The Principles of Psychology, Volume 1 · William James
THE MEASURE OF DISCRIMINATIVE SENSIBILITY.
"How much stronger or weaker one sensation is than another, we are never able to say. Whether the sun be a hundred or a thousand times brighter than the moon, a cannon a hundred or a thousand times louder than a pistol, is beyond our power to estimate. The natural measure of sensation which we possess enables us to judge of the equality, of the 'more' and of the 'less,' but not of 'how many times more or less.' This natural measure is, therefore, as good as no measure at all, whenever it becomes a question of accurately ascertaining intensities in the sensational sphere. Even though it may teach us in a general way that with the strength of the outward physical stimulus the strength of the concomitant sensation waxes or wanes, still it leaves us without the slightest knowledge of whether the sensation varies in exactly the same proportion as the stimulus itself, or at a slower or a more rapid rate. In a word, we know by our natural sensibility nothing of the _law_ that connects the sensation and its outward cause together. To find this law we must first find an exact measure for the sensation itself; we must be able to say: A stimulus of strength _one_ begets a sensation of strength _one_; a stimulus of strength _two_ begets a sensation of strength _two_, or _three_, or _four_, etc. But to do this we must first know what a sensation two, three, or four times greater than another signifies....
"Space magnitudes we soon learn to determine exactly because we only measure one space against another. The measure of mental magnitudes is far more difficult.... But the problem of measuring the magnitude of _sensations_ is the first step in the bold enterprise of making mental magnitudes altogether subject to exact measurement.... Were our whole knowledge limited to the fact that the sensation rises when the stimulus rises, and falls when the latter falls, much would not be gained. But even immediate unaided observation teaches us certain facts which, at least in a general way, suggest the law according to which the sensations vary with their outward cause. "Every one knows that in the stilly night we hear things unnoticed in the noise of day. The gentle ticking of the clock, the air circulating through the chimney, the cracking of the chairs in the room, and a thousand other slight noises, impress themselves upon our ear. It is equally well known that in the confused hubbub of the streets, or the clamor of a railway, we may lose not only what our neighbor says to us, but even not hear the sound of our own voice. The stars which are brightest at night are invisible by day; and although we see the moon then, she is far paler than at night. Everyone who has had to deal with weights knows that if to a pound in the hand a second pound be added, the difference is immediately felt; whilst if it be added to a hundredweight, we are not aware of the difference at all....
"The sound of the clock, the light of the stars, the pressure of the pound, these are all _stimuli_ to our senses, and stimuli whose outward amount remains the same. What then do these experiences teach? Evidently nothing but this, that one and the same stimulus, according to the circumstances under which it operates, will be felt either more or less intensely, or not felt at all. Of what sort now is the alteration in the circumstances, upon which this alteration in the feeling may depend? On considering the matter closely we see that it is everywhere of one and the same kind. The tick of the clock is a feeble stimulus for our auditory nerve, which we hear plainly when it is alone, but not when it is added to the strong stimulus of the carriage-wheels and other noises of the day. The light of the stars is a stimulus to the eye. But if the stimulation which this light exerts be added to the strong stimulus of daylight, we feel nothing of it, although we feel it distinctly when it unites itself with the feebler stimulation of the twilight. The pound-weight is a stimulus to our skin, which we feel when it joins itself to a preceding stimulus of equal strength, but which vanishes when it is combined with a stimulus a thousand times greater in amount.
"We may therefore lay it down as a general rule that a stimulus, in order to be felt, may be so much the smaller if the already pre-existing stimulation of the organ is small, but must be so much the larger, the greater the pre-existing stimulation is. From this in a general way we can perceive the connection between the stimulus and the feeling it excites. At least thus much appears, that the law of dependence is not as simple a one as might have been expected beforehand. The simplest relation would obviously be that the sensation should increase in identically the same ratio as the stimulus, thus that if a stimulus of strength _one_ occasioned a sensation _one_, a stimulus of _two_ should occasion sensation _two_, stimulus _three_, sensation _three_, etc. But if this simplest of all relations prevailed, a stimulus added to a pre-existing strong stimulus ought to provoke as great an increase of feeling as if it were added to a pre-existing weak stimulus; the light of the stars e.g., ought to make as great an addition to the daylight as it does to the darkness of the nocturnal sky. This we know not to be the case: the stars are invisible by day, the addition they make to our sensation then is unnoticeable, whereas the same addition to our feeling of the twilight is very considerable indeed. So it is clear that the strength of the sensations does not increase in proportion to the amount of the stimuli, but more slowly. And now comes the question, in what proportion does the increase of the sensation grow less as the increase of the stimulus grows greater. To answer this question, every-day experiences do not suffice. We need exact measurements both of the amounts of the various stimuli, and of the intensity of the sensations themselves.
