Rational basis test

“ The test of the degree in which the science affords safe ground for predicting (and consequently for practically dealing with) what has not yet happened, is the degree in which it would have enabled us to predict what has actually occurred. Before our theory of the influence of a particular cause, in a given state of circumstances, can be entirely trusted, we must be able to explain and account for the existing state of all that portion of the social phenomena which that cause has a tendency to influence. ”

“ We find that every fresh assumption of the postulate involves some risk of error; and, indeed, where the calculation is extremely intricate, and the assumptions therefore extremely numerous, our experience teaches us that the probability that there has been a wrong assumption is greater than the probability that there has not. So too in argument. We lose faith in a long series of steps, however logical they may seem, unless we can test the inference by appeal to fact—that is, unless we can get at the inference by a single use of the postulate. ”

“ An empirical law, then, is an observed uniformity, presumed to be resolvable into simpler laws, but not yet resolved into them. The ascertainment of the empirical laws of phenomena often precedes by a long interval the explanation of those laws by the Deductive Method ”

“ While, on the one hand, no finite number of accordances will give entire certainty, the probability of the hypothesis will increase very rapidly with the number of accordances.517 Almost every problem in science thus takes the form of a balance of probabilities. It is only when difficulty after difficulty has been successfully explained away, and decisive experimenta crucis have, time after time, resulted in favour of our theory, that we can venture to assert the falsity of all objections.
The sole real test of an hypothesis is its accordance with fact. ”

“ If, from observation and experiment, we can conclude to one new case, so may we to an indefinite number. If that which has held true in our past experience will therefore hold in time to come, it will hold not merely in some individual case, but in all cases of some given description. Every induction, therefore, which suffices to prove one fact, proves an indefinite multitude of facts: the experience which justifies a single prediction must be such as will suffice to bear out a general theorem. This theorem it is extremely important to ascertain and declare, in its broadest form of generality ”

“ What takes place in mathematics is thus no argument that the comparison is between the ideas only. It is always, either indirectly or directly, a comparison of the phenomena.
In cases in which we cannot bring the phenomena to the test of direct inspection at all, or not in a manner sufficiently precise, but must judge of their resemblance by inference from other resemblances or dissimilarities more accessible to observation, we of course require, as in all cases of ratiocination, generalizations or formulæ applicable to the subject. We must reason from laws of nature ”

“ The empirical law derives whatever truth it has, from the causal laws of which it is a consequence. If we know those laws, we know what are the limits to the derivative law: while, if we have not yet accounted for the empirical law—if it rest only on observation—there is no safety in applying it far beyond the limits of time, place, and circumstance, in which the observations were made. ”

“ It needs scarcely be observed, that,—if direct observation and collation of instances have furnished us with any empirical laws of the effect, whether true in all observed cases or only true for the most part,—the most effectual verification of which the theory could be susceptible would be, that it led deductively to those empirical laws; that the uniformities, whether complete or incomplete, which were observed to exist among the phenomena, were accounted for by the laws of the causes—were such as could not but exist if those be really the causes by which the phenomena are produced. ”

“ When a chemist announces the existence and properties of a newly-discovered substance, if we confide in his accuracy, we feel assured that the conclusions he has arrived at will hold universally, though the induction be founded but on a single instance. We do not withhold our assent, waiting for a repetition of the experiment; or if we do, it is from a doubt whether the one experiment was properly made, not whether if properly made it would be conclusive. Here, then, is a general law of nature, inferred without hesitation from a single instance; a universal proposition from a singular one. ”

“ Scientific inquirers give the name of Empirical Laws to those uniformities which observation or experiment has shown to exist, but on which they hesitate to rely in cases varying much from those which have been actually observed, for want of seeing any reason why such a law should exist. It is implied, therefore, in the notion of an empirical law, that it is not an ultimate law; that if true at all, its truth is capable of being, and requires to be, accounted for. ”

“ Since there is no case of legitimate inference from experience, in which the conclusion may not legitimately be a general proposition; an analysis of the process by which general truths are arrived at, is virtually an analysis of all induction whatever. Whether we are inquiring into a scientific principle or into an individual fact, and whether we proceed by experiment or by ratiocination, every step in the train of inferences is essentially inductive, and the legitimacy of the induction depends in both cases on the same conditions. ”

“ It is necessary, no doubt, to assure ourselves, by repetition of the observation or experiment, that no error has been committed concerning the individual facts observed; and until we have assured ourselves of this, instead of varying the circumstances, we can not too scrupulously repeat the same experiment or observation without any change. But when once this assurance has been obtained, the multiplication of instances which do not exclude any more circumstances is entirely useless, provided there have been already enough to exclude the supposition of Plurality of Causes. ”

“ In like manner, every because and every therefore, used in a mathematical argument, connotes a proposition of which the terms are absolutely coherent in the mode alleged: the proof being that an attempt to bring together in consciousness the terms of the opposite proposition is futile. And this method of testing, alike the fundamental propositions and all members of the fabrics of propositions raised upon them, is consistently pursued in verifying the conclusion. Inference and observation are compared; and when they agree, it is held inconceivable that the inference is other than true. ”

