Qualia and the Psychophysiological Explanation of Color Perception

Synthese, 65 (2), November 1985, 377-405.


Austen Clark
Department of Philosophy U-54
University of Connecticut
Storrs, CT 06269-2054


Abstract

Can psychology explain the qualitative content of experience? A persistent philosophical objection to that discipline is that it cannot. Qualitative states or "qualia" are argued to have characteristics which cannot be explained in terms of their relationships to other psychological states, stimuli, and behavior. Since psychology is confined to descriptions of such relationships, it seems that psychology cannot explain qualia.
A paradigm case of qualia is provided by simultaneous color contrast effects, in which (for example) a neutral grey patch is made to look reddish by being enclosed in a green surround. If the qualia based objections are sound, psychology ought not to be able to explain simultaneous color contrast; but psychology at least seems to be able quite successfully to explain those effects. This paper analyzes the logic of psychological explanations of simultaneous color contrast effects, and the import of various qualia based objections to those explanations. I argue that the qualia objections do not demonstrate any explanatory inadequacy in existing psychological explanations of "looks". Psychology succeeds in explaining at least some qualia. In a more positive vein, I argue that once the structure of such explanations is sufficiently understood, a place can be found for qualia within the emerging scientific account of color perception. The resulting account can deal with many of the traditional perplexities over qualia.

A neutral grey patch enclosed in a green surround looks somewhat red. It presents the appearance of a reddish grey, even though when one blocks the surround, one can see that the patch is not reddish at all, but merely grey. Furthermore, the same grey patch with a red surround will look somewhat green.

Such are the facts of "simultaneous color contrast". Can they be explained? Students of perception will find a remarkable divergence of opinion between psychologists and philosophers on this question. I think it fair to say that many psychologists would count the emerging explanation of color vision among the success stories of scientific psychology. The explanations therein proffered are simple yet remarkably powerful, and purportedly provide a systematic account not only of simultaneous color contrast, but also of color after-effects, mixing and matching, illumination and brightness effects, constancies of hue and brightness, abnormal color vision, and many others. Furthermore, some functions postulated by this theory have successfully been localized in the nervous system, so that theoretical terms such as "opponent process" have been given refreshingly tangible instantiations.

Many philosophers, on the other hand, will give a very different assessment of the prospects for explaining the reddish look of the grey patch. Far from accepting that there already exist successful explanations of such effects, many philosophers think that the reddish look of the grey patch is one phenomenon which psychology can never explain. Their objections derive from a cluster of philosophical problems concerning "qualitative content": the "qualia-based" objections to functional theories of mind. One needs to explain why the grey patch looks reddish, and not some other shade such as blue. While it is allowed that psychophysiologists can successfully detail the functional inter-relationships of perceptual states, stimuli, and discrimination behavior, some philosophers think that such relationships do not and cannot explain the qualitative content of experience. All such details concerning functional relationships are, on this view, equally consistent with the supposition that the patch looks blue (e.g., not red) and hence fail to entail that it looks reddish.

What are we to make of this difference of opinion concerning color perception? Do, or do not psychologists succeed (at least sometimes) in explaining how things look? If the qualia-based objections to functionalism are cogent, then psychology ought not to be able to explain any qualia, and in particular, it ought not to be able to explain simultaneous color contrast effects. So those objections imply that psychophysiological explanations of simultaneous color contrast are somehow inadequate. To assess this claim I shall analyze the logic of such explanations and consider the import of various qualia-based objections. I shall argue that the qualia objections do not demonstrate any explanatory inadequacy in psychophysiological accounts, and that the latter genuinely succeed in explaining how the grey patch looks. Such accounts are not even defeated by the possibility of "spectrum inversion"--that different qualia are engendered in different people by identical stimuli. Even if spectrum inversion obtains, current accounts succeed in explaining simultaneous color contrast, and so succeed in explaining at least some qualia. It follows that the qualia-based objections to functional theories of mind are not cogent. Furthermore, if a psychophysiological theory succeeds in explaining at least some qualia, then a functional definition of qualitative content ought to be possible, drawing upon the resources of that theory. In the course of the analysis I will suggest how this might be done.

I

How does the psychologist explain looks? Very roughly, and leaving aside many details, the story goes like this. (Endnote 1) There are three different types of color sensitive retinal receptors, which differ in the specific wavelengths to which each is optimally sensitive. Activity in any one type tends to inhibit activity from neighboring receptors of the same type--a process known as "lateral inhibition". For example, when a middle wavelength receptor is active, it somewhat inhibits the activity of neighboring middle wavelength receptors. A second relevant fact is that activity from the different receptors is combined in such a way that later cells function in a "spectrally opponent" fashion; that is, the cells are excited by presentation of a given color and inhibited by presentation of its complement. Certain cells, for instance, are excited maximally by monochromatic green light, and inhibited by red light. There are three such spectrally opponent channels: red-green, yellow-blue, and white-black.

These two facts are used to explain simultaneous color contrast. The green surround maximally excites middle wavelength receptors in the retina, which laterally inhibit their neighbors. Middle wavelength receptors stimulated by the central grey square are inhibited by some of such cells, and their activity is therefore reduced. That reduction yields a level of activity identical to that which would be given for a grey stimulus from which some middle wavelengths have been subtracted--that is, for a slightly reddish grey. Since the receptors are active at the rate they would be when stimulated by a reddish patch, the grey patch with green surround is encoded exactly the same way as a reddish patch. The visual system registers no difference between the grey patch and a reddish-grey one, and so the subject finds the grey patch to be indiscriminable from a reddish-grey one. Now the reddish-grey patch obviously looks reddish, and the grey patch is indiscriminable from it. Hence the grey patch looks reddish as well.

The explanation of simultaneous color contrast proceeds by showing how the particular stimulus configuration leads to a state of the visual system which is typically brought about by red things, and which leads the stimulus to be indiscriminable from red things. Roughly: the patch looks red (even though it is not) because it has an effect on the visual system identical to that of a patch which is red. Does demonstration of such identical effects explain why the grey patch looks red?

II

The negative philosophical response to this question derives from a certain account of "looks" statements, an account which introduces the technical terms "qualia", or "qualitative content". (Endnote 2.) The philosopher asks: what does it mean to say that the grey patch looks red, as opposed to its being red? Qualia are introduced to answer this question, and to explain how the patch looks red even though it is not.