"How to execute these measurements, however, is something which daily experience suggests. To measure the strength of sensations is, as we saw, impossible; we can only measure the difference of sensations. Experience showed us what very unequal differences of sensation might come from equal differences of outward stimulus. But all these experiences expressed themselves in one kind of fact, that the same difference of stimulus could in one case be felt, and in another case not felt at all--a pound felt if added to another pound, but not if added to a hundred-weight.... We can quickest reach a result with our observations if we start with an arbitrary strength of stimulus, notice what sensation it gives us, and then _see how much we can increase the stimulus without making the sensation seem to change_. If we carry out such observations with stimuli of varying absolute amounts, we shall be forced to choose in an equally varying way the amounts of addition to the stimulus which are capable of giving us a just barely perceptible feeling of _more_. A light, to be just perceptible in the twilight need not be near as bright as the starlight; it must be far brighter to be just perceived during the day. If now we institute such observations for all possible strengths of the various stimuli, and note for each strength the amount of addition of the latter required to produce a barely perceptible alteration of sensation, we shall have a series of figures in which is immediately expressed the law according to which the sensation alters when the stimulation is increased...."
Sensation of light, 1/100 Muscular sensation, 1/17 Feeling of pressure, 1/3 Feeling of warmth, 1/3 Feeling of sound, 1/3 "These figures are far from giving as accurate a measure as might be desired. But at least they are fit to convey a general notion of the relative discriminative susceptibility of the different senses.... The important law which gives in so simple a form the relation of the sensation to the stimulus that calls it forth was first discovered by the physiologist Ernst Heinrich Weber to obtain in special cases. Gustav Theodor Fechner first proved it to be a law for all departments of sensation. Psychology owes to him the first comprehensive investigation of sensations from a physical point of view, the first basis of an exact Theory of Sensibility."
So much for a general account of what Fechner calls Weber's law. The 'exactness' of the theory of sensibility to which it leads consists in the supposed fact that it gives the means of representing sensations by numbers. The _unit_ of any kind of sensation will be that increment which, when the stimulus is increased, we can just barely perceive to be added. The total number of units which any given sensation contains will consist of the total number of such increments which may be perceived in passing from no sensation of the kind to a sensation of the present amount. We cannot get at this number directly, but we can, now that we know Weber's law, get at it by means of the physical stimulus of which it is a function. For if we know how much of the stimulus it will take to give a barely perceptible sensation, and then what percentage of addition to the stimulus will constantly give a barely perceptible increment to the sensation, it is at bottom only a question of compound interest to compute, out of the total amount of stimulus which we may be employing at any moment, the number of such increments, or, in other words, of sensational units to which it may give rise. This number bears the same relation to the total stimulus which the time elapsed bears to the capital plus the compound interest accrued.
(1) _The Method of just-discernible Differences._ Take a standard sensation _S_, and add to it until you distinctly feel the addition _d_; then subtract from _S_ + _d_ until you distinctly feel the effect of the subtraction;[448] call the difference here _d'_. The least discernible difference sought is _d_ + _d'_/2; and the ratio of this quantity to the original _S_ (or rather to _S_ + _d_ - _d'_) is what Fechner calls the difference-threshold. _This difference-threshold should be a constant fraction_ (no matter what is the size of _S_) _if Weber's law holds universally true._ The difficulty in applying this method is that we are _so often in doubt_ whether anything has been added to _S_ or not. Furthermore, if we simply take the smallest _d_ about which we are _never_ in doubt or in error, we certainly get our least discernible difference larger than it ought theoretically to be.[449] Of course the _sensibility_ is small when the least discernible difference is large, and _vice versâ_; in other words, it and the difference-threshold are inversely related to each other. (2) _The Method of True and False Cases._ A sensation which is barely greater than another will, on account of accidental errors in a long series of experiments, sometimes be judged equal, and sometimes smaller; i.e., we shall make a certain number of false and a certain number of true judgments about the difference between the two sensations which we are comparing.
"But the larger this difference is, the more the number of the true judgments will increase at the expense of the false ones; or, otherwise expressed, the nearer to unity will be the fraction whose denominator represents the whole number of judgments, and whose numerator represents those which are true. If _m_ is a ratio of this nature, obtained by comparison of two stimuli, _A_ and _B_, we may seek another couple of stimuli, _a_ and _b_, which when compared will give the same ratio of true to false cases."[450] If this were done, and the ratio of _a_ to _b_ then proved to be equal to that of _A_ to _B_, that would prove that pairs of small stimuli and pairs of large stimuli may affect our discriminative sensibility similarly so long as the ratio of the components to each other within each pair is the same. In other words, it would in so far forth prove the Weberian law. Fechner made use of this method to ascertain his own power of discriminating differences of weight, recording no less than 24,576 separate judgments, and computing as a result that his discrimination for the same relative increase of weight was less good in the neighborhood of 500 than of 300 grams, but that after 500 grams it improved up to 3000, which was the highest weight he experimented with.
