placed in the forefront of the great seminal minds of recent times. Besides their inherent value as the expression of a highly original and fruitful mind, unusually well trained and informed in the exact sciences, these essays are also important as giving us the sources of a great deal of contemporary American philosophy. Because of this historical importance no omissions or changes have been made in the text beyond the correction of some obvious slips and the recasting of a few expressions in the interest of intelligibility.
In a subject which bristles with suggestions and difficulties the temptation to add notes of explanation or dissent is almost insuperable. But as such notes might easily have doubled the size of this volume I have refrained from all comment on the text except in a few footnotes (indicated, as usual, in brackets). The introduction is intended (and I hope it will) help the reader to concatenate the various lines of thought contained in these essays. I cannot pretend to have adequately indicated their significance. Great minds like those of James and Royce have been nourished by these writings and I am persuaded that they still offer mines of fruitful suggestion. Prof. Dewey’s supplementary essay indicates their value for the fundamental question of metaphysics, viz. the nature of reality.
Grateful acknowledgment is here made to Mrs. Paul Carus and to the Open Court Publishing Co. for permission to reprint the essays of Part II from the Monist. The late Paul Carus was one of the very few who not only gave Peirce an opportunity to publish, but publicly recognized the importance of his writings.
I must also acknowledge my obligation to Professor Dewey for kind permission to reprint his essay on the Pragmatism of Peirce from the Journal of Philosophy, and to the editors of that Journal, Professors Woodbridge and Bush, for permission to reprint some material of my own. Part V of the Bibliography was compiled by Mr. Irving Smith.
Morris R. Cohen
The College of the City of New York.
TABLE OF CONTENTS
[Proem. The Rules of Philosophy 1]
[Part I. Chance and Logic (Illustrations of the Logic of Science.)]
[2. How to Make Our Ideas Clear 32]
[3. The Doctrine of Chances 61]
[4. The Probability of Induction 82]
[6. Deduction, Induction and Hypothesis 131]
[1. The Architecture of Theories 157]
[2. The Doctrine of Necessity Examined 179]
[Supplementary Essay—The Pragmatism of Peirce, by John Dewey 301]
INTRODUCTION
Many and diverse are the minds that form the philosophic community. There are, first and foremost, the great masters, the system builders who rear their stately palaces towering to the moon. These architectonic minds are served by a varied host of followers and auxiliaries. Some provide the furnishings to make these mystic mansions of the mind more commodious, while others are engaged in making their façades more imposing. Some are busy strengthening weak places or building much-needed additions, while many more are engaged in defending these structures against the impetuous army of critics who are ever eager and ready to pounce down upon and destroy all that is new or bears the mortal mark of human imperfection. There are also the philologists, those who are in a more narrow sense scholars, who dig not only for facts or roots, but also for the stones which may serve either for building or as weapons of destruction. Remote from all these, however, are the intellectual rovers who, in their search for new fields, venture into the thick jungle that surrounds the little patch of cultivated science. They are not gregarious creatures, these lonely pioneers; and in their wanderings they often completely lose touch with those who tread the beaten paths. Those that return to the community often speak strangely of strange things; and it is not always that they arouse sufficient faith for others to follow them and change their trails into high roads.
Few nowadays question the great value of these pioneer minds; and it is often claimed that universities are established to facilitate their work, and to prevent it from being lost. But universities, like other well-managed institutions, can find place only for those who work well in harness. The restless, impatient minds, like the socially or conventionally unacceptable, are thus kept out, no matter how fruitful their originality. Charles S. Peirce was certainly one of these restless pioneer souls with the fatal gift of genuine originality. In his early papers, in the Journal of Speculative Philosophy, and later, in the Monist papers reprinted as [Part II] of this volume, we get glimpses of a vast philosophic system on which he was working with an unusual wealth of material and apparatus. To a rich imagination and extraordinary learning he added one of the most essential gifts of successful system builders, the power to coin an apt and striking terminology. But the admitted incompleteness of these preliminary sketches of his philosophic system is not altogether due to the inherent difficulty of the task and to external causes such as neglect and poverty. A certain inner instability or lack of self-mastery is reflected in the outer moral or conventional waywardness which, except for a few years at Johns Hopkins, caused him to be excluded from a university career, and thus deprived him of much needed stimulus to ordinary consistency and intelligibility. As the years advanced, bringing little general interest in, or recognition of, the brilliant logical studies of his early years, Peirce became more and more fragmentary, cryptic, and involved; so that James, the intellectual companion of his youth, later found his lectures on pragmatism, “flashes of brilliant light relieved against Cimmerian darkness”—a statement not to be entirely discounted by the fact that James had no interest in or aptitude for formal logical or mathematical considerations.
Despite these limitations, however, Peirce stands out as one of the great founders of modern scientific logic; and in the realm of general philosophy the development of some of his pregnant ideas has led to the pragmatism and radical empiricism of James, as well as to the mathematical idealism of Royce, and to the anti-nominalism which characterizes the philosophic movement known as Neo-Realism. At any rate, the work of James, Royce, and Russell, as well as that of logicians like Schroeder, brings us of the present generation into a better position to appreciate the significance of Peirce’s work, than were his contemporaries.
I
Peirce was by antecedents, training, and occupation a scientist. He was a son of Benjamin Peirce, the great Harvard mathematician, and his early environment, together with his training in the Lawrence Scientific School, justified his favorite claim that he was brought up in a laboratory. He made important contributions not only in mathematical logic but also in photometric astronomy, geodesy, and psychophysics, as well as in philology. For many years Peirce worked on the problems of geodesy, and his contribution to the subject, his researches on the pendulum, was at once recognized by European investigators in this field. The International Geodetic Congress, to which he was the first American representative, gave unusual attention to his paper, and men like Cellerier and Plantamour acknowledged their obligations to him.[[1]]
This and other scientific work involving fine measurement, with the correlative investigations into the theory of probable error, seem to have been a decisive influence in the development of Peirce’s philosophy of chance. Philosophers inexperienced in actual scientific measurement may naïvely accept as absolute truth such statements as “every particle of matter attracts every other particle directly as the product of their masses and inversely as the square of the distance,” or “when hydrogen and oxygen combine to form water the ratio of their weights is 1 : 8.” But to those who are actually engaged in measuring natural phenomena with instruments of precision, nature shows no such absolute constancy or simplicity. As every laboratory worker knows, no two observers, and no one observer in successive experiments, get absolutely identical results. To the men of the heroic period of science this was no difficulty. They held unquestioningly the Platonic faith that nature was created on simple geometric lines, and all the minute variations were attributable to the fault of the observer or the crudity of his instruments. This heroic faith was, and still is, a most powerful stimulus to scientific research and a protection against the incursions of supernaturalism. But few would defend it to-day in its explicit form, and there is little empirical evidence to show that while the observer and his instruments are always varying, the objects which he measures never deviate in the slightest from the simple law. Doubtless, as one becomes more expert in the manipulation of physical instruments, there is a noticeable diminution of the range of the personal “error,” but no amount of skill and no refinement of our instruments have ever succeeded in eliminating irregular, though small, variations. “Try to verify any law of nature and you will find that the more precise your observations, the more certain they will be to show irregular departure from the law.”[[2]] There is certainly nothing in our empirical information to prevent us from saying that all the so-called constants of nature are merely instances of variation between limits so near each other that their differences may be neglected for certain purposes. Moreover, the approach to constancy is observed only in mass phenomena, when we are dealing with very large numbers of particles; but social statistics also approach constant ratios when the numbers are very large. Hence, without denying discrepancies due solely to errors of observation, Peirce contends that “we must suppose far more minute discrepancies to exist owing to the imperfect cogency of the law itself, to a certain swerving of the facts from any definite formula.”[[3]]
It is usual to associate disbelief in absolute laws of nature with sentimental claims for freedom or theological miracles. It is, therefore, well to insist that Peirce’s attack is entirely in the interests of exact logic and a rational account of the physical universe. As a rigorous logician familiar with the actual procedures by which our knowledge of the various laws of nature is obtained, he could not admit that experience could prove their claim to absoluteness. All the physical laws actually known, like Boyle’s law or the law of gravitation, involve excessive simplification of the phenomenal course of events, and thus a large element of empirical inaccuracy. But a more positive objection against the traditional assumption of absolute or invariable laws of nature, is the fact that such assumption makes the regularities of the universe ultimate, and thus cuts us off from the possibility of ever explaining them or how there comes to be as much regularity in the universe as there is. But in ordinary affairs, the occurrence of any regularity is the very thing to be explained. Moreover, modern statistical mechanics and thermodynamics (theory of gases, entropy, etc.) suggest that the regularity in the universe is a matter of gradual growth; that the whole of physical nature is a growth from a chaos of diversity to a maximum of uniformity or entropy. A leading physicist of the 19th Century, Boltzmann, has suggested that the process of the whole physical universe is like that of a continuous shaking up of a hap-hazard or chance mixture of things, which thus gradually results in a progressively more uniform distribution. Since Duns Scotus, students of logic have known that every real entity has its individual character (its haecceitas or thisness) which cannot be explained or deduced from that which is uniform. Every explanation, for example, of the moon’s path must take particular existences for granted. Such original or underived individuality and diversity is precisely what Peirce means by chance; and from this point of view chance is prior to law.
All that is necessary to visualize this is to suppose that there is an infinitesimal tendency in things to acquire habits, a tendency which is itself an accidental variation grown habitual. We shall then be on the road to explain the evolution and existence of the limited uniformities actually prevailing in the physical world.
A good deal of the foregoing may sound somewhat mythologic. But even if it were so it would have the merit of offering a rational alternative to the mechanical mythology according to which all the atoms in the universe are to-day precisely in the same condition in which they were on the day of creation, a mythology which is forced to regard all the empirical facts of spontaneity and novelty as illusory, or devoid of substantial truth.
The doctrine of the primacy of chance naturally suggests the primacy of mind. Just as law is a chance habit so is matter inert mind. The principal law of mind is that ideas literally spread themselves continuously and become more and more general or inclusive, so that people who form communities of any sort develop general ideas in common. When this continuous reaching-out of feeling becomes nurturing love, such, e.g., which parents have for their offspring or thinkers for their ideas, we have creative evolution.
James and Royce have called attention to the similarity between Peirce’s doctrine of tychistic-agapism (chance and love) and the creative evolution of Bergson. But while both philosophies aim to restore life and growth in their account of the nature of things, Peirce’s approach seems to me to have marked advantages, owing to its being in closer touch with modern physics. Bergson’s procedure is largely based on the contention that mechanics cannot explain certain empirical facts, such as the supposed identity of the vertebrate eye and the eye of the scallop. But the fact here is merely one of a certain resemblance of pattern, which may well be explained by the mechanical principles of convergent evolution. Peirce’s account involves no rejection of the possibility of mechanical explanations. Indeed, by carrying chance into the laws of mechanics he is enabled to elaborate a positive and highly suggestive theory of protoplasm to explain the facts of plasticity and habit.[[4]] Instead of postulating with Spencer and Bergson a continuous growth of diversity, Peirce allows for growth of habits both in diversity and in uniformity. The Spencerian mechanical philosophy reduces all diversity to mere spatial differences. There can be no substantial novelty; only new forms or combinations can arise in time. The creative evolution of Bergson though intended to support the claims of spontaneity is still like the Spencerian in assuming all evolution as proceeding from the simple to the complex. Peirce allows for diversity and specificity as part of the original character or endowment of things, which in the course of time may increase in some respects and diminish in others. Mind acquires the habit both of taking on, and also of laying aside, habits. Evolution may thus lead to homogeneity or uniformity as well as to greater heterogeneity.
Not only has Peirce a greater regard than even Bergson for the actual diversity and spontaneity of things, but he is in a much better position than any other modern philosopher to explain the order and coherence of the world. This he effects by uniting the medieval regard for the reality of universals with the modern scientific use of the concept of continuity. The unfortunate war between the pioneers of modern science and the adherents of the scholastic doctrine of substantial forms, has been one of the great misfortunes of human thought, in that it made absolute atomism and nominalism the professed creed of physical science. Now, extreme nominalism, the insistence on the reality of the particular, leaves no room for the genuine reality of law. It leaves, as Hume had the courage to admit, nothing whereby the present can determine the future; so that anything is as likely to happen as not. From such a chaotic world, the procedure of modern natural and mathematical science has saved us by the persistent use of the principle of continuity; and no one has indicated this more clearly than Peirce who was uniquely qualified to do so by being a close student both of Duns Scotus and of modern scientific methods.
It is instructive in this respect to contrast the views of Peirce and James. James, who so generously indicated his indebtedness to Peirce for his pragmatism, was also largely indebted to Peirce for his doctrine of radical empiricism.[[5]] The latter doctrine seeks to rescue the continuity and fluidity of experience from the traditional British empiricism or nominalism, which had resolved everything into a number of mutually exclusive mental states. It is curious, however, that while in his psychology James made extensive use of the principle of continuity, he could not free himself from British nominalism in his philosophy—witness the extreme individualism of his social philosophy or the equally extreme anthropomorphism of his religion. Certain of Peirce’s suggestions as to the use of continuity in social philosophy have been developed by Royce in his theory of social consciousness and the nature of the community;[[6]] but much remains to be worked out and we can but repeat Peirce’s own hope: “May some future student go over this ground again and have the leisure to give his results to the world.”
It is well to note, however, that after writing the papers included in this volume Peirce continued to be occupied with the issues here raised. This he most significantly indicated in the articles on logical topics contributed to Baldwin’s Dictionary of Philosophy.[[7]]
In these articles it is naturally the logical bearing of the principles of tychism (chance), synechism (continuity), and agapism (love) that is stressed. To use the Kantian terminology, almost native to Peirce, the regulative rather than the constitutive aspect of these principles is emphasized. Thus the doctrine of chance is not only what it was for James’ radical empiricism, a release from the blind necessity of a “block universe,” but also a method of keeping open a possible explanation of the genesis of the laws of nature and an interpretation of them in accordance with the theorems of probability, so fruitful in physical science as well as in practical life. So the doctrine of love is not only a cosmologic one, showing how chance feeling generates order or rational diversity through the habit of generality or continuity, but it also gives us the meaning of truth in social terms, in showing that the test as to whether any proposition is true postulates an indefinite number of co-operating investigators. On its logical side the doctrine of love (agapism) also recognized the important fact that general ideas have a certain attraction which makes us divine their nature even though we cannot clearly determine their precise meaning before developing their possible consequences.
Of the doctrine of continuity we are told expressly[[8]] that “synechism is not an ultimate absolute metaphysical doctrine. It is a regulative principle of logic,” seeking the thread of identity in diverse cases and avoiding hypotheses that this or that is ultimate and, therefore, inexplicable. (Examples of such hypotheses are: the existence of absolutely accurate or uniform laws of nature, the eternity and absolute likeness of all atoms, etc.) To be sure, the synechist cannot deny that there is an element of the inexplicable or ultimate, since it is directly forced upon him. But he cannot regard it as a source of explanation. The assumption of an inexplicability is a barrier on the road to science. “The form under which alone anything can be understood is the form of generality which is the same thing as continuity.”[[9]] This insistence on the generality of intelligible form is perfectly consistent with due emphases on the reality of the individual, which to a Scotist realist connotes an element of will or will-resistence, but in logical procedure means that the test of the truth or falsity of any proposition refers us to particular perceptions.[[10]] But as no multitude of individuals can exhaust the meaning of a continuum, which includes also organizing relations of order, the full meaning of a concept cannot be in any individual reaction, but is rather to be sought in the manner in which all such reactions contribute to the development of the concrete reasonableness of the whole evolutionary process. In scientific procedure this means that integrity of belief in general is more important than, because it is the condition of, particular true beliefs.
II
This insistence on the continuity so effectually used as a heuristic principle in natural and mathematical science, distinguishes the pragmatism of Peirce from that of his follower James. Prof. Dewey has developed this point authoritatively in the supplementary essay; but in view of the general ignorance as to the sources of pragmatism which prevails in this incurious age, some remarks on the actual historical origin of pragmatism may be in order.