There is something similar between those situations in which one is presented a patch which merely looks red and those situations in which one is presented a patch which really is red. The difficulty in describing that common element is simply that the former patch is not red (but merely looks it), and so the similarity of the situations cannot be a matter of both patches sharing the same color. The proposed solution is that although the patches differ, both present the observer with the same "qualitative content" or "quale" concerning color. In one situation this content is satisfied--the patch is really red--while in the other it is not. That common content is described by saying they both present the observer with a reddish "quale" or sense datum.

If one were to develop this idea any further, problems would immediately arise concerning the relations of qualia to physical objects and to colors, their location, privacy, incorrigibility, and so on. It is not necessary to face those problems here. All we need to recognize is (1) that qualia are introduced in order to describe the looks or appearances of things--those cases where x "looks" P even though x is not P (see Sellars, 1963, p. 133); (2) that one describes this by proposing that both situations present one with the same quale; and (3) that the grey patch looks reddish even though it is not reddish, and so provides a paradigm case in which the observer is presented with a reddish quale. The explanation of why the patch looks red must explain whatever it is that is common to the experience of looking at the grey patch and the experience of seeing something reddish. That common content is the quale.

Why then are philosophers sceptical concerning explanations of how things look? Their claim is that nothing can ever explain why the grey patch looks just the way it does, and not some other way. Problems with qualia largely derive from the possibility of "spectrum inversion": that what looks reddish to one person may look greenish to another, even though all of their propensities to discriminate, verbalize, and so on are identical, so that the difference is undetectable. (Endnote 3) If one accepts the possibility of spectrum inversion (and I do not propose to challenge it here) then it seems that psychophysiological explanations of "looks" are faced with an impossible task. The possibility of inversion seems to show that the qualitative character of a psychological state is independent of all the relationships which it bears to other psychological states, stimuli, and behavior. But psychophysiological explanations of perception can only characterize relations between discrimination behavior, stimuli, and psychological states. The psychologists and physiologists may adduce considerations which show what the grey patch can and cannot be discriminated from, and why; but no such considerations can ever explain the particular qualitative content possessed by the items so discriminated. So psychology cannot explain qualia. As Ned Block says in "Troubles with Functionalism",

...psychophysics touches only the functional aspect of sensation, not its qualitative character. ... Indeed, on the basis of the kind of conceptual apparatus now available in psychology, I do not see how psychology in anything like its present incarnation could explain qualia. We cannot now conceive how psychology could explain qualia...(Block, 1980b, p. 289)

Block goes on to suggest that qualia may not be in the domain of psychology (Block, 1980b, pp. 288, 289). This conclusion is implicitly endorsed by all those philosophers who accept the claim that what makes two states instances of the same quale cannot be characterized functionally, but requires physiological details (see Putnam, 1981, p. 81).

These general assumptions might be applied to the specific case of simultaneous color contrast as follows. The supposition that the quale presented by the grey patch is bluish and not reddish is equally consistent with all possible data concerning discrimination behavior. Hence details concerning discriminability fail to explain why the patch has the particular qualitative content that it does. Since all such considerations are equally consistent with the supposition that the quale is a blue one, they fail to explain why the patch looks red. One can perhaps explain the conditions for discriminating stimuli x and y, but not the qualitative content associated with either one.

III

To assess that argument we must consider, in some detail, the structure of the psychophysiological explanation of contrast effects. The psychologist explains why the grey patch looks red, in the sense that he or she explains why the grey patch cannot be discriminated from a patch which is actually red. Is it true that the psychologist can only explain discrimination behavior, and never the qualitative content presented by the items discriminated? I shall argue that the latter is in fact achieved, although the achievement rests on a rather complicated pattern of inference involving discriminability, encoding, information, and "looks".

To say that colors x and y cannot be discriminated is to say something stronger than that they look the same when inspected, or that the subject calls them the same color. To show that the colors are indiscriminable, one must show that the subject literally registers no difference at all between the two, and cannot learn to identify which is which (Cornsweet, 1970, p. 218; Dember, 1960, pp. 52-60). For example, one member of the pair is labelled a "target", and the subject is asked to identify it over a sequence of randomized trials. The subject cannot discriminate the two patches if and only if performance on the identification task is not statistically distinct from a random distribution. If the subject operates at a close to random level, which is nevertheless statistically distinct from random, then information about the differences between the two patches must still be accessible to the subject in making the identification. To say the subject cannot discriminate the two patches means that discrimination between them is at the chance level, so that no difference between the patches can be identified.

Discrimination in this technical sense of the word is used to make two kinds of inferences about the processing underlying perception. Discrimination requires the subject to retain information concerning differences between stimuli. If one is able to perform at better than a chance level at the identification task, then information (in the technical sense of the word) concerning the difference between x and y is available to the subject. (Endnote 4) If information concerning the difference between x and y is available to the subject, then at no stage can the differences of effects of x and y be lost. So we get:

If at some stage of perceptual processing, x and y have identical effects--if, for example, x and y both lead to exactly the same rates of firing of all cells in all three color channels--then there is no way in which information about the differences between x and y can be retained, and so x and y cannot be discriminated (Cornsweet, 1970, p. 180). Discriminability implies differences in internal effects.

Now suppose that x and y cannot be discriminated: the subject operates at a chance level when trying to identify a patch. No information is accessible to the subject concerning the difference between x and y. But if no information about their difference registers, then there are no properties among the effects of x and y which make the pair discriminable. In particular:

Inability to discriminate does not imply that processing proceeds identically at all stages of analysis, but merely that at some stage the information is lost. (Thereafter, of course, it can never be regained.) That the information is lost means that effects of x and y do not differ in any property which would make them discriminable. Those effects can have many different properties, as long as none of the differences retain the information that x and y differ.

The two discrimination postulates are not true by definition or in virtue of the meaning of the term "discrimination"; they are merely true. P1 follows from the definition of "discriminable" conjoined with elementary information theory. The latter provides the rationale for the claim that if effects of x and y at some stage share all properties then information on the difference between x and y is lost and can never be regained in later stages (Dretske, 1981, p. 59, 104; Sayre, 1976, p. 29). This claim is not analytic, and so neither is P1. P2 provides a sufficient condition for discriminability. It is not analytic, since it is framed in terms of an information-theoretic model of the properties of effects of x and y. P1 and P2 play pivotal roles within such a theory.