(3) _The Method of Average Errors_ consists in taking a standard stimulus and then trying to make another one of the same sort exactly equal to it. There will in general be an error whose amount is large when the discriminative sensibility called in play is small, and _vice versâ_. The sum of the errors, no matter whether they be positive or negative, divided by their number, gives the average error. This, when certain corrections are made, is assumed by Fechner to be the 'reciprocal' of the discriminative sensibility in question. It should bear a constant proportion to the stimulus, no matter what the absolute size of the latter may be, if Weber's law hold true. * * * * * These methods deal with just perceptible differences. Delbœuf and Wundt have experimented with larger differences by means of what Wundt calls the _Méthode der mittleren Abstufungen_, and what we may call (4) _The Method of Equal-appearing Intervals._ This consists in so arranging three stimuli in a series that the intervals between the first and the second shall appear equal to that between the second and the third. At first sight there seems to be no direct logical connection between this method and the preceding ones. By them we compare equally _perceptible_ increments of stimulus in different regions of the latter's scale; but by the fourth method we compare increments which strike us as equally _big_. But what we can but just notice as an increment need not appear always of the same bigness after it is noticed. On the contrary, it will appear much bigger when we are dealing with stimuli that are already large.
(5) The method of doubling the _stimulus_ has been employed by Wundt's collaborator, Merkel, who tried to make one stimulus seem just double the other, and then measured the objective relation of the two. The remarks just made apply also to this case. * * * * * So much for the methods. The results differ in the hands of different observers. I will add a few of them, and will take first the _discriminative sensibility to light_. By the first method, Volkmann, Aubert, Masson, Helmholtz, and Kräpelin find figures varying from 1/3 or 1/4 to 1/195 of the original stimulus. The smaller fractional increments are discriminated when the light is already fairly strong, the larger ones when it is weak or intense. That is, the discriminative sensibility is low when weak or overstrong lights are compared, and at its best with a certain medium illumination. It is thus a function of the light's intensity; but throughout a certain range of the latter it keeps constant, and _in so far forth_ Weber's law is verified for light. Absolute figures cannot be given, but Merkel, by method 1, found that Weber's law held good for stimuli (measured by his arbitrary unit) between 96 and 4096, beyond which intensity no experiments were made.[451] König and Brodhun have given measurements by method 1 which cover the most extensive series, and moreover apply to six different colors of light. These experiments (performed in Helmholtz's laboratory, apparently,) ran from an intensity called 1 to one which was 100,000 times as great. From intensity 2000 to 20,000 Weber's law held good; below and above this range discriminative sensibility declined.
The increment discriminated here was the same for all colors of light, and lay (according to the tables) between 1 and 2 per cent of the stimulus.[452] Delbœuf had verified Weber's law for a certain range of luminous intensities by method 4; that is, he had found that the objective intensity of a light which appeared midway between two others was really the geometrical mean of the latter's intensities. But A. Lehmann and afterwards Neiglick, in Wundt's laboratory, found that effects of contrast played so large a part in experiments performed in this way that Delbœuf's results could not be held conclusive. Merkel, repeating the experiments still later, found that the objective intensity of the light which we judge to stand midway between two others neither stands midway nor is a geometric mean. The discrepancy from both figures is enormous, but is least large from the midway figure or arithmetical mean of the two extreme intensities.[453] Finally, the stars have from time immemorial been arranged in 'magnitudes' supposed to differ by equal-seeming intervals. Lately their intensities have been gauged photometrically, and the comparison of the subjective with the objective series has been made. Prof. J. Jastrow is the latest worker in this field. He finds, taking Pickering's Harvard photometric tables as a basis, that the ratio of the average intensity of each 'magnitude' to that below it decreases as we pass from lower to higher magnitudes, showing a uniform departure from Weber's law, if the method of equal-appearing intervals be held to have any direct relevance to the latter.[454]