There can be little doubt that Peirce was led to the formulation of the principle of pragmatism through the influence of Chauncey Wright.[[11]] Wright who had first hand acquaintance with creative scientific work in mathematics, physics, and botany was led by the study of Mill and Bain to reflect on the characteristics of scientific method. This reflection led him to draw a distinction between the use of popular scientific material, by men like Spencer, to construct a myth or picture of the world, and the scientific use of laws by men like Newton as means for extending our knowledge of phenomena. Gravitation as a general fact had interested metaphysicians long before Newton. What made Newton’s contribution scientific was the formulation of a mathematical law which has enabled us to deduce all the then known facts of the solar system and to anticipate or predict many more facts the existence of which would not otherwise be even suspected, e.g., the existence of the planet Neptune. Wright insists, therefore, that the principles of modern mathematical and physical science are the means through which nature is discovered, that scientific laws are the finders rather than merely the summaries of factual truths. This conception of the experimental scientist as translating general propositions into prescriptions for attaining new experimental truths, is the starting point of Peirce’s pragmatism. The latter is embodied in the principle that the meaning of a concept is to be found in “all the conceivable experimental phenomena which the affirmation or denial of the concept could imply.”[[12]]
In the earlier statement of the pragmatic maxim,[[13]] Peirce emphasized the consequences for conduct that follow from the acceptance or rejection of an idea; but the stoical maxim that the end of man is action did not appeal to him as much at sixty as it did at thirty.[[14]] Naturally also Peirce could not follow the development of pragmatism by Wm. James who, like almost all modern psychologists, was a thorough nominalist and always emphasized particular sensible experience.[[15]] It seemed to Peirce that such emphasis on particular experiences endangered the principle of continuity which in the hands of men like Weierstrass had reformed modern mathematics. For this reason he began to call his own doctrine pragmaticism, a sufficiently unattractive name, he thought, to save it from kidnappers and from popularity. He never, however, abandoned the principle of pragmatism, that the meaning of an idea is clarified (because constituted) by its conceivable experimental consequences. Indeed, if we want to clarify the meaning of the idea of pragmatism, let us apply the pragmatic test to it. What will be the effect of accepting it? Obviously it will be to develop certain general ideas or habits of looking at things.
Peirce’s pragmatism has, therefore, a decidedly intellectual cast. The meaning of an idea or proposition is found not by an intuition of it but by working out its implications. It admits that thought does not constitute reality. Categories can have no concrete being without action or immediate feeling. But thought is none the less an essential ingredient of reality; thought is “the melody running through the succession of our sensations.” Pragmatism, according to Peirce, seeks to define the rational purport, not the sensuous quality. It is interested not in the effect of our practical occupations or desires on our ideas, but in the function of ideas as guides of action. Whether a man is to pay damages in a certain lawsuit may depend, in fact, on a term in the Aristotelian logic such as proximate cause.
It is of interest to observe that though Peirce is an ardent admirer of Darwin’s method, his scientific caution makes him refuse to apply the analogy of biologic natural selection to the realm of ideas, in the wholesale and uncritical manner that has lately become fashionable. Natural selection may well favor the triumph of views which directly influence biologic survival. But the pleasure of entertaining congenial illusions may overbalance the inconvenience resulting from their deceptive character. Thus rhetorical appeals may long prevail over scientific evidence.
III
Peirce preferred to call himself a logician, and his contributions to logic have so far proved his most generally recognized achievement. For a right perspective of these contributions we may well begin with the observation that though few branches of philosophy have been cultivated as continuously as logic, Kant was able to affirm that the science of logic had made no substantial progress since the time of Aristotle. The reason for this is that Aristotle’s logic, the logic of classes, was based on his own scientific procedure as a zoologist, and is still in essence a valid method so far as classification is part of all rational procedure. But when we come to describe the mathematical method of physical science, we cannot cast it into the Aristotelian form without involving ourselves in such complicated artificialities as to reduce almost to nil the value of Aristotle’s logic as an organon. Aristotle’s logic enables us to make a single inference from two premises. But the vast multitude of theorems that modern mathematics has derived from a few premises as to the nature of number, shows the need of formulating a logic or theory of inference that shall correspond to the modern, more complicated, practice as Aristotle’s logic did to simple classificatory zoology. To do this effectively would require the highest constructive logical genius, together with an intimate knowledge of the methods of the great variety of modern sciences. This is in the nature of the case a very rare combination, since great investigators are not as critical in examining their own procedure as they are in examining the subject matter which is their primary scientific interest. Hence, when great investigators like Poincaré come to describe their own work, they fall back on the uncritical assumptions of the traditional logic which they learned in their school days. Moreover, “For the last three centuries thought has been conducted in laboratories, in the field, or otherwise in the face of the facts, while chairs of logic have been filled by men who breathe the air of the seminary.”[[16]] The great Leibnitz had the qualifications, but here, as elsewhere, his worldly occupations left him no opportunity except for very fragmentary contributions. It was not until the middle of the 19th century that two mathematicians, Boole and DeMorgan, laid the foundations for a more generalized logic. Boole developed a general logical algorithm or calculus, while DeMorgan called attention to non-syllogistic inference and especially to the importance of the logic of relations. Peirce’s great achievement is to have recognized the possibilities of both and to have generalized and developed them into a general theory of scientific inference. The extent and thoroughness of his achievement has been obscured by his fragmentary way of writing and by a rather unwieldy symbolism. Still, modern mathematical logic, such as that of Russell’s Principles of Mathematics, is but a development of Peirce’s logic of relatives.
This phase of Peirce’s work is highly technical and an account of it is out of place here. Such an account will be found in Lewis’ Survey of Symbolic Logic.[[17]] I refer to it here only to remind the reader that the Illustrations of the Logic of the Sciences ([Part I] of this volume) have a background of patient detailed work which is still being developed to-day.
Symbolic logic has been held in rather low esteem by the followers of the old classical methods in philosophy. Their stated objection to it has been mainly that it is concerned with the minutiae of an artificial language and is of no value as a guide to the interpretation of reality. Now it should be readily admitted that preoccupation with symbolic logic is rather apt to retard the irresponsible flight of philosophic fancy. Yet this is by no means always an evil. By insisting on an accuracy that is painful to those impatient to obtain sweeping and comforting, though hasty, conclusions, symbolic logic is well calculated to remove the great scandal of traditional philosophy—the claim of absolutely certain results in fields where there is the greatest conflict of opinion. This scandalous situation arises in part from the fact that in popular exposition we do not have to make our premises or assumptions explicit; hence all sorts of dubious prejudices are implicitly appealed to as absolutely necessary principles. Also, by the use of popular terms which have a variety of meanings, one easily slides from one meaning to another, so that the most improbable conclusions are thus derived from seeming truisms. By making assumptions and rules explicit, and by using technical terms that do not drag wide penumbras of meaning with them, the method of symbolic logic may cruelly reduce the sweeping pretensions of philosophy. But there is no reason for supposing that pretentiousness rather than humility is the way to philosophic salvation. Man is bound to speculate about the universe beyond the range of his knowledge, but he is not bound to indulge the vanity of setting up such speculations as absolutely certain dogmas.
There is, however, no reason for denying that greater rigor and accuracy of exposition can really help us to discern new truth. Modern mathematics since Gauss and Weierstrass has actually been led to greater fruitfulness by increased rigor which makes such procedure as the old proofs of Taylor’s theorem no longer possible. The substitution of rigorous analytic procedures for the old Euclidean proofs based on intuition, has opened up vast fields of geometry. Nor has this been without any effect on philosophy. Where formerly concepts like infinity and continuity were objects of gaping awe or the recurrent occasions for intellectual violence,[[18]] we are now beginning to use them, thanks to Peirce and Royce, in accurate and definable senses. Consider, for instance, the amount of a priori nonsense which Peirce eliminates by pointing out that the application of the concept of continuity to a span of consciousness removes the necessity for assuming a first or last moment; so likewise the range of vision on a large unobstructed ground has no line between the visible and the invisible. These considerations will be found utterly destructive of the force of the old arguments (fundamental to Kant and others) as to the necessary infinity of time and space. Similar enlightenment is soon likely to result from the more careful use of terms like relative and absolute, which are bones of contention in philosophy but Ariadne threads of exploration in theoretical physics, because of the definite symbolism of mathematics. Other important truths made clear by symbolic logic is the hypothetical character of universal propositions and the consequent insight that no particulars can be deduced from universals alone, since no number of hypotheses can without given data establish an existing fact.
There is, however, an even more positive direction in which symbolic logic serves the interest of philosophy, and that is in throwing light on the nature of symbols and on the relation of meaning. Philosophers have light-heartedly dismissed questions as to the nature of significant signs as ‘merely’ (most fatal word!) a matter of language. But Peirce in the paper on Man’s Glassy [Shakespearian for Mirror-Like] Essence, endeavors to exhibit man’s whole nature as symbolic.[[19]] This is closely connected with his logical doctrine which regards signs or symbols as one of the fundamental categories or aspects of the universe (Thoughts and things are the other two). Independently of Peirce but in line with his thought another great and neglected thinker, Santayana, has shown that the whole life of man that is bound up with the institutions of civilization, is concerned with symbols.
It is not altogether accidental that, since Boole and DeMorgan, those who have occupied themselves with symbolic logic have felt called upon to deal with the problem of probability. The reason is indicated by Peirce when he formulates the problem of probable inference in such a way as to make the old classic logic of absolutely true or false conclusions, a limiting case (i.e., of values 1 and 0) of the logic of probable inference whose values range all the way between these two limits. This technical device is itself the result of applying the principle of continuity to throw two hitherto distinct types of reasoning into the same class. The result is philosophically significant.
Where the classical logic spoke of major and minor premises without establishing any really important difference between the two, Peirce draws a distinction between the premises and the guiding principle of our argument. All reasoning is from some concrete situation to another. The propositions which represent the first are the premises in the strict sense of the word. But the feeling that certain conclusions follow from these premises is conditioned by an implicit or explicit belief in some guiding principle which connects the premises and the conclusions. When such a leading principle results in true conclusions in all cases of true premises, we have logical deduction of the orthodox type. If, however, such a principle brings about a true conclusion only in a certain proportion of cases, then we have probability.
This reduction of probability to the relative frequency of true propositions in a class of propositions, was suggested to Peirce by Venn’s Logic of Chance. Peirce uses it to establish some truths of greatest importance to logic and philosophy.
He eliminates the difficulties of the old conceptualist view, which made probability a measure of our ignorance and yet had to admit that almost all fruitfulness of our practical and scientific reasoning depended on the theorems of probability. How could we safely predict phenomena by measuring our ignorance?
Probability being reduced to a matter of the relative frequency of a class in a larger class or genus, it becomes, strictly speaking, inapplicable to single cases by themselves. A single penny will fall head or it will fall tail every time; to-morrow it will rain, or it will not rain at all. The probability of 1/2 or any other fraction means nothing in the single case. It is only because we feel the single event as representative of a class, as something which repeats itself, that we speak elliptically of the probability of a single event. Hence follows the important corollary that reasoning with respect to the probability of this or that arrangement of the universe would be valid only if universes were as plentiful as blackberries.
To be useful at all, theories must be simpler than the complex facts which they seek to explain. Hence, it is often convenient to employ a principle of certainty where the facts justify only a principle of some degree of probability. In such cases we must be cautious in accepting any extreme consequence of these principles, and also be on guard against apparent refutations based on such extreme consequences.
Finally I should like to emphasize the value of Peirce’s theory of inference for a philosophy of civilization. To the old argument that logic is of no importance because people learn to reason, as to walk, by instinct and habit and not by scientific instruction, Peirce admits[[20]] that “all human knowledge up to the highest flights of science is but the development of our inborn animal instincts.” But though logical rules are first felt implicitly, bringing them into explicit consciousness helps the process of analysis and thus makes possible the recognition of old principles in novel situations. This increases our range of adaptability to such an extent as to justify a general distinction between the slave of routine or habit and the freeman who can anticipate and control nature through knowledge of principles. Peirce’s analysis of the method of science as a method of attaining stability of beliefs by free inquiry inviting all possible doubt, in contrast with the methods of iteration (“will to believe”) and social authority, is one of the best introductions to a theory of liberal or Hellenic civilization, as opposed to those of despotic societies. Authority has its roots in the force of habit, but it cannot prevent new and unorthodox ideas from arising; and in the effort to defend authoritative social views men are apt to be far more ruthless than in defending their own personal convictions.
IV
Not only the pragmatism and the radical empiricism of James, but the idealism of Royce and the more recent movement of neo-realism are largely indebted to Peirce.
It may seem strange that the same thinker should be claimed as foster-father of both recent idealism and realism, and some may take it as another sign of his lack of consistency. But this seeming strangeness is really due to the looseness with which the antithesis between realism and idealism has generally been put. If by idealism we denote the nominalistic doctrine of Berkeley, then Peirce is clearly not an idealist; and his work in logic as a study of types of order (in which Royce followed him) is fundamental for a logical realism. But if idealism means the old Platonic doctrine that “ideas,” genera, or forms are not merely mental but the real conditions of existence, we need not wonder that Peirce was both idealist and realist.
Royce’s indebtedness to Peirce is principally in the use of modern mathematical material, such as the recent development of the concepts of infinity and continuity, to throw light on fundamental questions of philosophy, such as relation of the individual to God or the Universe. At the end of the nineteenth century mathematics had almost disappeared from the repertory of philosophy (cf. Külpe’s Introduction to Philosophy), and Peirce’s essay on the Law of Mind opened a new way which Royce followed in his World and the Individual, to the great surprise of his idealistic brethren. In his Problem of Christianity Royce has also indicated his indebtedness to Peirce for his doctrine of social consciousness, the mind of the community, and the process of interpretation. It may be that a great deal of the similarity between the thoughts of these two men is due to common sources, such as the works of Kant and Schelling; but it is well to note that not only in his later writings but also in his lectures and seminars Royce continually referred to Peirce’s views.
The ground for the neo-realist movement in American philosophy was largely prepared by the mathematical work of Russell and by the utilization of mathematics to which Royce was led by Peirce. The logic of Mr. Russell is based, as he himself has pointed out, on a combination of the work of Peirce and Peano. In this combination the notation of Peano has proved of greater technical fluency, but all of Peano’s results can also be obtained by Peirce’s method as developed by Schroeder and Mrs. Ladd-Franklin. But philosophically Peirce’s influence is far greater in insisting that logic is not a branch of psychology, that it is not concerned with merely mental processes, but with objective relations. To the view that the laws of logic represent “the necessities of thought,” that propositions are true because “we can not help thinking so,” he answers: “Exact logic will say that C’s following logically from A is a state of things which no impotence of thought alone can bring about.”[[21]] “The question of validity is purely one of fact and not of thinking.... It is not in the least the question whether, when the premises are accepted by the mind, we feel an impulse to accept the conclusion also. The true conclusion would remain true if we had no impulse to accept it, and the false one would remain false though we could not resist the tendency to believe in it.”[[22]]
Since the days of Locke modern philosophy has been almost entirely dominated by the assumption that one must study the process of knowing before one can find out the nature of things known; in other words, that psychology is the central philosophic science. The result of this has been an almost complete identification of philosophy with mental science. Nor did the influence of biologic studies of the middle of the nineteenth century shake the belief in that banal dictum of philosophic mediocrity: “The proper study of mankind is man.” The recent renaissance of logical studies, and the remarkable progress of physics in our own day bid fair to remind us that while the Lockian way has brought some gains to philosophy, the more ancient way of philosophy is by no means exhausted of promise. Man cannot lose his interest in the great cosmic play. Those who have faith in the ancient and fruitful approach to philosophy through the doors of mathematics and physics will find the writings of Charles S. Peirce full of suggestions. That such an approach can also throw light on the vexed problem of knowledge needs no assurance to those acquainted with Plato and Aristotle. But I may conclude by referring to Peirce’s doctrine of ideal as opposed to sensible experiment,[[23]] and to his treatment of the question how it is that in spite of an infinity of possible hypotheses, mankind has managed to make so many successful inductions.[[24]] And for the bearing of mathematical studies on the wisdom of life, the following is certainly worth serious reflection: “All human affairs rest upon probabilities. If man were immortal [on earth] he could be perfectly sure of seeing the day when everything in which he had trusted should betray his trust. He would break down, at last, as every great fortune, as every dynasty, as every civilization does. In place of this we have death.” The recognition that the death of the individual does not destroy the logical meaning of his utterances, that this meaning involves the ideal of an unlimited community, carries us into the heart of pure religion.
PROEM
THE RULES OF PHILOSOPHY[[25]]
Descartes is the father of modern philosophy, and the spirit of Cartesianism—that which principally distinguishes it from the scholasticism which it displaced—may be compendiously stated as follows:
1. It teaches that philosophy must begin with universal doubt; whereas scholasticism had never questioned fundamentals.
2. It teaches that the ultimate test of certainty is to be found in the individual consciousness; whereas scholasticism had rested on the testimony of sages and of the Catholic Church.
3. The multiform argumentation of the middle ages is replaced by a single thread of inference depending often upon inconspicuous premises.
4. Scholasticism had its mysteries of faith, but undertook to explain all created things. But there are many facts which Cartesianism not only does not explain but renders absolutely inexplicable, unless to say that “God makes them so” is to be regarded as an explanation.
In some, or all of these respects, most modern philosophers have been, in effect, Cartesians. Now without wishing to return to scholasticism, it seems to me that modern science and modern logic require us to stand upon a very different platform from this.