To bolster this claim it may help to give a sense of the scope of application of principles P1 and P2. Such principles underlie the inferences from color discrimination data to the number of color systems operative in human vision. The key fact of normal color discrimination is that any spectral color stimulus can be matched by suitable adjustment of the relative intensities of three arbitrarily chosen wavelengths (Cornsweet, 1970, p. 192; Haber and Hershenson, 1980, p. 99). P1 justifies the inference from this fact to the conclusion that there must be at least three independent color sensitive systems in human vision. P2 licenses the inference that there are no more than three. (Because identical effects would imply an inability to discriminate, if one could match any color with just two adjustments, a third system could not be involved.) The same reasoning is employed to show that the dichromat (who needs at most two intensity adjustments to match any color) must therefore have one color system inoperative (Cornsweet, 1970, pp. 218-219).

In order to account for the facts of color discrimination (or more broadly, for any kind of perceptual modality) one proposes a set of internal discriminanda. Events at sensory transducers are mapped into the set of discriminanda in some regular fashion. The internal states have various properties to which a discriminal process is sensitive, and so allow discrimination of stimuli by discrimination of properties of their effects. ("Discriminal process" is a term introduced by Thurstone (1927). See also Torgerson (1958), pp. 156-158.) If two transducer events both map into internal discriminanda with identical properties, then information on the difference between those events is lost, and the stimuli are indiscriminable. If the difference between internal discriminanda is sufficiently salient to engage the discriminal process, then the information that transducer events are distinct is retained, and the stimuli can be discriminated. Indiscriminability is not coextensive with identity of properties of internal discriminanda, but rather with their not failing to share any property whose absence would make the stimuli discriminable.

The proposed internal discriminanda are related to discrimination behavior as follows. Each is an encoding of some transducer event, and a pair of transducer events are indiscriminable if and only if their respective internal encodings share all properties whose absence would make the events discriminable. Since there can be indiscriminable differences in properties of the proposed internal events, to say that two discriminanda are identical with respect to all the properties to which the discriminal process is sensitive is to say something stronger, namely that the stimuli occasioning each are indiscriminable from the same set of stimuli. Indiscriminability of two stimuli does not establish identity of their encoding, since those encodings may have differences insufficient for discriminability.

IV

The psychophysiological explanation of simultaneous color contrast proceeds initially by showing how a grey patch with a green surround has an identical effect on human color systems as a reddish-grey patch. It can be summarized as follows:

----------------

We also derive:

L1 describes the different retinal receptors and their lateral inhibitory connections. L2 describes the three color systems and their opponent processing. As antecedent (A1) one describes the particular stimulus-surround combination. From L1, L2, and the stimulus description, one could derive the prediction (C2) that the grey patch would have identical effects on the three color systems as a reddish patch. Quantitative derivations of this sort--of which wavelength combinations will have identical effects, for example--are now quite common in visual psychophysics, and are unproblematic. Clearly psychophysiology can explain why x and y equally affect the visual system. But does demonstration of identity of effects between some x which is not P and some y which is P explain why x looks P?

As a first step in answering this question we should attempt to get a successful deduction whose conclusion is "the grey patch looks red to S." This clearly requires some generalization of the form:

L3 as it stands is only a schema for empirical generalizations; to fill out such a schema one needs to empirically determine (a) exactly which cells C have the requisite relations to indiscriminability; (b) what characteristics Q are necessary and sufficient for the occasioning stimuli to be indiscriminable, and (c) over what sorts of distal properties P such generalizations can be found. Such empirical determinations have been made for various sensory cells C, effects Q, and properties P. An example of one is:

The exact specification of effects Q requires details of the spectral sensitivity curves for the three color systems. Stimuli x and y with equal effects on the cells in the lateral geniculate are indiscriminable with respect to hue and look the same.

I shall call generalizations of this form "physio-phenomenological" generalizations. They correlate aspects of sensory physiology with descriptions of how a stimulus will "look" to the subject.

What role do such generalizations play in explaining looks? It seems there is no problem in understanding why the grey patch with a green surround has identical effects on the visual system as a reddish patch. The success of the psychophysiological explanation of looks therefore stands or falls with the physio-phenomenological correlations, which link such identity of effect to similarity of looks. That explanation can be detailed as follows:

-----------------

We first show that the grey patch has effects on the visual system which are such as to make it indiscriminable from a patch which is reddish. Since indiscriminable stimuli share the same appearance, and a reddish patch looks reddish, this entails our explanandum: the grey patch looks reddish. (Endnote 5)

Before proceeding further I should point out that this form of explanation is not confined to simultaneous color contrast effects, but is employed quite generally. One finds explanations of this form for color after effects, the Purkinje shift, Mach bands, brightness constancy, hue constancy, the Bezold-Brucke effect, brightness contrast effects, the Hermann grid, Chaldni figures, tilt after-effects, and many others. (Endnote 6) Each of these "effects" involves qualia: each involves some situation in which some physical thing x looks to be P, even though it is not. The explanatory strategy for each effect is the same. How do we explain why x looks to be P? Find some physiological mechanism which yields the same output when stimulated by the x which is not P (but looks it) and by a y which is P. One thereby "explains" the perceptual similarity between those situations.

But there are two immediate objections to physio-phenomenological generalizations with the form of L3. The first challenges their empirical adequacy. Some might be appalled at the inductive leap from lateral geniculate cells to phenomenology. This objection is easily met: as a matter of fact, generalizations of that form are empirically successful. They license predictions which are repeatedly confirmed. For example, one can predict which wavelengths in what quantities to subtract from grey to match the effect of surrounding the patch in green. Other generalizations yield successful predictions concerning which wavelength combinations will be indiscriminable, how hues shift with luminance, how anomalous observers will match colors, how changing illumination wavelengths will affect matching, and so on.

All such predictions are based on generalizations of the form L3: that if x and y both have effect Q on cells C, and y is P, then x will look P. Hue perception is perhaps the best understood, where one has indiscriminable hues if one has identical ratios of excitation in the three color channels (DeValois and DeValois, 1975). But other examples are available. The Purkinje shift is explained by reference to the spectral sensitivity curves of the different types of retinal receptors (cones and rods) active in day and night vision (Haber and Hershenson, 1980, p. 97). Mach bands can be predicted by the effect of an edge on lateral inhibitory cells in the retina (Cornsweet, 1970, pp. 302-303). Stimuli will show the same brightness if they have identical effects on the white-black channel, and brightness constancy effects can be predicted from retinal lateral inhibition. The Hermann grid and Chaldni figures can be predicted by reference to the width of on-center off-surround receptive fields for edges (Frisby, 1980, pp. 138-139). Spatial frequency channels have been isolated, and many contrast, edge, and contour phenomena (including tilt after-effects) can be predicted from equivalent encodings of those cells (Cornsweet, 1970, pp. 342 ff.).