_Sounds_ are less delicately discriminated in intensity than lights. A certain difficulty has come from disputes as to the measurement of the objective intensity of the stimulus. Earlier inquiries made the perceptible increase of the stimulus to be about 1/3 of the latter. Merkel's latest results of the method of just perceptible differences make it about 3/10 for that part of the scale of intensities during which Weber's law holds good, which is from 20 to 5000 of M.'s arbitrary unit.[455] Below this the fractional increment must be larger. Above it no measurements were made. For _pressure and muscular sense_ we have rather divergent results. Weber found by the method of just-perceptible differences that persons could distinguish an increase of weight of 1/40 when the two weights were successively lifted by the same hand. It took a much larger fraction to be discerned when the weights were laid on a hand which rested on the table. He seems to have verified his results for only two pairs of differing weights,[456] and on this founded his 'law.' Experiments in Hering's laboratory on lifting 11 weights, running from 250 to 2750 grams showed that the least perceptible increment varied from 1/21 for 250 grams to 1/114 for 2500. For 2750 it rose to 1/98 again. Merkel's recent and very careful experiments, in which the finger pressed down the beam of a balance counterweighted by from 25 to 8020 grams, showed that between 200 and 2000 grams a constant fractional increase of about 1/13 was felt when there was no movement of the finger, and of about 1/19 when there was movement. Above and below these limits the discriminative power grew less. It was greater when the pressure was upon one square millimeter of surface than when it was upon seven.[457]
_Theoretic Interpretation of Weber's Law_ that Fechner's originality exclusively consists, in his assumptions, namely, 1) that the just-perceptible increment is the _sensation-unit_, and is in all parts of the scale the same (mathematically expressed, Δ_s_= const.); 2) that all our sensations consist of sums of these units; and finally, 3) that the reason why it takes a constant fractional increase of the stimulus to awaken this unit lies in an ultimate law of the connection of mind with matter, whereby the quantities of our feelings are related logarithmically to the quantities of their objects. Fechner seems to find something inscrutably sublime in the existence of an ultimate 'psychophysic' law of this form. These assumptions are all peculiarly fragile. To begin with, the _mental fact_ which in the experiments corresponds to the increase of the stimulus is not an _enlarged sensation_, but a _judgment that the sensation is enlarged_. What Fechner calls the 'sensation' is what appears to the mind as the _objective phenomenon_ of light, warmth, weight, sound, impressed part of body, etc. Fechner tacitly if not openly assumes that such a _judgment of increase_ consists in the simple fact that an _increased number_ of sensation-units are present to the mind; and that the judgment is thus itself a quantitatively bigger mental thing when it judges large differences, or differences between large terms, than when it judges small ones. But these ideas are really absurd. The hardest sort of judgment, the judgment which strains the attention most (if _that_ be any criterion of the judgment's 'size'), is that about the _smallest_ things and differences. But really it has no meaning to talk about one judgment being bigger than another. And even if we leave out judgments and talk of sensations only, we have already found ourselves (in Chapter VI) quite unable to read any clear meaning into the notion that they are masses of units combined.
To introspection, our feeling of pink is surely not a portion of our feeling of scarlet; nor does the light of an electric arc seem to contain that of a tallow-candle in itself. Compound _things_ contain parts; and one such thing may have twice or three times as many parts as another. But when we take a simple sensible quality like light or sound, and say that there is now twice or thrice as much of it present as there was a moment ago, although we seem to mean the same thing as if we were talking of compound objects, we really mean something different. We mean that if we were to arrange the various possible degrees of the quality in a scale of serial increase, the _distance, interval_, or _difference_ between the stronger and the weaker specimen before us would seem about as great as that between the weaker one and the beginning of the scale. _It is these_ RELATIONS, _these_ DISTANCES, _which we are measuring and not the composition of the qualities themselves_, as Fechner thinks. Whilst if we turn to objects which _are_ divisible, surely a big object may be known in a little thought. Introspection shows moreover that in most sensations a new _kind_ of feeling invariably accompanies our judgment of an increased impression; and this is a fact which Fechner's formula disregards.[459]
But apart from these _a priori_ difficulties, and even supposing that sensations did consist of added units, Fechner's assumption that all _equally perceptible_ additions are _equally great_ additions is entirely arbitrary. Why might not a small addition to a small sensation be as _perceptible_ as a large addition to a large one? In this case Weber's law would apply not to the additions themselves, but only to their perceptibility. Our _noticing_ of a difference of units in two sensations would depend on the latter being in a fixed ratio. But the _difference itself_ would depend directly on that between their respective stimuli. So many units added to the stimulus, so many added to the sensation, and if the stimulus grew in a certain ratio, in exactly the same ratio would the sensation also grow, though its _perceptibility_ grew according to the logarithmic law.[460] If _A_ stand for the smallest difference which _we perceive_, then we should have, instead of the formula Δ_s_ = const., which is Fechner's, the formula Δ_s_/_s_ = const., a formula which interprets all the _facts_ of Weber's law, in an entirely different theoretic way from that adopted by Fechner.[461]
Text from Project Gutenberg, public domain in the United States.