1. We cannot begin with complete doubt. We must begin with all the prejudices which we actually have when we enter upon the study of philosophy. These prejudices are not to be dispelled by a maxim, for they are things which it does not occur to us can be questioned. Hence this initial skepticism will be a mere self-deception, and not real doubt; and no one who follows the Cartesian method will ever be satisfied until he has formally recovered all those beliefs which in form he has given up. It is, therefore, as useless a preliminary as going to the North Pole would be in order to get to Constantinople by coming down regularly upon a meridian. A person may, it is true, in the course of his studies, find reason to doubt what he began by believing; but in that case he doubts because he has a positive reason for it, and not on account of the Cartesian maxim. Let us not pretend to doubt in philosophy what we do not doubt in our hearts.
2. The same formalism appears in the Cartesian criterion, which amounts to this: “Whatever I am clearly convinced of, is true.” If I were really convinced, I should have done with reasoning and should require no test of certainty. But then to make single individuals absolute judges of truth is most pernicious. The result is that metaphysics has reached a pitch of certainty far beyond that of the physical sciences;—only they can agree upon nothing else. In sciences in which men come to agreement, when a theory has been broached it is considered to be on probation until this agreement is reached. After it is reached, the question of certainty becomes an idle one, because there is no one left who doubts it. We individually cannot reasonably hope to attain the ultimate philosophy which we pursue; we can only seek it, therefore, for the community of philosophers. Hence, if disciplined and candid minds carefully examine a theory and refuse to accept it, this ought to create doubts in the mind of the author of the theory himself.
3. Philosophy ought to imitate the successful sciences in its methods, so far as to proceed only from tangible premises which can be subjected to careful scrutiny, and to trust rather to the multitude and variety of its arguments than to the conclusiveness of any one. Its reasoning should not form a chain which is no stronger than its weakest link, but a cable whose fibers may be ever so slender, provided they are sufficiently numerous and intimately connected.
4. Every unidealistic philosophy supposes some absolutely inexplicable, unanalyzable ultimate; in short, something resulting from mediation itself not susceptible of mediation. Now that anything is thus inexplicable, can only be known by reasoning from signs. But the only justification of an inference from signs is that the conclusion explains the fact. To suppose the fact absolutely inexplicable, is not to explain it, and hence this supposition is never allowable.
PART I
CHANCE AND LOGIC
(ILLUSTRATIONS OF THE LOGIC OF SCIENCE)
CHANCE AND LOGIC
FIRST PAPER
THE FIXATION OF BELIEF[[26]]
I
Few persons care to study logic, because everybody conceives himself to be proficient enough in the art of reasoning already. But I observe that this satisfaction is limited to one’s own ratiocination, and does not extend to that of other men.
We come to the full possession of our power of drawing inferences the last of all our faculties, for it is not so much a natural gift as a long and difficult art. The history of its practice would make a grand subject for a book. The medieval schoolman, following the Romans, made logic the earliest of a boy’s studies after grammar, as being very easy. So it was as they understood it. Its fundamental principle, according to them, was, that all knowledge rests on either authority or reason; but that whatever is deduced by reason depends ultimately on a premise derived from authority. Accordingly, as soon as a boy was perfect in the syllogistic procedure, his intellectual kit of tools was held to be complete.
To Roger Bacon, that remarkable mind who in the middle of the thirteenth century was almost a scientific man, the schoolmen’s conception of reasoning appeared only an obstacle to truth. He saw that experience alone teaches anything—a proposition which to us seems easy to understand, because a distinct conception of experience has been handed down to us from former generations; which to him also seemed perfectly clear, because its difficulties had not yet unfolded themselves. Of all kinds of experience, the best, he thought, was interior illumination, which teaches many things about Nature which the external senses could never discover, such as the transubstantiation of bread.
Four centuries later, the more celebrated Bacon, in the first book of his “Novum Organum,” gave his clear account of experience as something which must be open to verification and reëxamination. But, superior as Lord Bacon’s conception is to earlier notions, a modern reader who is not in awe of his grandiloquence is chiefly struck by the inadequacy of his view of scientific procedure. That we have only to make some crude experiments, to draw up briefs of the results in certain blank forms, to go through these by rule, checking off everything disproved and setting down the alternatives, and that thus in a few years physical science would be finished up—what an idea! “He wrote on science like a Lord Chancellor,”[[27]] indeed.
The early scientists, Copernicus, Tycho Brahe, Kepler, Galileo and Gilbert, had methods more like those of their modern brethren. Kepler undertook to draw a curve through the places of Mars;[[28]] and his greatest service to science was in impressing on men’s minds that this was the thing to be done if they wished to improve astronomy; that they were not to content themselves with inquiring whether one system of epicycles was better than another but that they were to sit down by the figures and find out what the curve, in truth, was. He accomplished this by his incomparable energy and courage, blundering along in the most inconceivable way (to us), from one irrational hypothesis to another, until, after trying twenty-two of these, he fell, by the mere exhaustion of his invention, upon the orbit which a mind well furnished with the weapons of modern logic would have tried almost at the outset.[[29]]
In the same way, every work of science great enough to be remembered for a few generations affords some exemplification of the defective state of the art of reasoning of the time when it was written; and each chief step in science has been a lesson in logic. It was so when Lavoisier and his contemporaries took up the study of Chemistry. The old chemist’s maxim had been, “Lege, lege, lege, labora, ora, et relege.” Lavoisier’s method was not to read and pray, not to dream that some long and complicated chemical process would have a certain effect, to put it into practice with dull patience, after its inevitable failure to dream that with some modification it would have another result, and to end by publishing the last dream as a fact: his way was to carry his mind into his laboratory, and to make of his alembics and cucurbits instruments of thought, giving a new conception of reasoning as something which was to be done with one’s eyes open, by manipulating real things instead of words and fancies.
The Darwinian controversy is, in large part, a question of logic. Mr. Darwin proposed to apply the statistical method to biology. The same thing has been done in a widely different branch of science, the theory of gases. Though unable to say what the movement of any particular molecule of gas would be on a certain hypothesis regarding the constitution of this class of bodies, Clausius and Maxwell were yet able, by the application of the doctrine of probabilities, to predict that in the long run such and such a proportion of the molecules would, under given circumstances, acquire such and such velocities; that there would take place, every second, such and such a number of collisions, etc.; and from these propositions they were able to deduce certain properties of gases, especially in regard to their heat-relations. In like manner, Darwin, while unable to say what the operation of variation and natural selection in every individual case will be, demonstrates that in the long run they will adapt animals to their circumstances. Whether or not existing animal forms are due to such action, or what position the theory ought to take, forms the subject of a discussion in which questions of fact and questions of logic are curiously interlaced.
II
The object of reasoning is to find out, from the consideration of what we already know, something else which we do not know. Consequently, reasoning is good if it be such as to give a true conclusion from true premises, and not otherwise. Thus, the question of validity is purely one of fact and not of thinking. A being the premises and B being the conclusion, the question is, whether these facts are really so related that if A is B is. If so, the inference is valid; if not, not. It is not in the least the question whether, when the premises are accepted by the mind, we feel an impulse to accept the conclusion also. It is true that we do generally reason correctly by nature. But that is an accident; the true conclusion would remain true if we had no impulse to accept it; and the false one would remain false, though we could not resist the tendency to believe in it.
We are, doubtless, in the main logical animals, but we are not perfectly so. Most of us, for example, are naturally more sanguine and hopeful than logic would justify. We seem to be so constituted that in the absence of any facts to go upon we are happy and self-satisfied; so that the effect of experience is continually to counteract our hopes and aspirations. Yet a lifetime of the application of this corrective does not usually eradicate our sanguine disposition. Where hope is unchecked by any experience, it is likely that our optimism is extravagant. Logicality in regard to practical matters is the most useful quality an animal can possess, and might, therefore, result from the action of natural selection; but outside of these it is probably of more advantage to the animal to have his mind filled with pleasing and encouraging visions, independently of their truth; and thus, upon unpractical subjects, natural selection might occasion a fallacious tendency of thought.
That which determines us, from given premises, to draw one inference rather than another, is some habit of mind, whether it be constitutional or acquired. The habit is good or otherwise, according as it produces true conclusions from true premises or not; and an inference is regarded as valid or not, without reference to the truth or falsity of its conclusion specially, but according as the habit which determines it is such as to produce true conclusions in general or not. The particular habit of mind which governs this or that inference may be formulated in a proposition whose truth depends on the validity of the inferences which the habit determines; and such a formula is called a guiding principle of inference. Suppose, for example, that we observe that a rotating disk of copper quickly comes to rest when placed between the poles of a magnet, and we infer that this will happen with every disk of copper. The guiding principle is, that what is true of one piece of copper is true of another. Such a guiding principle with regard to copper would be much safer than with regard to many other substances—brass, for example.
A book might be written to signalize all the most important of these guiding principles of reasoning. It would probably be, we must confess, of no service to a person whose thought is directed wholly to practical subjects, and whose activity moves along thoroughly beaten paths. The problems which present themselves to such a mind are matters of routine which he has learned once for all to handle in learning his business. But let a man venture into an unfamiliar field, or where his results are not continually checked by experience, and all history shows that the most masculine intellect will ofttimes lose his orientation and waste his efforts in directions which bring him no nearer to his goal, or even carry him entirely astray. He is like a ship on the open sea, with no one on board who understands the rules of navigation. And in such a case some general study of the guiding principles of reasoning would be sure to be found useful.
The subject could hardly be treated, however, without being first limited; since almost any fact may serve as a guiding principle. But it so happens that there exists a division among facts, such that in one class are all those which are absolutely essential as guiding principles, while in the other are all those which have any other interest as objects of research. This division is between those which are necessarily taken for granted in asking whether a certain conclusion follows from certain premises, and those which are not implied in that question. A moment’s thought will show that a variety of facts are already assumed when the logical question is first asked. It is implied, for instance, that there are such states of mind as doubt and belief—that a passage from one to the other is possible, the object of thought remaining the same, and that this transition is subject to some rules which all minds are alike bound by. As these are facts which we must already know before we can have any clear conception of reasoning at all, it cannot be supposed to be any longer of much interest to inquire into their truth or falsity. On the other hand, it is easy to believe that those rules of reasoning which are deduced from the very idea of the process are the ones which are the most essential; and, indeed, that so long as it conforms to these it will, at least, not lead to false conclusions from true premises. In point of fact, the importance of what may be deduced from the assumptions involved in the logical question turns out to be greater than might be supposed, and this for reasons which it is difficult to exhibit at the outset. The only one which I shall here mention is, that conceptions which are really products of logical reflections, without being readily seen to be so, mingle with our ordinary thoughts, and are frequently the causes of great confusion. This is the case, for example, with the conception of quality. A quality as such is never an object of observation. We can see that a thing is blue or green, but the quality of being blue and the quality of being green are not things which we see; they are products of logical reflections. The truth is, that common-sense, or thought as it first emerges above the level of the narrowly practical, is deeply imbued with that bad logical quality to which the epithet metaphysical is commonly applied; and nothing can clear it up but a severe course of logic.
III
We generally know when we wish to ask a question and when we wish to pronounce a judgment, for there is a dissimilarity between the sensation of doubting and that of believing.
But this is not all which distinguishes doubt from belief. There is a practical difference. Our beliefs guide our desires and shape our actions. The Assassins, or followers of the Old Man of the Mountain, used to rush into death at his least command, because they believed that obedience to him would insure everlasting felicity. Had they doubted this, they would not have acted as they did. So it is with every belief, according to its degree. The feeling of believing is a more or less sure indication of there being established in our nature some habit which will determine our actions. Doubt never has such an effect.
Nor must we overlook a third point of difference. Doubt is an uneasy and dissatisfied state from which we struggle to free ourselves and pass into the state of belief; while the latter is a calm and satisfactory state which we do not wish to avoid, or to change to a belief in anything else.[[30]] On the contrary, we cling tenaciously, not merely to believing, but to believing just what we do believe.
Thus, both doubt and belief have positive effects upon us, though very different ones. Belief does not make us act at once, but puts us into such a condition that we shall behave in a certain way, when the occasion arises. Doubt has not the least effect of this sort, but stimulates us to action until it is destroyed. This reminds us of the irritation of a nerve and the reflex action produced thereby; while for the analogue of belief, in the nervous system, we must look to what are called nervous associations—for example, to that habit of the nerves in consequence of which the smell of a peach will make the mouth water.
IV
The irritation of doubt causes a struggle to attain a state of belief. I shall term this struggle inquiry, though it must be admitted that this is sometimes not a very apt designation.
The irritation of doubt is the only immediate motive for the struggle to attain belief. It is certainly best for us that our beliefs should be such as may truly guide our actions so as to satisfy our desires; and this reflection will make us reject any belief which does not seem to have been so formed as to insure this result. But it will only do so by creating a doubt in the place of that belief. With the doubt, therefore, the struggle begins, and with the cessation of doubt it ends. Hence, the sole object of inquiry is the settlement of opinion. We may fancy that this is not enough for us, and that we seek not merely an opinion, but a true opinion. But put this fancy to the test, and it proves groundless; for as soon as a firm belief is reached we are entirely satisfied, whether the belief be false or true. And it is clear that nothing out of the sphere of our knowledge can be our object, for nothing which does not affect the mind can be a motive for a mental effort. The most that can be maintained is, that we seek for a belief that we shall think to be true. But we think each one of our beliefs to be true, and, indeed, it is mere tautology to say so.
That the settlement of opinion is the sole end of inquiry is a very important proposition. It sweeps away, at once, various vague and erroneous conceptions of proof. A few of these may be noticed here.
1. Some philosophers have imagined that to start an inquiry it was only necessary to utter or question or set it down on paper, and have even recommended us to begin our studies with questioning everything! But the mere putting of a proposition into the interrogative form does not stimulate the mind to any struggle after belief. There must be a real and living doubt, and without all this discussion is idle.
2. It is a very common idea that a demonstration must rest on some ultimate and absolutely indubitable propositions. These, according to one school, are first principles of a general nature; according to another, are first sensations. But, in point of fact, an inquiry, to have that completely satisfactory result called demonstration, has only to start with propositions perfectly free from all actual doubt. If the premises are not in fact doubted at all, they cannot be more satisfactory than they are.
3. Some people seem to love to argue a point after all the world is fully convinced of it. But no further advance can be made. When doubt ceases, mental action on the subject comes to an end; and, if it did go on, it would be without a purpose.
V
If the settlement of opinion is the sole object of inquiry, and if belief is of the nature of a habit, why should we not attain the desired end, by taking any answer to a question, which we may fancy, and constantly reiterating it to ourselves, dwelling on all which may conduce to that belief, and learning to turn with contempt and hatred from anything which might disturb it? This simple and direct method is really pursued by many men. I remember once being entreated not to read a certain newspaper lest it might change my opinion upon free-trade. “Lest I might be entrapped by its fallacies and misstatements,” was the form of expression. “You are not,” my friend said, “a special student of political economy. You might, therefore, easily be deceived by fallacious arguments upon the subject. You might, then, if you read this paper, be led to believe in protection. But you admit that free-trade is the true doctrine; and you do not wish to believe what is not true.” I have often known this system to be deliberately adopted. Still oftener, the instinctive dislike of an undecided state of mind, exaggerated into a vague dread of doubt, makes men cling spasmodically to the views they already take. The man feels that, if he only holds to his belief without wavering, it will be entirely satisfactory. Nor can it be denied that a steady and immovable faith yields great peace of mind. It may, indeed, give rise to inconveniences, as if a man should resolutely continue to believe that fire would not burn him, or that he would be eternally damned if he received his ingesta otherwise than through a stomach-pump. But then the man who adopts this method will not allow that its inconveniences are greater than its advantages. He will say, “I hold steadfastly to the truth and the truth is always wholesome.” And in many cases it may very well be that the pleasure he derives from his calm faith overbalances any inconveniences resulting from its deceptive character. Thus, if it be true that death is annihilation, then the man who believes that he will certainly go straight to heaven when he dies, provided he have fulfilled certain simple observances in this life, has a cheap pleasure which will not be followed by the least disappointment. A similar consideration seems to have weight with many persons in religious topics, for we frequently hear it said, “Oh, I could not believe so-and-so, because I should be wretched if I did.” When an ostrich buries its head in the sand as danger approaches, it very likely takes the happiest course. It hides the danger, and then calmly says there is no danger; and, if it feels perfectly sure there is none, why should it raise its head to see? A man may go through life, systematically keeping out of view all that might cause a change in his opinions, and if he only succeeds—basing his method, as he does, on two fundamental psychological laws—I do not see what can be said against his doing so. It would be an egotistical impertinence to object that his procedure is irrational, for that only amounts to saying that his method of settling belief is not ours. He does not propose to himself to be rational, and indeed, will often talk with scorn of man’s weak and illusive reason. So let him think as he pleases.