We seem to have an empirically adequate explanation which logically entails that the grey patch looks red to the subject. A second objection to the explanation is that the physio-phenomenological generalizations fail to give any understanding of the relationship between visual physiology and how things look. We fail to explain why lateral geniculate spectral opponent cells are related to apparent hue. Why should it be the lateral geniculate? Why should that cell make x look green?

This objection can be answered, however, and in fact physio-phenomenological generalizations can be explained. We need to explain principles of the form:

This can be derived from two claims:

and

Again the presentation is schematic, and these claims only hold for some (empirically determined) distal properties P, cells C, and encodings Q. But when the schema is filled out, an empirical generalization of the form L3 can be explained by L4: a model of discrimination, tied to L5: a truism about looks. "Discrimination" is used here in the technical sense introduced above. Assume that stimuli x and y have identical effects at some stage in the visual system, where a "stage" is just a set of cells C such that all subsequent visual information is causally dependent on C. If those effects are identical then the information regarding the difference between x and y has been lost, and the subject will be unable to discriminate them. If other information is available besides that given in C, identity of effects on C will not assure indiscriminability. In that way one explains L4: how identical physiological effects lead to indiscriminability.

From indiscriminability it is a short step to phenomenal similarity. Notice that all the predictions concerning discriminations are intrasubjective and synchronic: we predict which simultaneously presented pairs of stimuli a given subject will find to be indiscriminable. This sense of discrimination clearly licenses the inference to L5. If a person cannot discriminate x and y, then x looks the same as y for that subject.

In short, our explanation of the reddish look of the grey patch has the following structure. We show how the grey patch x has identical effects on the three color systems as a reddish stimulus y. Since all visual hue information is causally dependent on those three systems, this shows that the subject has lost the information that x and y are different. The subject will therefore find x to be indiscriminable from y, and so x will look just like y. Since y (the reddish stimulus) looks reddish, so will x. Hence the grey patch looks reddish.

Together L4 and L5 have the interesting implication that one can explain how something phenomenally appears to be P by showing how that thing is encoded identically in the nervous system (at some stage--for thereafter the information on differences is lost) as a thing which really is P. Because x has effect Q, and y has effect Q, and y is P, x will look P. What is common, then, between stimulus situations x and y is that they both have effect Q, which makes them indiscriminable; so while y really is P (reddish) and x is not, the system is insensitive to that difference and both have result Q.

Now if x and y are indiscriminable, then by P2 (the second discrimination postulate given in III), the effects Q(x) and Q(y) of the respective stimuli share all properties whose absence would make them x and y discriminable. Furthermore, if x and y are discriminable, then information on their distinctness is preserved at every stage, and so Q(x) and Q(y) differ in some property which suffices to make x and y discriminable.

I argued above (in section II) that the concepts of "qualia" or "qualitative content" are introduced precisely to account for the similarity presented by an x which looks P, (even though it is not), and a y which really is P. What purportedly is common to those two situations is the presentation of the same qualitative content to the observer: the same "looking as if it is P" content, which is satisfied in the case of y, but not in the case of x.

The psychophysiologist explains why the grey patch looks reddish by showing how it has a physiological effect Q at some stage of visual processing which is also exactly the effect of a patch which really is reddish. I suggest that the Q states invoked in such psychophysiological explanations and the qualia postulated by philosophical accounts are one and the same. The functional role postulated for Q states, which make the x which looks P and the y which is P indiscriminable, is precisely the right sort of role for qualia to fulfill. Of course no such argument can be conclusive, since many different theories of qualia have been proposed. But insofar as philosophical discussions give a role to the notion of qualitative content in explanations of "looks", that content is satisfied by certain Q states invoked in psychophysiological explanations of "looks". Q states have all the properties of qualia.

In effect this is the key premise in the psycho-physiological explanation of why the grey patch looks reddish: that explaining the correspondence of psychophysiological effects of x and y can explain the phenomenal similarity of x and y. The fact that x looks reddish, after all, is an aspect of the immediate qualitative content of the experience of looking at x. Any argument that has as a conclusion that x looks red will have succeeded in explaining an aspect of the qualitative content of the experience of seeing x. If visual psychophysiology succeeds in explaining "x looks red", then it will have succeeded in explaining (at least some) qualia.

V

The main argument in favor of the proposed identification of qualia with Q states is that qualia and Q states play exactly the same role in explanations of visual discrimination. Principles relating sameness and difference of qualia to discriminability are precisely the ones applying as well to Q states. The key principles can be set out quickly. First:

"Indiscriminable" is here being used in the technical sense introduced above, meaning that the observer cannot differentiate the two. If two (visual) stimuli have absolutely identical qualitative content, then they look just the same, and it would follow that they are indiscriminable.

A complication arises concerning the converse. The indiscriminability of x and y does not suffice to show that they present the observer with identical qualia (see Goodman, 1977, p. 196). For suppose y is indiscriminable from some further z; we would then have x and z presenting identical qualia. However, the sum of differences from x to y, and from y to z, may suffice to make x and z discriminable, in which case they cannot be qualitatively identical. Hence we must allow indiscriminable qualitative differences; indiscriminability does not guarantee qualitative identity. What assures qualitative identity of presentations is rather that no discrimination reveal a difference between x and y: not only must x and y be indiscriminable, but there must be no z discriminable from one and not the other, for such a z would show that x and y differ. The indiscriminability of x and y guarantees at best a principle of the form:

Indiscriminability shows only that presentations of x and y match in all qualitative characteristics whose mismatch would make x and y discriminable.

These principles are framed exclusively in terms of judgments made by one observer at one point in time, (they are intra-subjective and synchronic) and do not directly imply anything about identity of qualia within one person across times (intrasubjective inversion) or across people (intersubjective inversion). But it is hard to see how within one observer at one time either conditional could be challenged without altering the concept of "qualitative content" or "quale".