“ We may say that we know some principle a-priori, without the need of empirical verification, e.g. that our sense-data have causes other than themselves, and that something can be known about these causes by inference from their effects. This way has been often adopted by philosophers. It may be necessary to adopt this way to some extent, but in so far as it is adopted physics ceases to be empirical or based upon experiment and observation alone. ”

“ In statistics, it is evident that empirical laws may sometimes be traced; and the tracing them forms an important part of that system of indirect observation on which we must often rely for the data of the Deductive Science. The process of the science consists in inferring effects from their causes; but we have often no means of observing the causes, except through the medium of their effects. In such cases the deductive science is unable to predict the effects, for want of the necessary data ”

“ Some such principle as this is required if the method of simple enumeration is to be valid.
But this brings us to our other question, namely, how is our principle known to be true? Obviously, since it is required to justify induction, it cannot be proved by induction; since it goes beyond the empirical data, it cannot be proved by them alone; since it is required to justify all inferences from empirical data to what goes beyond them, it cannot itself be even rendered in any degree probable by such data. ”

“ There are of course no means of applying such a test as that above described, unless we already possess a general knowledge of the prevalent character of the uniformities existing throughout nature. The indispensable foundation, therefore, of a scientific formula of induction, must be a survey of the inductions to which mankind have been conducted in unscientific practice ”

“ The reason is, that in all these classes of facts, the causes are either not amenable to direct observation at all, or not with the requisite precision, and we have no means of judging of their frequency except from the empirical law afforded by the frequency of the effects. The inference does not the less depend on causation alone. We reason from an effect to a similar effect by passing through the cause. ”

“ Coming more nearly to our own immediate subject, exactly the same difficulty exists in determining the law which certain things obey. Given a general mathematical expression, we can infallibly ascertain its value for any required value of the variable. But I am not aware that mathematicians have ever attempted to lay down the rules of a process by which, having given certain numbers, one might discover a rational or precise formula from which they proceed. ”

“ It deserves particular notice, that the theorem or speculative truth is not ripe for being turned into a precept, until the whole, and not a part merely, of the operation which belongs to science, has been performed. Suppose that we have completed the scientific process only up to a certain point; have discovered that a particular cause will produce the desired effect, but have not ascertained all the negative conditions which are necessary, that is, all the circumstances which, if present, would prevent its production. ”

“ Considering, however, in how accelerating a ratio the uncertainty of our conclusions increases, as we attempt to take the effect of a greater number of concurrent causes into our calculations; the hypothetical combinations of circumstances on which we construct the general theorems of the science, cannot be made very complex, without so rapidly-accumulating a liability to error as must soon deprive our conclusions of all value. ”

“ The whole process of scientific systematisation involves certain assumptions as to the ultimate nature of the facts which are systematised. Thus the very performance of an experiment for the purpose of verifying a suggested hypothesis involves the assumption that the facts with which the hypothesis is concerned conform to general laws, and that these laws are such as to be capable of formulation by human intelligence. If “nature” is not in some sense “uniform,” the conclusive force of a successful experiment is logically nil. ”

“ It is no disparagement, therefore, to the science of Human Nature, that those of its general propositions which descend sufficiently into detail to serve as a foundation for predicting phenomena in the concrete, are for the most part only approximately true. But in order to give a genuinely scientific character to the study, it is indispensable that these approximate generalizations, which in themselves would amount only to the lowest kind of empirical laws, should be connected deductively with the laws of nature from which they result ”

“ It is hardly necessary again to repeat, that, as in every other deductive science, verification à posteriori must proceed pari passu with deduction à priori. The inference given by theory as to the type of character which would be formed by any given circumstances, must be tested by specific experience of those circumstances whenever obtainable; and the conclusions of the science as a whole, must undergo a perpetual verification and correction from the general remarks afforded by common experience respecting human nature in our own age, and by history respecting times gone by. ”

“ But to extend our inference far beyond the limits of experience is exceedingly unsafe. Our knowledge is at the best only approximate, and takes no account of small tendencies. Now it usually happens that tendencies small within our limits of observation become perceptible or great under extreme circumstances. When the variable in our empirical formula is small, we are justified in overlooking the higher powers, and taking only two or three lower powers. But as the variable increases, the higher powers gain in importance, and in time yield the principal part of the value of the function. ”

“ To make sure, as far as possible, that it shall not be omitted, we make the testing operation a part of the reasoning process itself. We insist that the inference from particulars to particulars shall pass through a general proposition. But this is a security for good reasoning, not a condition of all reasoning; and in some cases not even a security. Our most familiar inferences are all made before we learn the use of general propositions ”

“ Such laws, which hold good, not for phenomena in their entirety, but for certain isolated groups of facts under narrowed conditions, are called laws of the factors of phenomena. And since the complexity of phenomena is such that it is [62] difficult for the human mind to grasp all the interlacing threads of causation at a single glance, men of science have endeavoured to isolate their several strands, and, applying the principle of analysis, without which reasoning is impossible, to separate out the factors and determine their laws. ”