But this method of fixing belief, which may be called the method of tenacity, will be unable to hold its ground in practice. The social impulse is against it. The man who adopts it will find that other men think differently from him, and it will be apt to occur to him in some saner moment that their opinions are quite as good as his own, and this will shake his confidence in his belief. This conception, that another man’s thought or sentiment may be equivalent to one’s own, is a distinctly new step, and a highly important one. It arises from an impulse too strong in man to be suppressed, without danger of destroying the human species. Unless we make ourselves hermits, we shall necessarily influence each other’s opinions; so that the problem becomes how to fix belief, not in the individual merely, but in the community.
Let the will of the state act, then, instead of that of the individual. Let an institution be created which shall have for its object to keep correct doctrines before the attention of the people, to reiterate them perpetually, and to teach them to the young; having at the same time power to prevent contrary doctrines from being taught, advocated, or expressed. Let all possible causes of a change of mind be removed from men’s apprehensions. Let them be kept ignorant, lest they should learn of some reason to think otherwise than they do. Let their passions be enlisted, so that they may regard private and unusual opinions with hatred and horror. Then, let all men who reject the established belief be terrified into silence. Let the people turn out and tar-and-feather such men, or let inquisitions be made into the manner of thinking of suspected persons, and, when they are found guilty of forbidden beliefs, let them be subjected to some signal punishment. When complete agreement could not otherwise be reached, a general massacre of all who have not thought in a certain way has proved a very effective means of settling opinion in a country. If the power to do this be wanting, let a list of opinions be drawn up, to which no man of the least independence of thought can assent, and let the faithful be required to accept all these propositions, in order to segregate them as radically as possible from the influence of the rest of the world.
This method has, from the earliest times, been one of the chief means of upholding correct theological and political doctrines, and of preserving their universal or catholic character. In Rome, especially, it has been practiced from the days of Numa Pompilius to those of Pius Nonus. This is the most perfect example in history; but wherever there is a priesthood—and no religion has been without one—this method has been more or less made use of. Wherever there is aristocracy, or a guild, or any association of a class of men whose interests depend or are supposed to depend on certain propositions, there will be inevitably found some traces of this natural product of social feeling. Cruelties always accompany this system; and when it is consistently carried out, they become atrocities of the most horrible kind in the eyes of any rational man. Nor should this occasion surprise, for the officer of a society does not feel justified in surrendering the interests of that society for the sake of mercy, as he might his own private interests. It is natural, therefore, that sympathy and fellowship should thus produce a most ruthless power.
In judging this method of fixing belief, which may be called the method of authority, we must in the first place, allow its immeasurable mental and moral superiority to the method of tenacity. Its success is proportionally greater; and in fact it has over and over again worked the most majestic results. The mere structures of stone which it has caused to be put together—in Siam, for example, in Egypt, and in Europe—have many of them a sublimity hardly more than rivaled by the greatest works of Nature. And, except the geological epochs, there are no periods of time so vast as those which are measured by some of these organized faiths. If we scrutinize the matter closely, we shall find that there has not been one of their creeds which has remained always the same; yet the change is so slow as to be imperceptible during one person’s life, so that individual belief remains sensibly fixed. For the mass of mankind, then, there is perhaps no better method than this. If it is their highest impulse to be intellectual slaves, then slaves they ought to remain.
But no institution can undertake to regulate opinions upon every subject. Only the most important ones can be attended to, and on the rest men’s minds must be left to the action of natural causes. This imperfection will be no source of weakness so long as men are in such a state of culture that one opinion does not influence another—that is, so long as they cannot put two and two together. But in the most priest-ridden states some individuals will be found who are raised above that condition. These men possess a wider sort of social feeling; they see that men in other countries and in other ages have held to very different doctrines from those which they themselves have been brought up to believe; and they cannot help seeing that it is the mere accident of their having been taught as they have, and of their having been surrounded with the manners and associations they have, that has caused them to believe as they do and not far differently. And their candor cannot resist the reflection that there is no reason to rate their own views at a higher value than those of other nations and other centuries; and this gives rise to doubts in their minds.
They will further perceive that such doubts as these must exist in their minds with reference to every belief which seems to be determined by the caprice either of themselves or of those who originated the popular opinions. The willful adherence to a belief, and the arbitrary forcing of it upon others, must, therefore, both be given up and a new method of settling opinions must be adopted, which shall not only produce an impulse to believe, but shall also decide what proposition it is which is to be believed. Let the action of natural preferences be unimpeded, then, and under their influence let men conversing together and regarding matters in different lights, gradually develop beliefs in harmony with natural causes. This method resembles that by which conceptions of art have been brought to maturity. The most perfect example of it is to be found in the history of metaphysical philosophy. Systems of this sort have not usually rested upon observed facts, at least not in any great degree. They have been chiefly adopted because their fundamental propositions seemed “agreeable to reason.” This is an apt expression; it does not mean that which agrees with experience, but that which we find ourselves inclined to believe. Plato, for example, finds it agreeable to reason that the distances of the celestial spheres from one another should be proportional to the different lengths of strings which produce harmonious chords. Many philosophers have been led to their main conclusions by considerations like this; but this is the lowest and least developed form which the method takes, for it is clear that another man might find Kepler’s [earlier] theory, that the celestial spheres are proportional to the inscribed and circumscribed spheres of the different regular solids, more agreeable to his reason. But the shock of opinions will soon lead men to rest on preferences of a far more universal nature. Take, for example, the doctrine that man only acts selfishly—that is, from the consideration that acting in one way will afford him more pleasure than acting in another. This rests on no fact in the world, but it has had a wide acceptance as being the only reasonable theory.
This method is far more intellectual and respectable from the point of view of reason than either of the others which we have noticed. But its failure has been the most manifest. It makes of inquiry something similar to the development of taste; but taste, unfortunately, is always more or less a matter of fashion, and accordingly, meta-physicians have never come to any fixed agreement, but the pendulum has swung backward and forward between a more material and a more spiritual philosophy, from the earliest times to the latest. And so from this, which has been called the a priori method, we are driven, in Lord Bacon’s phrase, to a true induction. We have examined into this a priori method as something which promised to deliver our opinions from their accidental and capricious element. But development, while it is a process which eliminates the effect of some casual circumstances, only magnifies that of others. This method, therefore, does not differ in a very essential way from that of authority. The government may not have lifted its finger to influence my convictions; I may have been left outwardly quite free to choose, we will say, between monogamy and polygamy, and appealing to my conscience only, I may have concluded that the latter practice is in itself licentious. But when I come to see that the chief obstacle to the spread of Christianity among a people of as high culture as the Hindoos has been a conviction of the immorality of our way of treating women, I cannot help seeing that, though governments do not interfere, sentiments in their development will be very greatly determined by accidental causes. Now, there are some people, among whom I must suppose that my reader is to be found, who, when they see that any belief of theirs is determined by any circumstance extraneous to the facts, will from that moment not merely admit in words that that belief is doubtful, but will experience a real doubt of it, so that it ceases to be a belief.
To satisfy our doubts, therefore, it is necessary that a method should be found by which our beliefs may be caused by nothing human, but by some external permanency—by something upon which our thinking has no effect. Some mystics imagine that they have such a method in a private inspiration from on high. But that is only a form of the method of tenacity, in which the conception of truth as something public is not yet developed. Our external permanency would not be external, in our sense, if it was restricted in its influence to one individual. It must be something which affects, or might affect, every man. And, though these affections are necessarily as various as are individual conditions, yet the method must be such that the ultimate conclusion of every man shall be the same. Such is the method of science. Its fundamental hypothesis, restated in more familiar language, is this: There are real things, whose characters are entirely independent of our opinions about them; whose realities affect our senses according to regular laws, and, though our sensations are as different as our relations to the objects, yet, by taking advantage of the laws of perception, we can ascertain by reasoning how things really are, and any man, if he have sufficient experience and reason enough about it, will be led to the one true conclusion. The new conception here involved is that of reality. It may be asked how I know that there are any realities. If this hypothesis is the sole support of my method of inquiry, my method of inquiry must not be used to support my hypothesis. The reply is this: 1. If investigation cannot be regarded as proving that there are real things, it at least does not lead to a contrary conclusion; but the method and the conception on which it is based remain ever in harmony. No doubts of the method, therefore, necessarily arise from its practice, as is the case with all the others. 2. The feeling which gives rise to any method of fixing belief is a dissatisfaction at two repugnant propositions. But here already is a vague concession that there is some one thing to which a proposition should conform. Nobody, therefore, can really doubt that there are realities, or, if he did, doubt would not be a source of dissatisfaction. The hypothesis, therefore, is one which every mind admits. So that the social impulse does not cause me to doubt it. 3. Everybody uses the scientific method about a great many things, and only ceases to use it when he does not know how to apply it. 4. Experience of the method has not led me to doubt it, but, on the contrary, scientific investigation has had the most wonderful triumphs in the way of settling opinion. These afford the explanation of my not doubting the method or the hypothesis which it supposes; and not having any doubt, nor believing that anybody else whom I could influence has, it would be the merest babble for me to say more about it. If there be anybody with a living doubt upon the subject, let him consider it.
To describe the method of scientific investigation is the object of this series of papers. At present I have only room to notice some points of contrast between it and other methods of fixing belief.
This is the only one of the four methods which presents any distinction of a right and a wrong way. If I adopt the method of tenacity and shut myself out from all influences, whatever I think necessary to doing this is necessary according to that method. So with the method of authority: the state may try to put down heresy by means which, from a scientific point of view, seems very ill-calculated to accomplish its purposes; but the only test on that method is what the state thinks, so that it cannot pursue the method wrongly. So with the a priori method. The very essence of it is to think as one is inclined to think. All metaphysicians will be sure to do that, however they may be inclined to judge each other to be perversely wrong. The Hegelian system recognizes every natural tendency of thought as logical, although it is certain to be abolished by counter-tendencies. Hegel thinks there is a regular system in the succession of these tendencies, in consequence of which, after drifting one way and the other for a long time, opinion will at last go right. And it is true that metaphysicians get the right ideas at last; Hegel’s system of Nature represents tolerably the science of that day; and one may be sure that whatever scientific investigation has put out of doubt will presently receive a priori demonstration on the part of the metaphysicians. But with the scientific method the case is different. I may start with known and observed facts to proceed to the unknown; and yet the rules which I follow in doing so may not be such as investigation would approve. The test of whether I am truly following the method is not an immediate appeal to my feelings and purposes, but, on the contrary, itself involves the application of the method. Hence it is that bad reasoning as well as good reasoning is possible; and this fact is the foundation of the practical side of logic.
It is not to be supposed that the first three methods of settling opinion present no advantage whatever over the scientific method. On the contrary, each has some peculiar convenience of its own. The a priori method is distinguished for its comfortable conclusions. It is the nature of the process to adopt whatever belief we are inclined to, and there are certain flatteries to one’s vanities which we all believe by nature, until we are awakened from our pleasing dream by rough facts. The method of authority will always govern the mass of mankind; and those who wield the various forms of organized force in the state will never be convinced that dangerous reasoning ought not to be suppressed in some way. If liberty of speech is to be untrammeled from the grosser forms of constraint, then uniformity of opinion will be secured by a moral terrorism to which the respectability of society will give its thorough approval. Following the method of authority is the path of peace. Certain non-conformities are permitted; certain others (considered unsafe) are forbidden. These are different in different countries and in different ages; but, wherever you are let it be known that you seriously hold a tabooed belief, and you may be perfectly sure of being treated with a cruelty no less brutal but more refined than hunting you like a wolf. Thus, the greatest intellectual benefactors of mankind have never dared, and dare not now, to utter the whole of their thought; and thus a shade of prima facie doubt is cast upon every proposition which is considered essential to the security of society. Singularly enough, the persecution does not all come from without; but a man torments himself and is oftentimes most distressed at finding himself believing propositions which he has been brought up to regard with aversion. The peaceful and sympathetic man will, therefore, find it hard to resist the temptation to submit his opinions to authority. But most of all I admire the method of tenacity for its strength, simplicity, and directness. Men who pursue it are distinguished for their decision of character, which becomes very easy with such a mental rule. They do not waste time in trying to make up their minds to what they want, but, fastening like lightning upon whatever alternative comes first, they hold to it to the end, whatever happens, without an instant’s irresolution. This is one of the splendid qualities which generally accompany brilliant, unlasting success. It is impossible not to envy the man who can dismiss reason, although we know how it must turn out at last.
Such are the advantages which the other methods of settling opinions have over scientific investigation. A man should consider well of them; and then he should consider that, after all, he wishes his opinions to coincide with the fact, and that there is no reason why the results of these three methods should do so. To bring about this effect is the prerogative of the method of science. Upon such considerations he has to make his choice—a choice which is far more than the adoption of any intellectual opinion, which is one of the ruling decisions of his life, to which when once made he is bound to adhere. The force of habit will sometimes cause a man to hold on to old beliefs, after he is in a condition to see that they have no sound basis. But reflection upon the state of the case will overcome these habits, and he ought to allow reflection full weight. People sometimes shrink from doing this, having an idea that beliefs are wholesome which they cannot help feeling rest on nothing. But let such persons suppose an analogous though different case from their own. Let them ask themselves what they would say to a reformed Mussulman who should hesitate to give up his old notions in regard to the relations of the sexes; or to a reformed Catholic who should still shrink from the Bible. Would they not say that these persons ought to consider the matter fully, and clearly understand the new doctrine, and then ought to embrace it in its entirety? But, above all, let it be considered that what is more wholesome than any particular belief, is integrity of belief; and that to avoid looking into the support of any belief from a fear that it may turn out rotten is quite as immoral as it is disadvantageous. The person who confesses that there is such a thing as truth, which is distinguished from falsehood simply by this, that if acted on it will carry us to the point we aim at and not astray, and then though convinced of this, dares not know the truth and seeks to avoid it, is in a sorry state of mind, indeed.
Yes, the other methods do have their merits: a clear logical conscience does cost something—just as any virtue, just as all that we cherish, costs us dear. But, we should not desire it to be otherwise. The genius of a man’s logical method should be loved and reverenced as his bride, whom he has chosen from all the world. He need not condemn the others; on the contrary, he may honor them deeply, and in doing so he only honors her the more. But she is the one that he has chosen, and he knows that he was right in making that choice. And having made it, he will work and fight for her, and will not complain that there are blows to take, hoping that there may be as many and as hard to give, and will strive to be the worthy knight and champion of her from the blaze of whose splendors he draws his inspiration and his courage.
SECOND PAPER
HOW TO MAKE OUR IDEAS CLEAR[[31]]
I
Whoever has looked into a modern treatise on logic of the common sort, will doubtless remember the two distinctions between clear and obscure conceptions, and between distinct and confused conceptions. They have lain in the books now for nigh two centuries, unimproved and unmodified, and are generally reckoned by logicians as among the gems of their doctrine.
A clear idea is defined as one which is so apprehended that it will be recognized wherever it is met with, and so that no other will be mistaken for it. If it fails of this clearness, it is said to be obscure.
This is rather a neat bit of philosophical terminology; yet, since it is clearness that they were defining, I wish the logicians had made their definition a little more plain. Never to fail to recognize an idea, and under no circumstances to mistake another for it, let it come in how recondite a form it may, would indeed imply such prodigious force and clearness of intellect as is seldom met with in this world. On the other hand, merely to have such an acquaintance with the idea as to have become familiar with it, and to have lost all hesitancy in recognizing it in ordinary cases, hardly seems to deserve the name of clearness of apprehension, since after all it only amounts to a subjective feeling of mastery which may be entirely mistaken. I take it, however, that when the logicians speak of “clearness,” they mean nothing more than such a familiarity with an idea, since they regard the quality as but a small merit, which needs to be supplemented by another, which they call distinctness.
A distinct idea is defined as one which contains nothing which is not clear. This is technical language; by the contents of an idea logicians understand whatever is contained in its definition. So that an idea is distinctly apprehended, according to them, when we can give a precise definition of it, in abstract terms. Here the professional logicians leave the subject; and I would not have troubled the reader with what they have to say, if it were not such a striking example of how they have been slumbering through ages of intellectual activity, listlessly disregarding the enginery of modern thought, and never dreaming of applying its lessons to the improvement of logic. It is easy to show that the doctrine that familiar use and abstract distinctness make the perfection of apprehension, has its only true place in philosophies which have long been extinct; and it is now time to formulate the method of attaining to a more perfect clearness of thought, such as we see and admire in the thinkers of our own time.
When Descartes set about the reconstruction of philosophy, his first step was to (theoretically) permit skepticism and to discard the practice of the schoolmen of looking to authority as the ultimate source of truth. That done, he sought a more natural fountain of true principles, and professed to find it in the human mind; thus passing, in the directest way, from the method of authority to that of apriority, as described in my first paper. Self-consciousness was to furnish us with our fundamental truths, and to decide what was agreeable to reason. But since, evidently, not all ideas are true, he was led to note, as the first condition of infallibility, that they must be clear. The distinction between an idea seeming clear and really being so, never occurred to him. Trusting to introspection, as he did, even for a knowledge of external things, why should he question its testimony in respect to the contents of our own minds? But then, I suppose, seeing men, who seemed to be quite clear and positive, holding opposite opinions upon fundamental principles, he was further led to say that clearness of ideas is not sufficient, but that they need also to be distinct, i.e., to have nothing unclear about them. What he probably meant by this (for he did not explain himself with precision) was, that they must sustain the test of dialectical examination; that they must not only seem clear at the outset, but that discussion must never be able to bring to light points of obscurity connected with them.