If qualia are to be located and identified within any psychological theory, it will be a theory of discrimination. All such theories have some analogue of Q states--states which are encodings of stimulus classes, and which differ in certain properties just in case the stimuli are discriminable. To explain why certain classes of events are classed as similar, and others are not, one must propose some internal machinery which works to classify and regroup transducer event-classes. One or another encoding function is proposed which shows how information on some physical differences of stimuli is lost while other bits are retained. At some stage in this encoding there must be a process of assessing differences between codes, and either accepting or rejecting a match. This discriminal process will be sensitive to some properties of codes, and not to others; the ones to which it is sensitive are the "critical" properties. Qualitative differences are differences in properties of internal states at that stage. If x and y are discriminable, then their encodings Q(x) and Q(y) will differ in some critical property. If their encodings differ in no critical property, then no discrimination will reveal a difference between x and y, and their presentations will be qualitatively identical.

Sensations are effects of things, and qualia are properties of those sensations. They are in particular those properties which enable one to discern similarities and differences: they engage discriminations. The way in which qualia have been thought to do this is isomorphic to the way critical properties engage an internal discriminal process. Items identically encoded yield qualitatively identical presentations, and differences at that stage occasion differences in qualia. In short, qualitative content can be identified with those properties of encodings which engage the discriminal process. (Endnote 7).

This identification can be bolstered by a second argument. Suppose some theory succeeds in explaining how some things sometimes look (as I have claimed for the psychophysiological account), and so succeeds in explaining at least some qualia. It follows that that theory provides the wherewithal to define the notions of "qualitative content" or "qualia", at least in the sense of providing non-circular coextensional terms. One could frame such definitions using the familiar technique of forming the Ramsey sentence of the theory, and choosing a Ramsey functional correlate for the term (see Block, 1980b, pp. 270-274). I have suggested possible Ramsey functional correlates for qualia: they are properties of encodings of things, which comport in a certain way with capacities of discernment.

VI

According to the account so far developed, qualia are explained somewhat the way one explains the appearance of lens flares or optical distortions in photographs. One explains such phenomena by first characterizing the external stimulus to the system, and then showing how the operation of the intervening camera-works modulates that input to produce the given result--be it lens flare or sensation of red. Explaining the lens flare is analogous to explaining looks, in that the lens flare gives the appearance of a bright line or speck at a given point when there is none there. We show how the optical properties of the lens system are such that given the incident light at such-and-such angles, a spot on the film is produced in just the place that it would be produced if there really were a bright point there. Similarly, in explaining the reddish look of the grey patch--a paradigm case of a quale--one shows how the visual mechanism is such that the stimulus of grey surrounded by green produces just the same effect as a reddish stimulus. To explain why x looks P, one describes certain properties of the visual mechanism which show how x has identical effects in the system as some stimulus y which is P. The suggestion is that visual experience is the output of processing, and that its features can be causally explained by reference to the mechanisms of that processing.

I noted above that if the qualia based objections to functionalism are cogent, it would follow that psychophysiology cannot explain any qualia, and so there must be some inadequacy in the proposed explanation of color contrast effects. The remainder of the paper will consider some ways in which the proposed explanatory strategy might be thought inadequate. Two preliminary objections to it have already been considered: that the required generalizations are empirically indefensible, and that they cannot be explained. Three further objections are qualia-based, and will occupy this and the next two sections.

The first and fundamental objection might be put as follows. "Something is missing from all your purported "explanations" of qualia. I understand why the grey patch is indiscriminable from a reddish patch, why it is discriminable from a neutral patch, and so on and so forth. But I still do not understand why it looks the way it does. Why does it present that particular qualitative content, and not some other one? Discrimination data do not answer this question. I want an explanation of the particular content of the experience, and not of its relations to other stimulus situations. Until you have explained why it looks just the way it does--reddish, and not greenish or some other appearance--you have not explained qualia."

A first response to this objection is to question exactly what is missing from the explanation. What aspect of qualitative content is left out, once conditions of discriminability are described? We can explain why the grey patch is indiscriminable from a reddish patch, but is discriminable from a greenish patch. We can explain why it looks redder than x, less saturated than y, brighter than z. Its exact place in a structure of discriminations--its coordinates in the color solid, if you will--can be predicted from the factors cited in early visual processing. Furthermore, according to the arguments in Section V, qualitative differences can be defined in terms of discriminable differences. Since the exact location of x in a discrimination structure has been explained, and discriminabilities suffice to identify qualitative differences, the qualia of x have been explained as well.

This line might be summarized by saying that one can explain exactly what x can and cannot be discriminated from, and by doing so one explains which quale x presents. The objector may admit all this, and yet still feel that the qualitative content of the sensation--what it looks like or seems like to have it--is totally missing from the explanation. Even after the conditions of discriminability are fully described, it seems there still remains a significant question concerning qualitative content, and that therefore the psychophysiological explanation fails to explain qualitative content.

One way this question can be raised is to invoke the possibility of "spectrum inversion": that what looks reddish to John may look greenish to Mary, yet they have learned to use the words differently, and all their behavioral dispositions agree. Since the proposed explanation of x looking P merely explains behavioral dispositions, and does not rule out the possibility of inversion, it does not explain the particular qualitative content of the Q state: reddish to John, greenish to Mary.

This objection has perhaps been thought sufficiently decisive to rule out any explanation of qualia. However, it shows no such thing, and its relevance is persistently mis-assessed. The possibility of spectrum inversion in no way impugns the adequacy of the proposed psychophysiological explanations. Because a condition of spectrum inversion is that different observers share the same behavioral dispositions (and in particular the same discriminations), the proposed explanation of qualitative content will in fact work just as well for John as it will for Mary, even if their qualia in some way differ. That is, it will correctly predict the qualitative content of John's experience, and it will correctly predict the qualitative content of Mary's experience, even if those contents somehow differ.

The reason a psychophysiological explanation emerges unscathed from inversion is that all of the generalizations it employs concerning the conditions under which a stimulus looks P remain true even under qualia inversion. For example, as the luminance of a stimulus is increased, its hues tend to shift, so that it looks less reddish or greenish and more yellowish or bluish (see Burnham et al., 1963, p. 54). This description of the so called "Bezold-Bruecke" effect is true of all observers with normal discriminations, even if spectrum inversion occurs. One can describe the phenomenon (and give quantitative predictions of it) in terms of the changes in classes of standard color chips with which a given target will be found indiscriminable as the luminance of the target is increased. Now a condition of spectrum inversion is that John and Mary share all those discriminations; they will both pick out the same sequence of color chips to match the increasingly bright target. Both agree that the qualia engendered at the end of that sequence are less reddish. Hence whatever "less reddish" means to John, and whatever it means to Mary, that x looks less reddish as it gets brighter is true of the experiences of both, even if we assume the respective qualitative contents somehow differ. Predictions of when x will look less reddish will work correctly for both observers and describe for each the content of their experience. If John desires a theory which will correctly predict when his qualia shift from the reddish--as he means the term "reddish"--he had best employ something like the opponent process theory--a theory which will do the same job for Mary as well.

