The Particulate Instantiation of Homogeneous Pink

Synthese, 80 (2), 1989, pp. 277-304.

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

If one examines the sky at sunset on a clear night, one seems to see a continuum of colors from reds, oranges and yellows to a deep blue-black. Between any two colored points in the sky there seem to be other colored points. Furthermore, the changes in color across the sky appear to be continuous. Although the colors at the zenith and the horizon are obviously distinct, nowhere in the sky can one see any color borders, and every sufficiently small region of the sky is made up of regions that all seem to be of the same color.

How can this apparent continuity be explained? One classic answer is simply to admit an infinite number of colors, and an infinite number of sense data, and allow that a glimpse of the sky comprises a completed infinity--a continuous series of colors-at-phenomenal-places. The question of whether the set of sense data (or sense impressions) is finite or infinite created some interesting divisions among classic theorists. Endnote1.

More recently the issue has been invoked as an objection to any attempt to identify sensory experiences with brain states. Wilfrid Sellars is a proponent of a subtle variant of this objection. Sellars' celebrated 'grain' argument begins with the example of a solid pink ice cube:

The manifest ice cube presents itself to us as something which is pink through and through, as a pink continuum, all the regions of which, however small, are pink. It presents itself to us as ultimately homogeneous; and an ice cube variegated in colour is, though not homogeneous in its specific colour, 'ultimately homogeneous', in the sense to which I am calling attention, with respect to the generic trait of being coloured (Sellars 1963b, p. 26).

The homogeneity in question is not uniformity of color, but rather refers to the fact that pink is a simple logical quality: all its parts, no matter how small, are pink.

Ultimate homogeneity is initially treated as a characteristic that the 'manifest image' ascribes to colors of ordinary objects. According to Sellars, that image embraces a naive realist view of colors, and ascribes ultimate homogeneity to those colors. Indeed, it is precisely the difficulties in providing any scientific account of such homogeneity that led Sellars to deny that ordinary things as conceived in the manifest image are literally identical to systems of scientific objects (clouds of particles). Sellars also ascribes ultimate homogeneity to sensations of things: what he calls 'sense impressions' (or, adverbially construed, 'manners of sensing'). Color predicates come to play a second role--an analogous but non-synonymous role--as sortal predicates characterizing sense impressions. In that second use, colors, now construed as manners of sensing, are also ultimately homogeneous:

Colors as manners of sensing form a logical space modeled on colors as attributes of the physical objects of the Manifest Image. They inherit the "ultimate homogeneity" of the latter (Sellars 1971, p. 408). Endnote 2.

Furthermore, just as a cloud of elementary particles cannot (according to Sellars) be pink in the naive realist sense of 'pink', so a collection of particles cannot instantiate the sense impression of pink (or a pinkly sensing), with its homogeneous character. Sellars asks whether one can, within the framework of current neurophysiology, plausibly identify any brain states as sensations. He answers negatively, since

Putting it crudely, colour expanses in the manifest world consist of regions which are themselves colour expanses, and these consist in their turn of regions which are colour expanses, and so on; whereas the state of a group of neurons, though it has regions which are also states of groups of neurons, has ultimate regions which are not states of groups of neurons... (Sellars 1963b, p. 35).

Here Sellars distinguishes between a narrower and a broader class of physical terms. Physical2 terms are those sufficient to provide a theoretical description and explanation of non-living matter, and so would include, for example, the terms of physics. Physical1 terms are a broader class including any which fit within a "spatio-temporal-nomological framework of scientific explanation", hence including those terms (if any) which may be required for the theoretical description of living things but which are not needed to describe non-living things (see Sellars 1965, p. 447). Sense impressions (or manners of sensing) cannot be identified with any physical2 properties, although they can (ultimately) be identified with physical1 properties. The current framework of neurophysiology presumably fails to identify such properties, and indeed it must fail if it proceeds in purely physical2 terms. Any such identity is barred by the ultimately homogeneous character of colors (see Sellars 1971, pp. 406-407, 415; 1963b, pp. 35-36).

If colors (in the naive realist sense) cannot be identified with properties of objects, and also (as manners of sensing) cannot be identified with physical2 characteristics of the neurophysiological states of perceivers, what is one to do with them? Sellars argues that to account for the ultimately homogeneous character of manifest colors and of sensory consciousness, one must introduce

a new domain of scientific objects to be the subjects of these successor color predicates. ...such a successor concept must involve the "ultimate homogeneity" of color, for it is to be the final home, the ultimate "transposition" of the colors of the Manifest Image. In this final transposition color would exist as colored particulars... Thus, at the end as at the beginning of the journey, our image of man-in-the-world would include color with its ultimate homogeneity as an occurrent attribute of actually existing particulars (Sellars 1971, p. 410).

These new scientific particulars are sensa. They are ingredients or constituents of the sensory states of perceivers. Sense impressions can ultimately be identified with neurophysiological states, but only if those states include sensa as constituent elements. Sensa are particulars that exist only in the context of neurophysiological processes. For this reason sensings will be identical to physical1 states of perceivers, but not with physical2 states (Sellars 1971, p. 410; 1981a, p. 87).

1. Ultimate homogeneity

How can we explain the apparent homogeneity of colors and of sensations of colors? Must we introduce a new class of scientific entity to do so?