Such was the distinction of Descartes, and one sees that it was precisely on the level of his philosophy. It was somewhat developed by Leibnitz. This great and singular genius was as remarkable for what he failed to see as for what he saw. That a piece of mechanism could not do work perpetually without being fed with power in some form, was a thing perfectly apparent to him; yet he did not understand that the machinery of the mind can only transform knowledge, but never originate it, unless it be fed with facts of observation. He thus missed the most essential point of the Cartesian philosophy, which is, that to accept propositions which seem perfectly evident to us is a thing which, whether it be logical or illogical, we cannot help doing. Instead of regarding the matter in this way, he sought to reduce the first principles of science to formulas which cannot be denied without self-contradiction, and was apparently unaware of the great difference between his position and that of Descartes. So he reverted to the old formalities of logic, and, above all, abstract definitions played a great part in his philosophy. It was quite natural, therefore, that on observing that the method of Descartes labored under the difficulty that we may seem to ourselves to have clear apprehensions of ideas which in truth are very hazy, no better remedy occurred to him than to require an abstract definition of every important term. Accordingly, in adopting the distinction of clear and distinct notions, he described the latter quality as the clear apprehension of everything contained in the definition; and the books have ever since copied his words. There is no danger that his chimerical scheme will ever again be over-valued. Nothing new can ever be learned by analyzing definitions. Nevertheless, our existing beliefs can be set in order by this process, and order is an essential element of intellectual economy, as of every other. It may be acknowledged, therefore, that the books are right in making familiarity with a notion the first step toward clearness of apprehension, and the defining of it the second. But in omitting all mention of any higher perspicuity of thought, they simply mirror a philosophy which was exploded a hundred years ago. That much-admired “ornament of logic”—the doctrine of clearness and distinctness—may be pretty enough, but it is high time to relegate to our cabinet of curiosities the antique bijou, and to wear about us something better adapted to modern uses.
The very first lesson that we have a right to demand that logic shall teach us is, how to make our ideas clear; and a most important one it is, depreciated only by minds who stand in need of it. To know what we think, to be masters of our own meaning, will make a solid foundation for great and weighty thought. It is most easily learned by those whose ideas are meagre and restricted; and far happier they than such as wallow helplessly in a rich mud of conceptions. A nation, it is true, may, in the course of generations, overcome the disadvantage of an excessive wealth of language and its natural concomitant, a vast, unfathomable deep of ideas. We may see it in history, slowly perfecting its literary forms, sloughing at length its metaphysics, and, by virtue of the untirable patience which is often a compensation, attaining great excellence in every branch of mental acquirement. The page of history is not yet unrolled which is to tell us whether such a people will or will not in the long run prevail over one whose ideas (like the words of their language) are few, but which possesses a wonderful mastery over those which it has. For an individual, however, there can be no question that a few clear ideas are worth more than many confused ones. A young man would hardly be persuaded to sacrifice the greater part of his thoughts to save the rest; and the muddled head is the least apt to see the necessity of such a sacrifice. Him we can usually only commiserate, as a person with a congenital defect. Time will help him, but intellectual maturity with regard to clearness comes rather late, an unfortunate arrangement of Nature, inasmuch as clearness is of less use to a man settled in life, whose errors have in great measure had their effect, than it would be to one whose path lies before him. It is terrible to see how a single unclear idea, a single formula without meaning, lurking in a young man’s head, will sometimes act like an obstruction of inert matter in an artery, hindering the nutrition of the brain, and condemning its victim to pine away in the fullness of his intellectual vigor and in the midst of intellectual plenty. Many a man has cherished for years as his hobby some vague shadow of an idea, too meaningless to be positively false; he has, nevertheless, passionately loved it, has made it his companion by day and by night, and has given to it his strength and his life, leaving all other occupations for its sake, and in short has lived with it and for it, until it has become, as it were, flesh of his flesh and bone of his bone; and then he has waked up some bright morning to find it gone, clean vanished away like the beautiful Melusina of the fable, and the essence of his life gone with it. I have myself known such a man; and who can tell how many histories of circle-squarers, metaphysicians, astrologers, and what not, may not be told in the old German story?
II
The principles set forth in the first of these papers lead, at once, to a method of reaching a clearness of thought of a far higher grade than the “distinctness” of the logicians. We have there found that the action of thought is excited by the irritation of doubt, and ceases when belief is attained; so that the production of belief is the sole function of thought. All these words, however, are too strong for my purpose. It is as if I had described the phenomena as they appear under a mental microscope. Doubt and Belief, as the words are commonly employed, relate to religious or other grave discussions. But here I use them to designate the starting of any question, no matter how small or how great, and the resolution of it. If, for instance, in a horse-car, I pull out my purse and find a five-cent nickel and five coppers, I decide, while my hand is going to the purse, in which way I will pay my fare. To call such a question Doubt, and my decision Belief, is certainly to use words very disproportionate to the occasion. To speak of such a doubt as causing an irritation which needs to be appeased, suggests a temper which is uncomfortable to the verge of insanity. Yet, looking at the matter minutely, it must be admitted that, if there is the least hesitation as to whether I shall pay the five coppers or the nickel (as there will be sure to be, unless I act from some previously contracted habit in the matter), though irritation is too strong a word, yet I am excited to such small mental activity as may be necessary to deciding how I shall act. Most frequently doubts arise from some indecision, however momentary, in our action. Sometimes it is not so. I have, for example, to wait in a railway-station, and to pass the time I read the advertisements on the walls, I compare the advantages of different trains and different routes which I never expect to take, merely fancying myself to be in a state of hesitancy, because I am bored with having nothing to trouble me. Feigned hesitancy, whether feigned for mere amusement or with a lofty purpose, plays a great part in the production of scientific inquiry. However the doubt may originate, it stimulates the mind to an activity which may be slight or energetic, calm or turbulent. Images pass rapidly through consciousness, one incessantly melting into another, until at last, when all is over—it may be in a fraction of a second, in an hour, or after long years—we find ourselves decided as to how we should act under such circumstances as those which occasioned our hesitation. In other words, we have attained belief.
In this process we observe two sorts of elements of consciousness, the distinction between which may best be made clear by means of an illustration. In a piece of music there are the separate notes, and there is the air. A single tone may be prolonged for an hour or a day, and it exists as perfectly in each second of that time as in the whole taken together; so that, as long as it is sounding, it might be present to a sense from which everything in the past was as completely absent as the future itself. But it is different with the air, the performance of which occupies a certain time, during the portions of which only portions of it are played. It consists in an orderliness in the succession of sounds which strike the ear at different times; and to perceive it there must be some continuity of consciousness which makes the events of a lapse of time present to us. We certainly only perceive the air by hearing the separate notes; yet we cannot be said to directly hear it, for we hear only what is present at the instant, and an orderliness of succession cannot exist in an instant. These two sorts of objects, what we are immediately conscious of and what we are mediately conscious of, are found in all consciousness. Some elements (the sensations) are completely present at every instant so long as they last, while others (like thought) are actions having beginning, middle, and end, and consist in a congruence in the succession of sensations which flow through the mind. They cannot be immediately present to us, but must cover some portion of the past or future. Thought is a thread of melody running through the succession of our sensations.
We may add that just as a piece of music may be written in parts, each part having its own air, so various systems of relationship of succession subsist together between the same sensations. These different systems are distinguished by having different motives, ideas, or functions. Thought is only one such system; for its sole motive, idea, and function is to produce belief, and whatever does not concern that purpose belongs to some other system of relations. The action of thinking may incidentally have other results. It may serve to amuse us, for example, and among dilettanti it is not rare to find those who have so perverted thought to the purposes of pleasure that it seems to vex them to think that the questions upon which they delight to exercise it may ever get finally settled; and a positive discovery which takes a favorite subject out of the arena of literary debate is met with ill-concealed dislike. This disposition is the very debauchery of thought. But the soul and meaning of thought, abstracted from the other elements which accompany it, though it may be voluntarily thwarted, can never be made to direct itself toward anything but the production of belief. Thought in action has for its only possible motive the attainment of thought at rest; and whatever does not refer to belief is no part of the thought itself.
And what, then, is belief? It is the demi-cadence which closes a musical phrase in the symphony of our intellectual life. We have seen that it has just three properties: First, it is something that we are aware of; second, it appeases the irritation of doubt; and, third, it involves the establishment in our nature of a rule of action, or, say for short, a habit. As it appeases the irritation of doubt, which is the motive for thinking, thought relaxes, and comes to rest for a moment when belief is reached. But, since belief is a rule for action, the application of which involves further doubt and further thought, at the same time that it is a stopping-place, it is also a new starting-place for thought. That is why I have permitted myself to call it thought at rest, although thought is essentially an action. The final upshot of thinking is the exercise of volition, and of this thought no longer forms a part; but belief is only a stadium of mental action, an effect upon our nature due to thought, which will influence future thinking.
Figure 1.
Figure 2.
The essence of belief is the establishment of a habit, and different beliefs are distinguished by the different modes of action to which they give rise. If beliefs do not differ in this respect, if they appease the same doubt by producing the same rule of action, then no mere differences in the manner of consciousness of them can make them different beliefs, any more than playing a tune in different keys is playing different tunes. Imaginary distinctions are often drawn between beliefs which differ only in their mode of expression;—the wrangling which ensues is real enough, however. To believe that any objects are arranged as in Fig. 1, and to believe that they are arranged as in Fig. 2, are one and the same belief; yet it is conceivable that a man should assert one proposition and deny the other. Such false distinctions do as much harm as the confusion of beliefs really different, and are among the pitfalls of which we ought constantly to beware, especially when we are upon metaphysical ground. One singular deception of this sort, which often occurs, is to mistake the sensation produced by our own unclearness of thought for a character of the object we are thinking. Instead of perceiving that the obscurity is purely subjective, we fancy that we contemplate a quality of the object which is essentially mysterious; and if our conception be afterward presented to us in a clear form we do not recognize it as the same, owing to the absence of the feeling of unintelligibility. So long as this deception lasts, it obviously puts an impassable barrier in the way of perspicuous thinking; so that it equally interests the opponents of rational thought to perpetuate it, and its adherents to guard against it.
Another such deception is to mistake a mere difference in the grammatical construction of two words for a distinction between the ideas they express. In this pedantic age, when the general mob of writers attend so much more to words than to things, this error is common enough. When I just said that thought is an action, and that it consists in a relation, although a person performs an action but not a relation, which can only be the result of an action, yet there was no inconsistency in what I said, but only a grammatical vagueness.
From all these sophisms we shall be perfectly safe so long as we reflect that the whole function of thought is to produce habits of action; and that whatever there is connected with a thought, but irrelevant to its purpose, is an accretion to it, but no part of it. If there be a unity among our sensations which has no reference to how we shall act on a given occasion, as when we listen to a piece of music, why we do not call that thinking. To develop its meaning, we have, therefore, simply to determine what habits it produces, for what a thing means is simply what habits it involves. Now, the identity of a habit depends on how it might lead us to act, not merely under such circumstances as are likely to arise, but under such as might possibly occur, no matter how improbable they may be. What the habit is depends on when and how it causes us to act. As for the when, every stimulus to action is derived from perception; as for the how, every purpose of action is to produce some sensible result. Thus, we come down to what is tangible and practical, as the root of every real distinction of thought, no matter how subtile it may be; and there is no distinction of meaning so fine as to consist in anything but a possible difference of practice.
To see what this principle leads to, consider in the light of it such a doctrine as that of transubstantiation. The Protestant churches generally hold that the elements of the sacrament are flesh and blood only in a tropical sense; they nourish our souls as meat and the juice of it would our bodies. But the Catholics maintain that they are literally just that; although they possess all the sensible qualities of wafer-cakes and diluted wine. But we can have no conception of wine except what may enter into a belief, either—
1. That this, that, or the other, is wine; or,
2. That wine possesses certain properties.
Such beliefs are nothing but self-notifications that we should, upon occasion, act in regard to such things as we believe to be wine according to the qualities which we believe wine to possess. The occasion of such action would be some sensible perception, the motive of it to produce some sensible result. Thus our action has exclusive reference to what affects the senses, our habit has the same bearing as our action, our belief the same as our habit, our conception the same as our belief; and we can consequently mean nothing by wine but what has certain effects, direct or indirect, upon our senses; and to talk of something as having all the sensible characters of wine, yet being in reality blood, is senseless jargon. Now, it is not my object to pursue the theological question; and having used it as a logical example I drop it, without caring to anticipate the theologian’s reply. I only desire to point out how impossible it is that we should have an idea in our minds which relates to anything but conceived sensible effects of things. Our idea of anything is our idea of its sensible effects; and if we fancy that we have any other we deceive ourselves, and mistake a mere sensation accompanying the thought for a part of the thought itself. It is absurd to say that thought has any meaning unrelated to its only function. It is foolish for Catholics and Protestants to fancy themselves in disagreement about the elements of the sacrament, if they agree in regard to all their sensible effects, here or hereafter.
It appears, then, that the rule for attaining the third grade of clearness of apprehension is as follows: Consider what effects, which might conceivably have practical bearings, we conceive the object of our conception to have. Then, our conception of these effects is the whole of our conception of the object.
III
Let us illustrate this rule by some examples; and, to begin with the simplest one possible, let us ask what we mean by calling a thing hard. Evidently that it will not be scratched by many other substances. The whole conception of this quality, as of every other, lies in its conceived effects. There is absolutely no difference between a hard thing and a soft thing so long as they are not brought to the test. Suppose, then, that a diamond could be crystallized in the midst of a cushion of soft cotton, and should remain there until it was finally burned up. Would it be false to say that that diamond was soft? This seems a foolish question, and would be so, in fact, except in the realm of logic. There such questions are often of the greatest utility as serving to bring logical principles into sharper relief than real discussions ever could. In studying logic we must not put them aside with hasty answers, but must consider them with attentive care, in order to make out the principles involved. We may, in the present case, modify our question, and ask what prevents us from saying that all hard bodies remain perfectly soft until they are touched, when their hardness increases with the pressure until they are scratched. Reflection will show that the reply is this: there would be no falsity in such modes of speech. They would involve a modification of our present usage of speech with regard to the words hard and soft, but not of their meanings. For they represent no fact to be different from what it is; only they involve arrangements of facts which would be exceedingly maladroit. This leads us to remark that the question of what would occur under circumstances which do not actually arise is not a question of fact, but only of the most perspicuous arrangement of them. For example, the question of free-will and fate in its simplest form, stripped of verbiage, is something like this: I have done something of which I am ashamed; could I, by an effort of the will, have resisted the temptation, and done otherwise? The philosophical reply is, that this is not a question of fact, but only of the arrangement of facts. Arranging them so as to exhibit what is particularly pertinent to my question—namely, that I ought to blame myself for having done wrong—it is perfectly true to say that, if I had willed to do otherwise than I did, I should have done otherwise. On the other hand, arranging the facts so as to exhibit another important consideration, it is equally true that, when a temptation has once been allowed to work, it will, if it has a certain force, produce its effect, let me struggle how I may. There is no objection to a contradiction in what would result from a false supposition. The reductio ad absurdum consists in showing that contradictory results would follow from a hypothesis which is consequently judged to be false. Many questions are involved in the free-will discussion, and I am far from desiring to say that both sides are equally right. On the contrary, I am of opinion that one side denies important facts, and that the other does not. But what I do say is, that the above single question was the origin of the whole doubt; that, had it not been for this question, the controversy would never have arisen; and that this question is perfectly solved in the manner which I have indicated.
Let us next seek a clear idea of Weight. This is another very easy case. To say that a body is heavy means simply that, in the absence of opposing force, it will fall. This (neglecting certain specifications of how it will fall, etc., which exist in the mind of the physicist who uses the word) is evidently the whole conception of weight. It is a fair question whether some particular facts may not account for gravity; but what we mean by the force itself is completely involved in its effects.
This leads us to undertake an account of the idea of Force in general. This is the great conception which, developed in the early part of the seventeenth century from the rude idea of a cause, and constantly improved upon since, has shown us how to explain all the changes of motion which bodies experience, and how to think about all physical phenomena; which has given birth to modern science, and changed the face of the globe; and which, aside from its more special uses, has played a principal part in directing the course of modern thought, and in furthering modern social development. It is, therefore, worth some pains to comprehend it. According to our rule, we must begin by asking what is the immediate use of thinking about force; and the answer is, that we thus account for changes of motion. If bodies were left to themselves, without the intervention of forces, every motion would continue unchanged both in velocity and in direction. Furthermore, change of motion never takes place abruptly; if its direction is changed, it is always through a curve without angles; if its velocity alters, it is by degrees. The gradual changes which are constantly taking place are conceived by geometers to be compounded together according to the rules of the parallelogram of forces. If the reader does not already know what this is, he will find it, I hope, to his advantage to endeavor to follow the following explanation; but if mathematics are insupportable to him, pray let him skip three paragraphs rather than that we should part company here.