The key is that the relevant psychophysiological generalizations all employ the intrasubjective and synchronic sense of "discriminability". For this reason they include the sensations of everyone (with normal discriminations) in their scope, even if spectrum inversion obtains. Consider a second example. The Munsell color solid attaches color chips to physical descriptions of stimuli (the relative intensities of three reference wavelengths providing its coordinates). Even if spectrum inversion obtains, the color solid still accurately models the color experiences of every person with normal discriminations. Because all discriminations are presumed unchanged, for every observer any stimulus with physical parameters described by the coordinates [x,y,z] will still look like the color chip attached to that point. No movement or rearrangement of color chips is necessary (or possible) even if spectrum inversion obtains. Oddly enough, when one "inverts" the spectrum, one must leave every chip exactly where it is.

The generalizations needed for a psychophysiological explanation of qualia survive inversion in the same way as does the color solid. For each person they correctly describe for that person the extension of the relationship "indiscriminable", and hence of the relationship "presents the same qualia as". Hence for no person is there some quale falling outside the scope of the explanation. Whatever it is that comprises reddish sensations in John, and whatever it is in Mary, the explanation accurately describes both. So spectrum inversion, even if it actually obtains, would in no way contradict the adequacy of the proposed explanations.

This argument does not challenge the possibility of spectrum inversion, but its relevance. Since the psychophysiological generalizations employ an intrasubjective sense of discriminability, they include everyone's sensations in their scope, even if spectrum inversion obtains. The possibility of inversion does not imply that qualia are outside the domain of psychological explanation, or that psychology cannot explain qualia.

VII

The second qualia-based objection to explanation via discriminability is that surely something is left out of such an explanation if inversion obtains. Although one might admit that it can explain from John's point of view when his sensations go reddish, and from Mary's point of view when her sensations go reddish, it fails to explain the difference between the respective reddish sensations of John and Mary. The reddish sensations of John and of Mary both fall within the scope of psycho-physiological generalizations only in the Pickwickian sense that those sensations are called the same thing. Surely they have different properties--reddish in John, greenish in Mary--and it is that difference which remains unexplained. Call that difference a "qualitative difference". It remains unexplained and outside the scope of current psychological theory.

To one aspect of this objection the psychologist can readily agree. The qualitative difference between John and Mary--if such there be--would remain unexplained, because all current psycho-physiological generalizations employ an intra-subjective concept of "sameness" or "difference" in qualia. To explain the difference between John and Mary would require an inter-subjective concept of qualitative difference which is simply not available in any current accounts. Current psychophysical methods can of course describe interpersonal differences insofar as they are revealed in differences in discriminations or thresholds. They use the ordinary concept of "looks", in which if some thing x looks different to subject p than to q, then the set of things p judges to look the same as x will be different from the respective set for q. But spectrum inversion proposes a different sense of inter-subjective qualitative difference. It proposes qualitative differences across subjects in exactly the case where their discriminations and thresholds are identical. For this there is no psychophysical concept, and hence inter-subjective qualitative differences would not be in the domain of current theories.

The psychologist can admit this yet deny that there is therefore something "left out" by current theories. To demonstrate the latter it must first be shown that there is some qualitative difference left unexplained; that is, that there is a difference in the sensations engendered in different subjects by an identical stimulus. To convict current accounts of explanatory inadequacy, it must be shown that there is something they cannot explain. The mere possibility of inversion does not suffice. One must produce some data which the opponent process theory cannot explain.

This shifts the burden of proof onto the inversion theorist, who must show that there is some intersubjective qualitative difference which cannot be explained by current theories. Intriguingly, the definition of spectrum inversion makes it impossible to ever obtain a conviction. The supposed qualitative difference across subjects is by definition functionally inconsequential, and makes no difference at all to the rest of their psychology or to their behavior. Given an identity of behavioral dispositions and functional roles, the hypothesis of intersubjective spectrum inversion is one for which no evidence could be obtained. So one cannot show that there is some qualitative difference between observers which is left unexplained.

In short, if the inter-subjective qualitative differences proposed in spectrum inversion could be shown to exist, then current theories could not account for them; but such differences cannot be shown to exist.

The conclusion of this argument is something stronger than stalemate. The possibility of inversion does not show what it was thought to show. It was thought to rule out psychophysiological explanations of qualitative content. Once it is allowed that qualitative differences can be defined in terms of discriminatory differences (as argued in Section V), then identifying the place of a stimulus within the structure of an observer's discriminations suffices to identify its qualitative content for that observer. Now a condition of spectrum inversion is that different observers share discriminations; and it follows that psychophysiological generalizations will accurately identify for each observer that observer's qualia. It may then be insisted that those qualia differ across observers; but again this claim is consistent with successful psychophysiological explanation. Not only does the latter employ an intrasubjective (and not intersubjective) concept of qualitative differences, but the very definition of spectrum inversion rules out the possibility of demonstrating the existence of the requisite sorts of intersubjective qualitative differences. Any differences which manifest themselves in altered discriminations can of course be accommodated by the psychophysiological framework. In short, the possibility of spectrum inversion in no way impugns the adequacy of the proposed psychophysiological explanations of looks.

VIII

One may abandon the idea of discrimination-independent qualitative content, and yet still feel dissatisfied with the sort of explanation of phenomenal properties which has been given. This dissatisfaction might be expressed as follows: "I understand the idea of visual encoding, and of stimulus x having effects within the system which lead x to be indiscriminable from stimulus y which is P. But such visual encodings occur inside the head, and are therefore invisible, colorless, and have no visual qualities at all. How can citing such states explain why some external x looks to have a particular color? In particular, since sensations of red are not literally red, how can citing them explain why the patch out there looks red?"

In many ways this is the basic problem of qualia. The fact that some thing x may look red even when it is not is after all the initial reason for postulating qualia. I suggest the best way to answer the query is to insist that codes for color are genuine codes, having representational character. Qualia are properties of effects of things. Because the mapping from transducer events to internal discriminanda is a regular one, however, they are also codes for things. They are codes in at least the minimal and non-metaphysical sense provided by information theory: an ensemble of events bearing significant contingent probabilities with transducer events (see Dretske, 1981, pp. 21-22). Just like any code, the association between the token doing the representing and the feature represented can be arbitrary; a representation of red need not be red. A token need not have the properties of its referent. It can represent a large red bulgy thing without being large, red, or bulgy. The three adjectives themselves provide examples. One hypothesis explaining how an internal qualitative state can represent a quality which it fails to share is that it is a representation, and the quality satisfies that representation.

Of course the relationship between the internal state and the sensory quality of redness is not like the relationship between the adjective "red" and redness. The former relationship is established by the causal regularities of visual processing and the vagaries of a discriminal process, rather than by linguistic tokens and convention. Nevertheless, such causal regularities can (as information theory shows) make Q states codes for qualities, and many of the perplexities of qualitative content can be resolved by assimilating it to a kind of representational content.

The qualitative content of a Q state is, for every observer, given by its place in a relational structure of discriminations. To specify the qualitative content of a particular state for a particular individual is a matter of placing it within the discriminal ordering for that individual. Even if there is some difference across individuals in the qualitative content of a reddish quale (if, that is, qualia inversion is possible), nevertheless within each person's putatively private qualia set the only way to specify the qualitative content of a particular member is to give its discriminative relations to other members. Remarkably enough, the relations needed to do this ("redder than", "brighter than", etc.) succeed intersubjectively, even if inversion obtains. They succeed in picking out, for each person, the qualitative content of a reddish quale, even if that content in some way differs across people.

A sensation of red can be ascribed the content "red" in the following way. There is a class of objects P which are mutually indiscriminable; they all affect the visual system identically in certain respects (which can be specified in terms of identity of critical properties) and are all encoded by the same Q state. Q is then a pointer to a place within a relational structure of discriminations. It is a "sensation of redness" because it is produced by perceptual encoding mechanisms whenever one encounters a stimulus indiscriminable from members of that class. Ultimately to eliminate "red" from the definiens, (and to pin down which class of things is the class of red things), ostension to a paradigm case is required. For example: "red" names the class of things which are indiscriminable from that (pointing). (Endnote 8) The definition of "indiscriminable" contains no color terms, and ostension is perfectly acceptable in the definition of a natural kind term (see Putnam, 1973). A state has qualitative content insofar as it encodes and is occasioned by members of some class of indiscriminably similar things. To identify the specific content of that state, one must identify the encoded class.

The quale itself is not red, or indeed any color at all. It is not seen; it occurs inside the head. But it is a code for the class of things having just that place within the structure of discriminations. The representation itself need not have any visual qualities; it simply encodes the class of stimuli with which x is indiscriminable. But showing how x is encoded as a member of that class shows why x is indiscriminable from reddish-looking things, and therefore why x looks reddish.

IX

Discussion of the representational character of qualia--and indeed, even the use of the term "qualia"--may give the impression that psychophysiological explanations of appearances are committed to a representationalist (or indirect realist) theory of perception. While the proposed account is clearly of a realist bent, it is thus far unclear whether it bends more toward the direct or the indirect variety. To conclude I shall sketch its relations to those two views.

Many of the claims making up a psychophysiological account of qualia are consistent with a representational account of perception. Schematically, the representationalist analyzes "S sees x" in terms of S having some mental state or entity y which is distinct from x and which bears a certain relation R to x (see Jackson, 1977, pp. 19-23; Perkins, 1983, pp. 13-19). The relation R which y bears to x is typically some sort of causal relation, which is used to explain how sense data "belong to" physical objects. Many of the claims made above accord with this schema. On the proposed account, Q states are psychological states of the observer, and so are not identical to any physical object situated before a sense organ. They are causally dependent upon states of the observer, and so would not occur or have the properties which they do unless the observer is in a certain state. They are related to physical objects as being among their effects. Indeed, it is only in virtue of having such Q states that one perceives physical objects; without any internal discriminandum, for example, one would have no perception at all of the grey patch. Most importantly, the proposed account accepts the representationalist claim that the entities y (in this case, Q states) represent physical things. All these claims are consistent with broad features of a representationalist theory.

It may seem therefore as if a psychophysiological account of color perception commits one to a representational theory of perception. Indeed, causal and scientific accounts of perception are often thought to have that implication.

Direct realists differ with representationalists over the identity of the direct (or immediate) objects of perception. The direct realist claims that the immediate object of perception is always some physical object (or state or event) which is situated before the sense organs. The representationalist, on the other hand, claims that physical objects situated before the sense organs are never the direct objects of perception; that role is rather played by mental entities y caused by objects x. The physical object x is at best an indirect object, represented by y.

Rather than adjudicate between those views, I will simply show that the psychophysiological account is neutral between them. First, while all the features of the psychophysiological account of color perception so far mentioned are consistent with representationalism, they are consistent as well with direct realism. The direct realist can easily admit that perception is causally dependent upon various receptor and brain mechanisms. Such dependencies do not oblige one to adopt the representationalist viewpoint concerning the immediate objects of perception. One can allow that one sees in virtue of having qualia, and that qualia causally depend upon states of the observer, while denying that qualia are therefore the direct objects of perception. A simple model can show these claims consistent.

If qualitative content is genuinely a kind of content, then we have for qualia, as for any kind of representation, a use/mention or token/referent distinction. Suppose that to be directly aware of x as being P is to have a code representing x as P in working memory of whatever processing units subserve conscious processing. In this case, the representation of x as P is a Q state caused by some member of the set of indiscriminable P members. The object of the perception (or the awareness) is the referent of the code. The item processed, however, is the code itself, not the thing. The fact that only codes can be processed (and not things) does not prevent them from representing those things.

To say x is the immediate object of perception is on this line to say it is the referent of a representation occupying immediate awareness. Qualia encode properties of things. The quale itself is not the immediate object of perception; its object is, and it is by having the quale that its object is the "immediate" object of perception.

On the psychophysiological account, as in representationalism, there is indeed a y not identical to x in virtue of which we are aware of any physical x before the sense organs, and such a y is causally dependent upon states of the observer. Item y is a quale. One can however deny that y is perceived: it is merely had, and it is in virtue of having it that a subject perceives objects. Such a view would hold, with the direct realist, that the immediate object of perception is always a physical object (property, event, etc.) in front of the sense organs. But this is consistent with a complicated causal story of how one sees; what it means to say x is the direct object of perception is that one has some representation of x in immediate awareness, and one sees in virtue of having that representation in awareness.