A path is a line whose beginning and end are distinguished. Two paths are considered to be equivalent, which, beginning at the same point, lead to the same point. Thus the two paths, A B C D E and A F G H E (Fig. 3), are equivalent. Paths which do not begin at the same point are considered to be equivalent, provided that, on moving either of them without turning it, but keeping it always parallel to its original position, [so that] when its beginning coincides with that of the other path, the ends also coincide. Paths are considered as geometrically added together, when one begins where the other ends; thus the path A E is conceived to be a sum of A B, B C, C D, and D E. In the parallelogram of Fig. 4 the diagonal A C is the sum of A B and B C; or, since A D is geometrically equivalent to B C, A C is the geometrical sum of A B and A D.
Figure 3.
Figure 4.
All this is purely conventional. It simply amounts to this: that we choose to call paths having the relations I have described equal or added. But, though it is a convention, it is a convention with a good reason. The rule for geometrical addition may be applied not only to paths, but to any other things which can be represented by paths. Now, as a path is determined by the varying direction and distance of the point which moves over it from the starting-point, it follows that anything which from its beginning to its end is determined by a varying direction and a varying magnitude is capable of being represented by a line. Accordingly, velocities may be represented by lines, for they have only directions and rates. The same thing is true of accelerations, or changes of velocities. This is evident enough in the case of velocities; and it becomes evident for accelerations if we consider that precisely what velocities are to positions—namely, states of change of them—that accelerations are to velocities.
Figure 5.
The so-called “parallelogram of forces” is simply a rule for compounding accelerations. The rule is, to represent the accelerations by paths, and then to geometrically add the paths. The geometers, however, not only use the “parallelogram of forces” to compound different accelerations, but also to resolve one acceleration into a sum of several. Let A B (Fig. 5) be the path which represents a certain acceleration—say, such a change in the motion of a body that at the end of one second the body will, under the influence of that change, be in a position different from what it would have had if its motion had continued unchanged, such that a path equivalent to A B would lead from the latter position to the former. This acceleration may be considered as the sum of the accelerations represented by A C and C B. It may also be considered as the sum of the very different accelerations represented by A D and D B, where A D is almost the opposite of A C. And it is clear that there is an immense variety of ways in which A B might be resolved into the sum of two accelerations.
After this tedious explanation, which I hope, in view of the extraordinary interest of the conception of force, may not have exhausted the reader’s patience, we are prepared at last to state the grand fact which this conception embodies. This fact is that if the actual changes of motion which the different particles of bodies experience are each resolved in its appropriate way, each component acceleration is precisely such as is prescribed by a certain law of Nature, according to which bodies in the relative positions which the bodies in question actually have at the moment,[[32]] always receive certain accelerations, which, being compounded by geometrical addition, give the acceleration which the body actually experiences.
This is the only fact which the idea of force represents, and whoever will take the trouble clearly to apprehend what this fact is, perfectly comprehends what force is. Whether we ought to say that a force is an acceleration, or that it causes an acceleration, is a mere question of propriety of language, which has no more to do with our real meaning than the difference between the French idiom “Il fait froid” and its English equivalent “It is cold.” Yet it is surprising to see how this simple affair has muddled men’s minds. In how many profound treatises is not force spoken of as a “mysterious entity,” which seems to be only a way of confessing that the author despairs of ever getting a clear notion of what the word means! In a recent admired work on Analytic Mechanics it is stated that we understand precisely the effect of force, but what force itself is we do not understand! This is simply a self-contradiction. The idea which the word force excites in our minds has no other function than to affect our actions, and these actions can have no reference to force otherwise than through its effects. Consequently, if we know what the effects of force are, we are acquainted with every fact which is implied in saying that a force exists, and there is nothing more to know. The truth is, there is some vague notion afloat that a question may mean something which the mind cannot conceive; and when some hair-splitting philosophers have been confronted with the absurdity of such a view, they have invented an empty distinction between positive and negative conceptions, in the attempt to give their non-idea a form not obviously nonsensical. The nullity of it is sufficiently plain from the considerations given a few pages back; and, apart from those considerations, the quibbling character of the distinction must have struck every mind accustomed to real thinking.
IV
Let us now approach the subject of logic, and consider a conception which particularly concerns it, that of reality. Taking clearness in the sense of familiarity, no idea could be clearer than this. Every child uses it with perfect confidence, never dreaming that he does not understand it. As for clearness in its second grade, however, it would probably puzzle most men, even among those of a reflective turn of mind, to give an abstract definition of the real. Yet such a definition may perhaps be reached by considering the points of difference between reality and its opposite, fiction. A figment is a product of somebody’s imagination; it has such characters as his thought impresses upon it. That those characters are independent of how you or I think is an external reality. There are, however, phenomena within our own minds, dependent upon our thought, which are at the same time real in the sense that we really think them. But though their characters depend on how we think, they do not depend on what we think those characters to be. Thus, a dream has a real existence as a mental phenomenon, if somebody has really dreamt it; that he dreamt so and so, does not depend on what anybody thinks was dreamt, but is completely independent of all opinion on the subject. On the other hand, considering, not the fact of dreaming, but the thing dreamt, it retains its peculiarities by virtue of no other fact than that it was dreamt to possess them. Thus we may define the real as that whose characters are independent of what anybody may think them to be.
But, however satisfactory such a definition may be found, it would be a great mistake to suppose that it makes the idea of reality perfectly clear. Here, then, let us apply our rules. According to them, reality, like every other quality, consists in the peculiar sensible effects which things partaking of it produce. The only effect which real things have is to cause belief, for all the sensations which they excite emerge into consciousness in the form of beliefs. The question, therefore, is, how is true belief (or belief in the real) distinguished from false belief (or belief in fiction). Now, as we have seen in the former paper, the ideas of truth and falsehood, in their full development, appertain exclusively to the scientific method of settling opinion. A person who arbitrarily chooses the propositions which he will adopt can use the word truth only to emphasize the expression of his determination to hold on to his choice. Of course, the method of tenacity never prevailed exclusively; reason is too natural to men for that. But in the literature of the dark ages we find some fine examples of it. When Scotus Erigena is commenting upon a poetical passage in which hellebore is spoken of as having caused the death of Socrates, he does not hesitate to inform the inquiring reader that Helleborus and Socrates were two eminent Greek philosophers, and that the latter having been overcome in argument by the former took the matter to heart and died of it! What sort of an idea of truth could a man have who could adopt and teach, without the qualification of a perhaps, an opinion taken so entirely at random? The real spirit of Socrates, who I hope would have been delighted to have been “overcome in argument,” because he would have learned something by it, is in curious contrast with the naïve idea of the glossist, for whom discussion would seem to have been simply a struggle. When philosophy began to awake from its long slumber, and before theology completely dominated it, the practice seems to have been for each professor to seize upon any philosophical position he found unoccupied and which seemed a strong one, to intrench himself in it, and to sally forth from time to time to give battle to the others. Thus, even the scanty records we possess of those disputes enable us to make out a dozen or more opinions held by different teachers at one time concerning the question of nominalism and realism. Read the opening part of the Historia Calamitatum of Abelard, who was certainly as philosophical as any of his contemporaries, and see the spirit of combat which it breathes. For him, the truth is simply his particular stronghold. When the method of authority prevailed, the truth meant little more than the Catholic faith. All the efforts of the scholastic doctors are directed toward harmonizing their faith in Aristotle and their faith in the Church, and one may search their ponderous folios through without finding an argument which goes any further. It is noticeable that where different faiths flourish side by side, renegades are looked upon with contempt even by the party whose belief they adopt; so completely has the idea of loyalty replaced that of truth-seeking. Since the time of Descartes, the defect in the conception of truth has been less apparent. Still, it will sometimes strike a scientific man that the philosophers have been less intent on finding out what the facts are, than on inquiring what belief is most in harmony with their system. It is hard to convince a follower of the a priori method by adducing facts; but show him that an opinion he is defending is inconsistent with what he has laid down elsewhere, and he will be very apt to retract it. These minds do not seem to believe that disputation is ever to cease; they seem to think that the opinion which is natural for one man is not so for another, and that belief will, consequently, never be settled. In contenting themselves with fixing their own opinions by a method which would lead another man to a different result, they betray their feeble hold of the conception of what truth is.
On the other hand, all the followers of science are fully persuaded that the processes of investigation, if only pushed far enough, will give one certain solution to every question to which they can be applied. One man may investigate the velocity of light by studying the transits of Venus and the aberration of the stars; another by the oppositions of Mars and the eclipses of Jupiter’s satellites; a third by the method of Fizeau; a fourth by that of Foucault; a fifth by the motions of the curves of Lissajoux; a sixth, a seventh, an eighth, and a ninth, may follow the different methods of comparing the measures of statical and dynamical electricity. They may at first obtain different results, but, as each perfects his method and his processes, the results will move steadily together toward a destined center. So with all scientific research. Different minds may set out with the most antagonistic views, but the progress of investigation carries them by a force outside of themselves to one and the same conclusion. This activity of thought by which we are carried, not where we wish, but to a fore-ordained goal, is like the operation of destiny. No modification of the point of view taken, no selection of other facts for study, no natural bent of mind even, can enable a man to escape the predestinate opinion. This great law is embodied in the conception of truth and reality. The opinion which is fated[[33]] to be ultimately agreed to by all who investigate, is what we mean by the truth, and the object represented in this opinion is the real. That is the way I would explain reality.
But it may be said that this view is directly opposed to the abstract definition which we have given of reality, inasmuch as it makes the characters of the real depend on what is ultimately thought about them. But the answer to this is that, on the one hand, reality is independent, not necessarily of thought in general, but only of what you or I or any finite number of men may think about it; and that, on the other hand, though the object of the final opinion depends on what that opinion is, yet what that opinion is does not depend on what you or I or any man thinks. Our perversity and that of others may indefinitely postpone the settlement of opinion; it might even conceivably cause an arbitrary proposition to be universally accepted as long as the human race should last. Yet even that would not change the nature of the belief, which alone could be the result of investigation carried sufficiently far; and if, after the extinction of our race, another should arise with faculties and disposition for investigation, that true opinion must be the one which they would ultimately come to. “Truth crushed to earth shall rise again,” and the opinion which would finally result from investigation does not depend on how anybody may actually think. But the reality of that which is real does depend on the real fact that investigation is destined to lead, at last, if continued long enough, to a belief in it.
But I may be asked what I have to say to all the minute facts of history, forgotten never to be recovered, to the lost books of the ancients, to the buried secrets.
“Full many a gem of purest ray serene
The dark, unfathomed caves of ocean bear;
Full many a flower is born to blush unseen,
And waste its sweetness on the desert air.”
Do these things not really exist because they are hopelessly beyond the reach of our knowledge? And then, after the universe is dead (according to the prediction of some scientists), and all life has ceased forever, will not the shock of atoms continue though there will be no mind to know it? To this I reply that, though in no possible state of knowledge can any number be great enough to express the relation between the amount of what rests unknown to the amount of the known, yet it is unphilosophical to suppose that, with regard to any given question (which has any clear meaning), investigation would not bring forth a solution of it, if it were carried far enough. Who would have said, a few years ago, that we could ever know of what substances stars are made whose light may have been longer in reaching us than the human race has existed? Who can be sure of what we shall not know in a few hundred years? Who can guess what would be the result of continuing the pursuit of science for ten thousand years, with the activity of the last hundred? And if it were to go on for a million, or a billion, or any number of years you please, how is it possible to say that there is any question which might not ultimately be solved?
But it may be objected, “Why make so much of these remote considerations, especially when it is your principle that only practical distinctions have a meaning?” Well, I must confess that it makes very little difference whether we say that a stone on the bottom of the ocean, in complete darkness, is brilliant or not—that is to say, that it probably makes no difference, remembering always that that stone may be fished up to-morrow. But that there are gems at the bottom of the sea, flowers in the untraveled desert, etc., are propositions which, like that about a diamond being hard when it is not pressed, concern much more the arrangement of our language than they do the meaning of our ideas.
It seems to me, however, that we have, by the application of our rule, reached so clear an apprehension of what we mean by reality, and of the fact which the idea rests on, that we should not, perhaps, be making a pretension so presumptuous as it would be singular, if we were to offer a metaphysical theory of existence for universal acceptance among those who employ the scientific method of fixing belief. However, as metaphysics is a subject much more curious than useful, the knowledge of which, like that of a sunken reef, serves chiefly to enable us to keep clear of it, I will not trouble the reader with any more Ontology at this moment. I have already been led much further into that path than I should have desired; and I have given the reader such a dose of mathematics, psychology, and all that is most abstruse, that I fear he may already have left me, and that what I am now writing is for the compositor and proofreader exclusively. I trusted to the importance of the subject. There is no royal road to logic, and really valuable ideas can only be had at the price of close attention. But I know that in the matter of ideas the public prefer the cheap and nasty; and in my next paper I am going to return to the easily intelligible, and not wander from it again. The reader who has been at the pains of wading through this paper, shall be rewarded in the next one by seeing how beautifully what has been developed in this tedious way can be applied to the ascertainment of the rules of scientific reasoning.
We have, hitherto, not crossed the threshold of scientific logic. It is certainly important to know how to make our ideas clear, but they may be ever so clear without being true. How to make them so, we have next to study. How to give birth to those vital and procreative ideas which multiply into a thousand forms and diffuse themselves everywhere, advancing civilization and making the dignity of man, is an art not yet reduced to rules, but of the secret of which the history of science affords some hints.
THIRD PAPER
THE DOCTRINE OF CHANCES[[34]]
I
It is a common observation that a science first begins to be exact when it is quantitatively treated. What are called the exact sciences are no others than the mathematical ones. Chemists reasoned vaguely until Lavoisier showed them how to apply the balance to the verification of their theories, when chemistry leaped suddenly into the position of the most perfect of the classificatory sciences. It has thus become so precise and certain that we usually think of it along with optics, thermotics, and electrics. But these are studies of general laws, while chemistry considers merely the relations and classification of certain objects; and belongs, in reality, in the same category as systematic botany and zoölogy. Compare it with these last, however, and the advantage that it derives from its quantitative treatment is very evident.
The rudest numerical scales, such as that by which the mineralogists distinguish the different degrees of hardness, are found useful. The mere counting of pistils and stamens sufficed to bring botany out of total chaos into some kind of form. It is not, however, so much from counting as from measuring, not so much from the conception of number as from that of continuous quantity, that the advantage of mathematical treatment comes. Number, after all, only serves to pin us down to a precision in our thoughts which, however beneficial, can seldom lead to lofty conceptions, and frequently descends to pettiness. Of those two faculties of which Bacon speaks, that which marks differences and that which notes resemblances, the employment of number can only aid the lesser one; and the excessive use of it must tend to narrow the powers of the mind. But the conception of continuous quantity has a great office to fulfill, independently of any attempt at precision. Far from tending to the exaggeration of differences, it is the direct instrument of the finest generalizations. When a naturalist wishes to study a species, he collects a considerable number of specimens more or less similar. In contemplating them, he observes certain ones which are more or less alike in some particular respect. They all have, for instance, a certain S-shaped marking. He observes that they are not precisely alike, in this respect; the S has not precisely the same shape, but the differences are such as to lead him to believe that forms could be found intermediate between any two of those he possesses. He, now, finds other forms apparently quite dissimilar—say a marking in the form of a C—and the question is, whether he can find intermediate ones which will connect these latter with the others. This he often succeeds in doing in cases where it would at first be thought impossible; whereas he sometimes finds those which differ, at first glance, much less, to be separated in Nature by the non-occurrence of intermediaries. In this way, he builds up from the study of Nature a new general conception of the character in question. He obtains, for example, an idea of a leaf which includes every part of the flower, and an idea of a vertebra which includes the skull. I surely need not say much to show what a logical engine there is here. It is the essence of the method of the naturalist.[[35]] How he applies it first to one character, and then to another, and finally obtains a notion of a species of animals, the differences between whose members, however great, are confined within limits, is a matter which does not here concern us. The whole method of classification must be considered later; but, at present, I only desire to point out that it is by taking advantage of the idea of continuity, or the passage from one form to another by insensible degrees, that the naturalist builds his conceptions. Now, the naturalists are the great builders of conceptions; there is no other branch of science where so much of this work is done as in theirs; and we must, in great measure, take them for our teachers in this important part of logic. And it will be found everywhere that the idea of continuity is a powerful aid to the formation of true and fruitful conceptions. By means of it, the greatest differences are broken down and resolved into differences of degree, and the incessant application of it is of the greatest value in broadening our conceptions. I propose to make a great use of this idea in the present series of papers; and the particular series of important fallacies, which, arising from a neglect of it, have desolated philosophy, must further on be closely studied. At present, I simply call the reader’s attention to the utility of this conception.