C. D. Broad suggested (1923, p. 96) that the relationship between sense data and things was similar to the relationship between written words and the things they describe. Just as a written word need not have the properties of the thing it is representing, so the quale need not share the properties of the physical thing it represents. Furthermore, as Broad pointed out, even though the printed word is in one sense the immediate content of consciousness, it is not the object of awareness (except in rare cases of proofreading or checking typography). Instead one attends to the referent of the word. Similarly, the qualitative state may be the immediate occupant of the processor underlying consciousness, but is not its object, which resolutely remains the referent of the code. Its properties code classes of things, and allow one to make the appropriate discriminations between that class and other classes. It thereby has the qualitative content of, for example, a sensation of red.

The word-qualia analogy is strong then on two counts, but weak on a third: the relationship between word and referent is not at all like the relationship between neural discriminanda and sensory qualities. The latter is purely a matter of an encoding function mapping characteristics of things into an internal ensemble of events. Detailing the vagaries of that mechanism can, at least sometimes, explain how some things look.

Endnotes

1. The explanation derives from the opponent process theory proposed by Hurvich and Jameson (1957). See Boynton (1979) and Cornsweet (1970) for careful analyses of the theory, and DeValois and DeValois (1975) for physiological details.

2. C. I. Lewis (1929) introduced the term "qualia". C. D. Broad (1923) presents a classic version of a qualia theory. There are many different qualia theories; the "qualia based" objections borrow the term but do not commit themselves to any particular version. Swartz (1965) is a useful collection of variants.

3. A stimulating exchange on the relevance of inverted spectra to psychological explanation is found in Block (1980a, 1980b) and Shoemaker (1975, 1982). Shoemaker (1982) has extensive references to the philosophical literature on the inverted spectrum.

4. If the output ensemble differs significantly from random then some of its members bear significant correlations with some input events. The output ensemble somewhat reduces uncertainty over which input event occurred, and so it satisfies the technical definition of information. See Dretske (1981), pp. 23-26; Sayre, (1976), pp. 26-30.

5. It should be noted that there remain several obscurities and omissions in this explanation. First, the exact retinal mechanisms underlying lateral inhibition are unclear, and one needs to explain why it is only the red-green channel which is differentially affected by the green surround. Second, lateral inhibition would seem to explain only colors induced along edges of the square, and perhaps other mechanisms are required to explain why the entire patch takes on the reddish hue. Third, while lateral inhibition and spectral opponent processes are clearly present, their exact inter-relations are obscure and still controversial. In short, while the general principles of the explanation are well founded, currently one can only speculate on some of the details.

6. Cornsweet (1970) remains an excellent reference on these phenomena. See also Frisby (1980) and Kaufman (1974).

7. Qualia will be construed as universals and not particulars; qualia are properties of Q states, while a quale is a token of a Q state. Note that Q states may have properties which are not qualia, and they may have differences which do not suffice to make them discriminable.

8. To avoid the need for an enormous number of paradigms, it is better to use relative similarity as opposed to indiscriminability. Red things are those which are relatively more similar to that exemplar (pointing) than to any of the other paradigms.

References

Block, N.: 1980a, "Are Absent Qualia Impossible?", The Philosophical Review 89(2), 257-274.

Block, N.: 1980b, "Troubles with Functionalism", in N. Block, ed. Readings in the Philosophy of Psychology, vol. 1, Harvard University Press, 268-305.

Boynton, R.M.: 1979, Human Color Vision, Holt Rinehart and Winston, New York.

Broad, C.D.: 1927, Scientific Thought, Routledge and Kegan Paul, London.

Burnham, R.W., Hanes, R.M., and Bartleson, C.J.: 1963, Color: A Guide To Basic Facts and Concepts, John Wiley & Sons, New York.

Cornsweet, T.N.: 1970, Visual Perception, Academic Press, New York.

Dember, W.N.: 1960, The Psychology of Perception, Holt Rinehart and Winston, New York.

DeValois, R.L. and DeValois, K.K.: 1975, "Neural Coding of Color", in E.C. Carterette and M.P. Friedman, eds., Handbook of Perception. Volume 5: Seeing, Academic Press, New York.

Dretske, F.I.: 1981, Knowledge and the Flow of Information, MIT Press, Cambridge, Massachusetts.

Frisby, J.P.: 1980, Seeing: Illusion, Brain and Mind, Oxford University Press, Oxford.

Goodman, N.: 1977, The Structure of Appearance, 3rd edition, Dordrecht Reidel Publishing, Boston.

Haber, R.N. and Hershenson, M.: 1980, The Psychology of Visual Perception, 2nd edition, Holt Rinehart and Winston, New York.

Hurvich, L.M. and Jameson, D.: 1957, "An Opponent-Process Theory of Color Vision", Psychological Review 64(6), 384-404.

Jackson, F.: 1977, Perception: A Representative Theory, Cambridge University Press, Cambridge.

Kaufman, L.: 1974, Sight and Mind, Oxford University Press, New York.

Lewis, C.I.: 1929, Mind and the World Order, Charles Scribners" Sons, New York.

Perkins, M.: 1983, Sensing the World, Hackett Publishing Company, Indianapolis.

Putnam, H.: 1973, "Meaning and Reference", Journal of Philosophy 70, 699-711.

Putnam, H.: 1981, Reason, Truth and History, Cambridge University Press, Cambridge.

Sayre, K.M.: 1976, Cybernetics and the Philosophy of Mind, Routledge & Kegan Paul, London.

Sellars, W.: 1963, "Empiricism and the Philosophy of Mind", in W. Sellars, Science, Perception and Reality, Routledge and Kegan Paul, London, 127-196.

Shoemaker, S.: 1975, "Functionalism and Qualia", Philosophical Studies 27, 291-315.

Shoemaker, S.: 1982, "The Inverted Spectrum", Journal of Philosophy 79(7), 357-381.

Swartz, R.J. (ed.): 1965, Perceiving, Sensing, and Knowing, Anchor Books, Garden City, New York.

Thurstone, L.L.: 1927, "A Law of Comparative Judgment", Psychological Review 34, 273-286.

Torgerson, W.S.: 1958, Theory and Methods of Scaling, John Wiley & Sons, New York.


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