In studies of numbers, the idea of continuity is so indispensable, that it is perpetually introduced even where there is no continuity in fact, as where we say that there are in the United States 10.7 inhabitants per square mile, or that in New York 14.72 persons live in the average house.[[36]] Another example is that law of the distribution of errors which Quetelet, Galton, and others, have applied with so much success to the study of biological and social matters. This application of continuity to cases where it does not really exist illustrates, also, another point which will hereafter demand a separate study, namely, the great utility which fictions sometimes have in science.
II
The theory of probabilities is simply the science of logic quantitatively treated. There are two conceivable certainties with reference to any hypothesis, the certainty of its truth and the certainty of its falsity. The numbers one and zero are appropriated, in this calculus, to marking these extremes of knowledge; while fractions having values intermediate between them indicate, as we may vaguely say, the degrees in which the evidence leans toward one or the other. The general problem of probabilities is, from a given state of facts, to determine the numerical probability of a possible fact. This is the same as to inquire how much the given facts are worth, considered as evidence to prove the possible fact. Thus the problem of probabilities is simply the general problem of logic.
Probability is a continuous quantity, so that great advantages may be expected from this mode of studying logic. Some writers have gone so far as to maintain that, by means of the calculus of chances, every solid inference may be represented by legitimate arithmetical operations upon the numbers given in the premises. If this be, indeed, true, the great problem of logic, how it is that the observation of one fact can give us knowledge of another independent fact, is reduced to a mere question of arithmetic. It seems proper to examine this pretension before undertaking any more recondite solution of the paradox.
But, unfortunately, writers on probabilities are not agreed in regard to this result. This branch of mathematics is the only one, I believe, in which good writers frequently get results entirely erroneous. In elementary geometry the reasoning is frequently fallacious, but erroneous conclusions are avoided; but it may be doubted if there is a single extensive treatise on probabilities in existence which does not contain solutions absolutely indefensible. This is partly owing to the want of any regular method of procedure; for the subject involves too many subtilties to make it easy to put its problems into equations without such an aid. But, beyond this, the fundamental principles of its calculus are more or less in dispute. In regard to that class of questions to which it is chiefly applied for practical purposes, there is comparatively little doubt; but in regard to others to which it has been sought to extend it, opinion is somewhat unsettled.
This last class of difficulties can only be entirely overcome by making the idea of probability perfectly clear in our minds in the way set forth in our last paper.
III
To get a clear idea of what we mean by probability, we have to consider what real and sensible difference there is between one degree of probability and another.
The character of probability belongs primarily, without doubt, to certain inferences. Locke explains it as follows: After remarking that the mathematician positively knows that the sum of the three angles of a triangle is equal to two right angles because he apprehends the geometrical proof, he thus continues: “But another man who never took the pains to observe the demonstration, hearing a mathematician, a man of credit, affirm the three angles of a triangle to be equal to two right ones, assents to it; i.e., receives it for true. In which case the foundation of his assent is the probability of the thing, the proof being such as, for the most part, carries truth with it; the man on whose testimony he receives it not being wont to affirm anything contrary to, or besides his knowledge, especially in matters of this kind.” The celebrated Essay concerning Human Understanding contains many passages which, like this one, make the first steps in profound analyses which are not further developed. It was shown in the first of these papers that the validity of an inference does not depend on any tendency of the mind to accept it, however strong such tendency may be; but consists in the real fact that, when premises like those of the argument in question are true, conclusions related to them like that of this argument are also true. It was remarked that in a logical mind an argument is always conceived as a member of a genus of arguments all constructed in the same way, and such that, when their premises are real facts, their conclusions are so also. If the argument is demonstrative, then this is always so; if it is only probable, then it is for the most part so. As Locke says, the probable argument is “such as for the most part carries truth with it.”
According to this, that real and sensible difference between one degree of probability and another, in which the meaning of the distinction lies, is that in the frequent employment of two different modes of inference, one will carry truth with it oftener than the other. It is evident that this is the only difference there is in the existing fact. Having certain premises, a man draws a certain conclusion, and as far as this inference alone is concerned the only possible practical question is whether that conclusion is true or not, and between existence and non-existence there is no middle term. “Being only is and nothing is altogether not,” said Parmenides; and this is in strict accordance with the analysis of the conception of reality given in the last paper. For we found that the distinction of reality and fiction depends on the supposition that sufficient investigation would cause one opinion to be universally received and all others to be rejected. That presupposition, involved in the very conceptions of reality and figment, involves a complete sundering of the two. It is the heaven-and-hell idea in the domain of thought. But, in the long run, there is a real fact which corresponds to the idea of probability, and it is that a given mode of inference sometimes proves successful and sometimes not, and that in a ratio ultimately fixed. As we go on drawing inference after inference of the given kind, during the first ten or hundred cases the ratio of successes may be expected to show considerable fluctuations; but when we come into the thousands and millions, these fluctuations become less and less; and if we continue long enough, the ratio will approximate toward a fixed limit. We may, therefore, define the probability of a mode of argument as the proportion of cases in which it carries truth with it.
The inference from the premise, A, to the conclusion, B, depends, as we have seen, on the guiding principle, that if a fact of the class A is true, a fact of the class B is true. The probability consists of the fraction whose numerator is the number of times in which both A and B are true, and whose denominator is the total number of times in which A is true, whether B is so or not. Instead of speaking of this as the probability of the inference, there is not the slightest objection to calling it the probability that, if A happens, B happens. But to speak of the probability of the event B, without naming the condition, really has no meaning at all. It is true that when it is perfectly obvious what condition is meant, the ellipsis may be permitted. But we should avoid contracting the habit of using language in this way (universal as the habit is), because it gives rise to a vague way of thinking, as if the action of causation might either determine an event to happen or determine it not to happen, or leave it more or less free to happen or not, so as to give rise to an inherent chance in regard to its occurrence.[[37]] It is quite clear to me that some of the worst and most persistent errors in the use of the doctrine of chances have arisen from this vicious mode of expression.[[38]]
IV
But there remains an important point to be cleared up. According to what has been said, the idea of probability essentially belongs to a kind of inference which is repeated indefinitely. An individual inference must be either true or false, and can show no effect of probability; and, therefore, in reference to a single case considered in itself, probability can have no meaning. Yet if a man had to choose between drawing a card from a pack containing twenty-five red cards and a black one, or from a pack containing twenty-five black cards and a red one, and if the drawing of a red card were destined to transport him to eternal felicity, and that of a black one to consign him to everlasting woe, it would be folly to deny that he ought to prefer the pack containing the larger portion of red cards, although, from the nature of the risk, it could not be repeated. It is not easy to reconcile this with our analysis of the conception of chance. But suppose he should choose the red pack, and should draw the wrong card, what consolation would he have? He might say that he had acted in accordance with reason, but that would only show that his reason was absolutely worthless. And if he should choose the right card, how could he regard it as anything but a happy accident? He could not say that if he had drawn from the other pack, he might have drawn the wrong one, because an hypothetical proposition such as, “if A, then B,” means nothing with reference to a single case. Truth consists in the existence of a real fact corresponding to the true proposition. Corresponding to the proposition, “if A, then B,” there may be the fact that whenever such an event as A happens such an event as B happens. But in the case supposed, which has no parallel as far as this man is concerned, there would be no real fact whose existence could give any truth to the statement that, if he had drawn from the other pack, he might have drawn a black card. Indeed, since the validity of an inference consists in the truth of the hypothetical proposition that if the premises be true the conclusion will also be true, and since the only real fact which can correspond to such a proposition is that whenever the antecedent is true the consequent is so also, it follows that there can be no sense in reasoning in an isolated case, at all.
These considerations appear, at first sight, to dispose of the difficulty mentioned. Yet the case of the other side is not yet exhausted. Although probability will probably manifest its effect in, say, a thousand risks, by a certain proportion between the numbers of successes and failures, yet this, as we have seen, is only to say that it certainly will, at length, do so. Now the number of risks, the number of probable inferences, which a man draws in his whole life, is a finite one, and he cannot be absolutely certain that the mean result will accord with the probabilities at all. Taking all his risks collectively, then, it cannot be certain that they will not fail, and his case does not differ, except in degree, from the one last supposed. It is an indubitable result of the theory of probabilities that every gambler, if he continues long enough, must ultimately be ruined. Suppose he tries the martingale, which some believe infallible, and which is, as I am informed, disallowed in the gambling-houses. In this method of playing, he first bets say $1; if he loses it he bets $2; if he loses that he bets $4; if he loses that he bets $8; if he then gains he has lost 1 + 2 + 4 = 7, and he has gained $1 more; and no matter how many bets he loses, the first one he gains will make him $1 richer than he was in the beginning. In that way, he will probably gain at first; but, at last, the time will come when the run of luck is so against him that he will not have money enough to double, and must, therefore, let his bet go. This will probably happen before he has won as much as he had in the first place, so that this run against him will leave him poorer than he began; some time or other it will be sure to happen. It is true that there is always a possibility of his winning any sum the bank can pay, and we thus come upon a celebrated paradox that, though he is certain to be ruined, the value of his expectation calculated according to the usual rules (which omit this consideration) is large. But, whether a gambler plays in this way or any other, the same thing is true, namely, that if he plays long enough he will be sure some time to have such a run against him as to exhaust his entire fortune. The same thing is true of an insurance company. Let the directors take the utmost pains to be independent of great conflagrations and pestilences, their actuaries can tell them that, according to the doctrine of chances, the time must come, at last, when their losses will bring them to a stop. They may tide over such a crisis by extraordinary means, but then they will start again in a weakened state, and the same thing will happen again all the sooner. An actuary might be inclined to deny this, because he knows that the expectation of his company is large, or perhaps (neglecting the interest upon money) is infinite. But calculations of expectations leave out of account the circumstance now under consideration, which reverses the whole thing. However, I must not be understood as saying that insurance is on this account unsound, more than other kinds of business. All human affairs rest upon probabilities, and the same thing is true everywhere. If man were immortal he could be perfectly sure of seeing the day when everything in which he had trusted should betray his trust, and, in short, of coming eventually to hopeless misery. He would break down, at last, as every good fortune, as every dynasty, as every civilization does. In place of this we have death.
But what, without death, would happen to every man, with death must happen to some man. At the same time, death makes the number of our risks, of our inferences, finite, and so makes their mean result uncertain. The very idea of probability and of reasoning rests on the assumption that this number is indefinitely great. We are thus landed in the same difficulty as before, and I can see but one solution of it. It seems to me that we are driven to this, that logicality inexorably requires that our interests shall not be limited. They must not stop at our own fate, but must embrace the whole community. This community, again, must not be limited, but must extend to all races of beings with whom we can come into immediate or mediate intellectual relation. It must reach, however vaguely, beyond this geological epoch, beyond all bounds. He who would not sacrifice his own soul to save the whole world, is, as it seems to me, illogical in all his inferences, collectively. Logic is rooted in the social principle.
To be logical men should not be selfish; and, in point of fact, they are not so selfish as they are thought. The willful prosecution of one’s desires is a different thing from selfishness. The miser is not selfish; his money does him no good, and he cares for what shall become of it after his death. We are constantly speaking of our possessions on the Pacific, and of our destiny as a republic, where no personal interests are involved, in a way which shows that we have wider ones. We discuss with anxiety the possible exhaustion of coal in some hundreds of years, or the cooling-off of the sun in some millions, and show in the most popular of all religious tenets that we can conceive the possibility of a man’s descending into hell for the salvation of his fellows.
Now, it is not necessary for logicality that a man should himself be capable of the heroism of self-sacrifice. It is sufficient that he should recognize the possibility of it, should perceive that only that man’s inferences who has it are really logical, and should consequently regard his own as being only so far valid as they would be accepted by the hero. So far as he thus refers his inferences to that standard, he becomes identified with such a mind.
This makes logicality attainable enough. Sometimes we can personally attain to heroism. The soldier who runs to scale a wall knows that he will probably be shot, but that is not all he cares for. He also knows that if all the regiment, with whom in feeling he identifies himself, rush forward at once, the fort will be taken. In other cases we can only imitate the virtue. The man whom we have supposed as having to draw from the two packs, who if he is not a logician will draw from the red pack from mere habit, will see, if he is logician enough, that he cannot be logical so long as he is concerned only with his own fate, but that that man who should care equally for what was to happen in all possible cases of the sort could act logically, and would draw from the pack with the most red cards, and thus, though incapable himself of such sublimity, our logician would imitate the effect of that man’s courage in order to share his logicality.
But all this requires a conceived identification of one’s interests with those of an unlimited community. Now, there exist no reasons, and a later discussion will show that there can be no reasons, for thinking that the human race, or any intellectual race, will exist forever. On the other hand, there can be no reason against it;[[39]] and, fortunately, as the whole requirement is that we should have certain sentiments, there is nothing in the facts to forbid our having a hope, or calm and cheerful wish, that the community may last beyond any assignable date.
It may seem strange that I should put forward three sentiments, namely, interest in an indefinite community, recognition of the possibility of this interest being made supreme, and hope in the unlimited continuance of intellectual activity, as indispensable requirements of logic. Yet, when we consider that logic depends on a mere struggle to escape doubt, which, as it terminates in action, must begin in emotion, and that, furthermore, the only cause of our planting ourselves on reason is that other methods of escaping doubt fail on account of the social impulse, why should we wonder to find social sentiment presupposed in reasoning? As for the other two sentiments which I find necessary, they are so only as supports and accessories of that. It interests me to notice that these three sentiments seem to be pretty much the same as that famous trio of Charity, Faith, and Hope, which, in the estimation of St. Paul, are the finest and greatest of spiritual gifts. Neither Old nor New Testament is a textbook of the logic of science, but the latter is certainly the highest existing authority in regard to the dispositions of heart which a man ought to have.
V
Such average statistical numbers as the number of inhabitants per square mile, the average number of deaths per week, the number of convictions per indictment, or, generally speaking, the numbers of x’s per y, where the x’s are a class of things some or all of which are connected with another class of things, their y’s, I term relative numbers. Of the two classes of things to which a relative number refers, that one of which it is a number may be called its relate, and that one per which the numeration is made may be called its correlate.
Probability is a kind of relative number; namely, it is the ratio of the number of arguments of a certain genus which carry truth with them to the total number of arguments of that genus, and the rules for the calculation of probabilities are very easily derived from this consideration. They may all be given here, since they are extremely simple, and it is sometimes convenient to know something of the elementary rules of calculation of chances.
Rule I. Direct Calculation.—To calculate, directly, any relative number, say for instance the number of passengers in the average trip of a street-car, we must proceed as follows:
Count the number of passengers for each trip; add all these numbers, and divide by the number of trips. There are cases in which this rule may be simplified. Suppose we wish to know the number of inhabitants to a dwelling in New York. The same person cannot inhabit two dwellings. If he divide his time between two dwellings he ought to be counted a half-inhabitant of each. In this case we have only to divide the total number of the inhabitants of New York by the number of their dwellings, without the necessity of counting separately those which inhabit each one. A similar proceeding will apply wherever each individual relate belongs to one individual correlate exclusively. If we want the number of x’s per y, and no x belongs to more than one y, we have only to divide the whole number of x’s of y’s by the number of y’s. Such a method would, of course, fail if applied to finding the average number of street-car passengers per trip. We could not divide the total number of travelers by the number of trips, since many of them would have made many passages.
To find the probability that from a given class of premises, A, a given class of conclusions, B, follow, it is simply necessary to ascertain what proportion of the times in which premises of that class are true, the appropriate conclusions are also true. In other words, it is the number of cases of the occurrence of both the events A and B, divided by the total number of cases of the occurrence of the event A.
Rule II. Addition of Relative Numbers.—Given two relative numbers having the same correlate, say the number of x’s per y, and the number of z’s per y; it is required to find the number of x’s and z’s together per y. If there is nothing which is at once an x and a z to the same y, the sum of the two given numbers would give the required number. Suppose, for example, that we had given the average number of friends that men have, and the average number of enemies, the sum of these two is the average number of persons interested in a man. On the other hand, it plainly would not do to add the average number of persons having constitutional diseases and over military age, to the average number exempted by each special cause from military service, in order to get the average number exempt in any way, since many are exempt in two or more ways at once.
This rule applies directly to probabilities, given the probability that two different and mutually exclusive events will happen under the same supposed set of circumstances. Given, for instance, the probability that if A then B, and also the probability that if A then C, then the sum of these two probabilities is the probability that if A then either B or C, so long as there is no event which belongs at once to the two classes B and C.
Rule III. Multiplication of Relative Numbers.—Suppose that we have given the relative number of x’s per y; also the relative number of z’s per x of y; or, to take a concrete example, suppose that we have given, first, the average number of children in families living in New York; and, second, the average number of teeth in the head of a New York child—then the product of these two numbers would give the average number of children’s teeth in a New York family. But this mode of reckoning will only apply in general under two restrictions. In the first place, it would not be true if the same child could belong to different families, for in that case those children who belonged to several different families might have an exceptionally large or small number of teeth, which would affect the average number of children’s teeth in a family more than it would affect the average number of teeth in a child’s head. In the second place, the rule would not be true if different children could share the same teeth, the average number of children’s teeth being in that case evidently something different from the average number of teeth belonging to a child.
In order to apply this rule to probabilities, we must proceed as follows: Suppose that we have given the probability that the conclusion B follows from the premise A, B and A representing as usual certain classes of propositions. Suppose that we also knew the probability of an inference in which B should be the premise, and a proposition of a third kind, C, the conclusion. Here, then, we have the materials for the application of this rule. We have, first, the relative number of B’s per A. We next should have the relative number of C’s per B following from A. But the classes of propositions being so selected that the probability of C following from any B in general is just the same as the probability of C’s following from one of those B’s which is deducible from an A, the two probabilities may be multiplied together, in order to give the probability of C following from A. The same restrictions exist as before. It might happen that the probability that B follows from A was affected by certain propositions of the class B following from several different propositions of the class A. But, practically speaking, all these restrictions are of very little consequence, and it is usually recognized as a principle universally true that the probability that, if A is true, B is, multiplied by the probability that, if B is true, C is, gives the probability that, if A is true, C is.
There is a rule supplementary to this, of which great use is made. It is not universally valid, and the greatest caution has to be exercised in making use of it—a double care, first, never to use it when it will involve serious error; and, second, never to fail to take advantage of it in cases in which it can be employed. This rule depends upon the fact that in very many cases the probability that C is true if B is, is substantially the same as the probability that C is true if A is. Suppose, for example, we have the average number of males among the children born in New York; suppose that we also have the average number of children born in the winter months among those born in New York. Now, we may assume without doubt, at least as a closely approximate proposition (and no very nice calculation would be in place in regard to probabilities), that the proportion of males among all the children born in New York is the same as the proportion of males born in summer in New York; and, therefore, if the names of all the children born during a year were put into an urn, we might multiply the probability that any name drawn would be the name of a male child by the probability that it would be the name of a child born in summer, in order to obtain the probability that it would be the name of a male child born in summer. The questions of probability, in the treatises upon the subject, have usually been such as relate to balls drawn from urns, and games of cards, and so on, in which the question of the independence of events, as it is called—that is to say, the question of whether the probability of C, under the hypothesis B, is the same as its probability under the hypothesis A, has been very simple; but, in the application of probabilities to the ordinary questions of life, it is often an exceedingly nice question whether two events may be considered as independent with sufficient accuracy or not. In all calculations about cards it is assumed that the cards are thoroughly shuffled, which makes one deal quite independent of another. In point of fact the cards seldom are, in practice, shuffled sufficiently to make this true; thus, in a game of whist, in which the cards have fallen in suits of four of the same suit, and are so gathered up, they will lie more or less in sets of four of the same suit, and this will be true even after they are shuffled. At least some traces of this arrangement will remain, in consequence of which the number of “short suits,” as they are called—that is to say, the number of hands in which the cards are very unequally divided in regard to suits—is smaller than the calculation would make it to be; so that, when there is a misdeal, where the cards, being thrown about the table, get very thoroughly shuffled, it is a common saying that in the hands next dealt out there are generally short suits. A few years ago a friend of mine, who plays whist a great deal, was so good as to count the number of spades dealt to him in 165 hands, in which the cards had been, if anything, shuffled better than usual. According to calculation, there should have been 85 of these hands in which my friend held either three or four spades, but in point of fact there were 94, showing the influence of imperfect shuffling.
According to the view here taken, these are the only fundamental rules for the calculation of chances. An additional one, derived from a different conception of probability, is given in some treatises, which if it be sound might be made the basis of a theory of reasoning. Being, as I believe it is, absolutely absurd, the consideration of it serves to bring us to the true theory; and it is for the sake of this discussion, which must be postponed to the next number, that I have brought the doctrine of chances to the reader’s attention at this early stage of our studies of the logic of science.
FOURTH PAPER
THE PROBABILITY OF INDUCTION[[40]]
I
We have found that every argument derives its force from the general truth of the class of inferences to which it belongs; and that probability is the proportion of arguments carrying truth with them among those of any genus. This is most conveniently expressed in the nomenclature of the medieval logicians. They called the fact expressed by a premise an antecedent, and that which follows from it its consequent; while the leading principle, that every (or almost every) such antecedent is followed by such a consequent, they termed the consequence. Using this language, we may say that probability belongs exclusively to consequences, and the probability of any consequence is the number of times in which antecedent and consequent both occur divided by the number of all the times in which the antecedent occurs. From this definition are deduced the following rules for the addition and multiplication of probabilities:
Rule for the Addition of Probabilities.—Given the separate probabilities of two consequences having the same antecedent and incompatible consequents. Then the sum of these two numbers is the probability of the consequence, that from the same antecedent one or other of those consequents follows.
Rule for the Multiplication of Probabilities.—Given the separate probabilities of the two consequences, “If A then B,” and “If both A and B, then C.” Then the product of these two numbers is the probability of the consequence, “If A, then both B and C.”
Special Rule for the Multiplication of Independent Probabilities.—Given the separate probabilities of two consequences having the same antecedents, “If A, then B,” and “If A, then C.” Suppose that these consequences are such that the probability of the second is equal to the probability of the consequence, “If both A and B, then C.” Then the product of the two given numbers is equal to the probability of the consequence, “If A, then both B and C.”
To show the working of these rules we may examine the probabilities in regard to throwing dice. What is the probability of throwing a six with one die? The antecedent here is the event of throwing a die; the consequent, its turning up a six. As the die has six sides, all of which are turned up with equal frequency, the probability of turning up any one is 1/6. Suppose two dice are thrown, what is the probability of throwing sixes? The probability of either coming up six is obviously the same when both are thrown as when one is thrown—namely, 1/6. The probability that either will come up six when the other does is also the same as that of its coming up six whether the other does or not. The probabilities are, therefore, independent; and, by our rule, the probability that both events will happen together is the product of their several probabilities, or 1/6 x 1/6. What is the probability of throwing deuce-ace? The probability that the first die will turn up ace and the second deuce is the same as the probability that both will turn up sixes—namely, 1/36; the probability that the second will turn up ace and the first deuce is likewise 1/36; these two events—first, ace; second, deuce; and, second, ace; first, deuce—are incompatible. Hence the rule for addition holds, and the probability that either will come up ace and the other deuce is 1/36 + 1/36, or 1/18.
In this way all problems about dice, etc., may be solved. When the number of dice thrown is supposed very large, mathematics (which may be defined as the art of making groups to facilitate numeration) comes to our aid with certain devices to reduce the difficulties.
II
The conception of probability as a matter of fact, i.e., as the proportion of times in which an occurrence of one kind is accompanied by an occurrence of another kind, is termed by Mr. Venn the materialistic view of the subject. But probability has often been regarded as being simply the degree of belief which ought to attach to a proposition, and this mode of explaining the idea is termed by Venn the conceptualistic view. Most writers have mixed the two conceptions together. They, first, define the probability of an event as the reason we have to believe that it has taken place, which is conceptualistic; but shortly after they state that it is the ratio of the number of cases favorable to the event to the total number of cases favorable or contrary, and all equally possible. Except that this introduces the thoroughly unclear idea of cases equally possible in place of cases equally frequent, this is a tolerable statement of the materialistic view. The pure conceptualistic theory has been best expounded by Mr. De Morgan in his Formal Logic: or, the Calculus of Inference, Necessary and Probable.
The great difference between the two analyses is, that the conceptualists refer probability to an event, while the materialists make it the ratio of frequency of events of a species to those of a genus over that species, thus giving it two terms instead of one. The opposition may be made to appear as follows:
Suppose that we have two rules of inference, such that, of all the questions to the solution of which both can be applied, the first yields correct answers to 81/100, and incorrect answers to the remaining 19/100; while the second yields correct answers to 93/100, and incorrect answers to the remaining 7/100. Suppose, further, that the two rules are entirely independent as to their truth, so that the second answers correctly 93/100 of the questions which the first answers correctly, and also 93/100 of the questions which the first answers incorrectly, and answers incorrectly the remaining 7/100 of the questions which the first answers correctly, and also the remaining 7/100 of the questions which the first answers incorrectly. Then, of all the questions to the solution of which both rules can be applied—
both answer correctly 93/100 of 81/100 or 93/100 x 81/100;
the second answers correctly and the first incorrectly 93/100 of 19/100 or 93/100 x 19/100;
the second answers incorrectly and the first correctly 7/100 of 81/100 or 7/100 x 81/100;
and both answer incorrectly 7/100 of 19/100 or 7/100 x 19/100;
Suppose, now, that, in reference to any question, both give the same answer. Then (the questions being always such as are to be answered by yes or no), those in reference to which their answers agree are the same as those which both answer correctly together with those which both answer falsely, or 93/100 x 81/100 + 7/100 x 19/100 of all. The proportion of those which both answer correctly out of those their answers to which agree is, therefore—
((93 × 81)/(100 × 100))/((93 × 81)/(100 × 100)) + ((7 × 19)/(100 × 100)) or (93 × 81)/((93 × 81) + (7 × 19)).
This is, therefore, the probability that, if both modes of inference yield the same result, that result is correct. We may here conveniently make use of another mode of expression. Probability is the ratio of the favorable cases to all the cases. Instead of expressing our result in terms of this ratio, we may make use of another—the ratio of favorable to unfavorable cases. This last ratio may be called the chance of an event. Then the chance of a true answer by the first mode of inference is 81/19 and by the second is 93/7; and the chance of a correct answer from both, when they agree, is—
(81 × 93)/(19 × 7) or 81/19 × 93/7,
or the product of the chances of each singly yielding a true answer.
It will be seen that a chance is a quantity which may have any magnitude, however great. An event in whose favor there is an even chance, or 1/1, has a probability of 1/2. An argument having an even chance can do nothing toward re-enforcing others, since according to the rule its combination with another would only multiply the chance of the latter by 1.
Probability and chance undoubtedly belong primarily to consequences, and are relative to premises; but we may, nevertheless, speak of the chance of an event absolutely, meaning by that the chance of the combination of all arguments in reference to it which exist for us in the given state of our knowledge. Taken in this sense it is incontestable that the chance of an event has an intimate connection with the degree of our belief in it. Belief is certainly something more than a mere feeling; yet there is a feeling of believing, and this feeling does and ought to vary with the chance of the thing believed, as deduced from all the arguments. Any quantity which varies with the chance might, therefore, it would seem, serve as a thermometer for the proper intensity of belief. Among all such quantities there is one which is peculiarly appropriate. When there is a very great chance, the feeling of belief ought to be very intense. Absolute certainty, or an infinite chance, can never be attained by mortals, and this may be represented appropriately by an infinite belief. As the chance diminishes the feeling of believing should diminish, until an even chance is reached, where it should completely vanish and not incline either toward or away from the proposition. When the chance becomes less, then a contrary belief should spring up and should increase in intensity as the chance diminishes, and as the chance almost vanishes (which it can never quite do) the contrary belief should tend toward an infinite intensity. Now, there is one quantity which, more simply than any other, fulfills these conditions; it is the logarithm of the chance. But there is another consideration which must, if admitted, fix us to this choice for our thermometer. It is that our belief ought to be proportional to the weight of evidence, in this sense, that two arguments which are entirely independent, neither weakening nor strengthening each other, ought, when they concur, to produce a belief equal to the sum of the intensities of belief which either would produce separately. Now, we have seen that the chances of independent concurrent arguments are to be multiplied together to get the chance of their combination, and, therefore, the quantities which best express the intensities of belief should be such that they are to be added when the chances are multiplied in order to produce the quantity which corresponds to the combined chance. Now, the logarithm is the only quantity which fulfills this condition. There is a general law of sensibility, called Fechner’s psychophysical law. It is that the intensity of any sensation is proportional to the logarithm of the external force which produces it. It is entirely in harmony with this law that the feeling of belief should be as the logarithm of the chance, this latter being the expression of the state of facts which produces the belief.
The rule for the combination of independent concurrent arguments takes a very simple form when expressed in terms of the intensity of belief, measured in the proposed way. It is this: Take the sum of all the feelings of belief which would be produced separately by all the arguments pro, subtract from that the similar sum for arguments con, and the remainder is the feeling of belief which we ought to have on the whole. This is a proceeding which men often resort to, under the name of balancing reasons.
These considerations constitute an argument in favor of the conceptualistic view. The kernel of it is that the conjoint probability of all the arguments in our possession, with reference to any fact, must be intimately connected with the just degree of our belief in that fact; and this point is supplemented by various others showing the consistency of the theory with itself and with the rest of our knowledge.
But probability, to have any value at all, must express a fact. It is, therefore, a thing to be inferred upon evidence. Let us, then, consider for a moment the formation of a belief of probability. Suppose we have a large bag of beans from which one has been secretly taken at random and hidden under a thimble. We are now to form a probable judgment of the color of that bean, by drawing others singly from the bag and looking at them, each one to be thrown back, and the whole well mixed up after each drawing. Suppose the first drawing is white and the next black. We conclude that there is not an immense preponderance of either color, and that there is something like an even chance that the bean under the thimble is black. But this judgment may be altered by the next few drawings. When we have drawn ten times, if 4, 5, or 6, are white, we have more confidence that the chance is even. When we have drawn a thousand times, if about half have been white, we have great confidence in this result. We now feel pretty sure that, if we were to make a large number of bets upon the color of single beans drawn from the bag, we could approximately insure ourselves in the long run by betting each time upon the white, a confidence which would be entirely wanting if, instead of sampling the bag by 1,000 drawings, we had done so by only two. Now, as the whole utility of probability is to insure us in the long run, and as that assurance depends, not merely on the value of the chance, but also on the accuracy of the evaluation, it follows that we ought not to have the same feeling of belief in reference to all events of which the chance is even. In short, to express the proper state of our belief, not one number but two are requisite, the first depending on the inferred probability, the second on the amount of knowledge on which that probability is based.[[41]] It is true that when our knowledge is very precise, when we have made many drawings from the bag, or, as in most of the examples in the books, when the total contents of the bag are absolutely known, the number which expresses the uncertainty of the assumed probability and its liability to be changed by further experience may become insignificant, or utterly vanish. But, when our knowledge is very slight, this number may be even more important than the probability itself; and when we have no knowledge at all this completely overwhelms the other, so that there is no sense in saying that the chance of the totally unknown event is even (for what expresses absolutely no fact has absolutely no meaning), and what ought to be said is that the chance is entirely indefinite. We thus perceive that the conceptualistic view, though answering well enough in some cases, is quite inadequate.
Suppose that the first bean which we drew from our bag were black. That would constitute an argument, no matter how slender, that the bean under the thimble was also black. If the second bean were also to turn out black, that would be a second independent argument reënforcing the first. If the whole of the first twenty beans drawn should prove black, our confidence that the hidden bean was black would justly attain considerable strength. But suppose the twenty-first bean were to be white and that we were to go on drawing until we found that we had drawn 1,010 black beans and 990 white ones. We should conclude that our first twenty beans being black was simply an extraordinary accident, and that in fact the proportion of white beans to black was sensibly equal, and that it was an even chance that the hidden bean was black. Yet according to the rule of balancing reasons, since all the drawings of black beans are so many independent arguments in favor of the one under the thimble being black, and all the white drawings so many against it, an excess of twenty black beans ought to produce the same degree of belief that the hidden bean was black, whatever the total number drawn.
In the conceptualistic view of probability, complete ignorance, where the judgment ought not to swerve either toward or away from the hypothesis, is represented by the probability 1/2.[[42]]
But let us suppose that we are totally ignorant what colored hair the inhabitants of Saturn have. Let us, then, take a color-chart in which all possible colors are shown shading into one another by imperceptible degrees. In such a chart the relative areas occupied by different classes of colors are perfectly arbitrary. Let us inclose such an area with a closed line, and ask what is the chance on conceptualistic principles that the color of the hair of the inhabitants of Saturn falls within that area? The answer cannot be indeterminate because we must be in some state of belief; and, indeed, conceptualistic writers do not admit indeterminate probabilities. As there is no certainty in the matter, the answer lies between zero and unity. As no numerical value is afforded by the data, the number must be determined by the nature of the scale of probability itself, and not by calculation from the data. The answer can, therefore, only be one-half, since the judgment should neither favor nor oppose the hypothesis. What is true of this area is true of any other one; and it will equally be true of a third area which embraces the other two. But the probability for each of the smaller areas being one-half, that for the larger should be at least unity, which is absurd.