I shall argue that the homogeneity of manifest colors can in principle be explained using the current collection of scientific particulars. No new entities are required. To do this I need merely show how there could be (for example) a particulate instantiation of the experience of homogeneous pink. I will propose a hypothesis that employs only current neurophysiological principles and entities and that can in principle explain the ultimate homogeneity of colors. I need not (and do not) claim that this hypothesis is true, but only that it could be true.

This proposal will be seen to be consistent with the major tenets of Sellarsian scientific realism, and is perhaps best viewed as a speculation on what sensa might be. That is, if 'sensa' are those states which explain manifest homogeneity and provide it with a final resting place, then this paper gives an account of 'sensa'. It is non-Sellarsian only in the claim that 'sensa' are, at least in principle, within the purview of current neurophysiology.

The crux of the argument concerns the notion of ultimate homogeneity, and the sense in which the colors of objects or the counterpart properties of sensations are ultimately homogeneous. I shall begin by adopting a deliberately over-simplified analysis of this notion. This analysis yields an argument which Sellars clearly does not endorse, and which should not be attributed to him. Nevertheless this argument is suggested by some of his writings, and it is interesting in its own right. An analysis which is closer to Sellars' intention will be considered in the last section of the paper.

The above citations initially suggest an analysis of 'homogeneity' and 'non-homogeneity' in terms of continuity. Recall that the pink ice cube presents itself "as a pink continuum, all of the regions of which, however small, are pink" (Sellars 1963b, p. 26). The 'homogeneity' here suggested is a species of continuity (the species known as density, or what is sometimes called 'compactness'): between any two pink points, there seems to be more pink. Let us initially construe 'ultimate homogeneity' as 'continuity'. Suppose a series is generated by a transitive and asymmetric relation. Given such a relation R, y is between x and z if Rxy and Ryz. Endnote 3 A series is 'dense' if between any two elements there is a third. As a first pass, we will say a system has 'continuous' internal states if those states form a dense series.

The pinkness of the ice cube is ultimately homogeneous just in case between any two pink points on the cube there is a third pink point. Let us grant (with some provisos to be mentioned below) that colors of things appear to be ultimately homogeneous in this sense--that one has experiences of (what appear to be) continuous colors.

How is ultimate homogeneity to be analyzed when it is ascribed to sense impressions (or sensations)? Characteristics of sense impressions are invoked to explain aspects of perceptual propositional attitudes. One way to explain experiences of what appear to be continuous colors is in effect to posit continuous experiences. That is, the sense impression of the cube is ultimately homogeneous if between any two pink points of that impression there is a sense impression of a third pink point. Note that without this strong and rather strange claim, the gappiness of neurons would not bar identities between sense impressions and neurophysiological states. Between two impressions of pink, there is yet another impression of pink; but ultimately between two neurons there is no neuron.

One should note that "between" does not necessarily have the same sense when applied to sense impressions as when applied to things. Sellars notes that spatial predicates, just like color predicates, may well acquire a new sense when applied to sensations. For example, when one senses a red square adjoining a green square, the sense impression of the red square is not literally adjoining the sense impression of the green square (Sellars 1968, pp. 25-26, 30). Instead there is some counterpart predicate "adjoin*" relating the impressions. Similarly, when applied to sense impressions, "between" should presumably be construed in the sense of some counterpart predicate "between*" (Sellars 1968, p. 29). The sense impression of the pink ice cube is ultimately homogeneous if and only if whenever one has an impression of a point y between points x and z on the cube, the impression of y is between* the impression of x and the impression of z.

If one notes the use of this counterpart predicate "between*", one can easily avoid a bad formulation of the difficulty engendered by homogeneity. One might think the problem lies with the spatial gappiness of neurons: that between neurons there are ultimately regions that are not neurons. This is not a real problem, however, since "between" is not used in the same sense when applied to sensations. One is not committed to the claim that a sense impression of a third pink point is literally spatially located between the sense impressions of an enclosing pair. The counterpart predicate "between*" need not imply spatial betwixtness.

What then is the difficulty in ascribing homogeneous sensations to a particulate system? Even if one allows "between*" as ascribed to sense impressions to have a different meaning than "between" as ascribed to things, the proposed analysis of homogeneous sense impressions yields a dense series. Whatever "between*" means when applied to sense impressions, between* any two there is a third.

Take a 'non-homogeneous' or 'particulate' system to be a system that does not have dense internal states, so that no matter which class of internal states one chooses, there are states x and y in it between which there is no distinct member state z. Instead x has y as a neighbor. Such a system will also be called 'digital'. Any system with only a finite number of distinct internal states is digital. The spike action of neurons has a digital character, in that the neuron either fires or does not, and it fires some integral number of times per second. The computationally relevant states of digital computers also satisfy the definition.

This analysis yields the required incompatibility between homogeneous sensings and particulate systems. Any digital system has a (finite) set of discrete internal states. Although different combinations of switch settings may give it an enormous repertoire of different internal states, nevertheless any element in any series of such states will always have a neighbor; hence no such series is dense. The cardinality of the set of such states will be less than--infinitely less than--that of the purported continuous experiences. How then could the experience of the sky at sunset ever be instantiated in a digital system? The hypothesis seems a non-starter. It seems that to explain experiences of colors which appear continuous, one must invoke experiences which in a certain sense are continuous. Sensa will indeed have some property which corresponds to--is a scientific counterpart of--homogeneity. Endnote 4. Our initial analysis of that homogeneity is that some counterpart predicate "between*" will apply to sensa in such a way as to generate dense series. Without such continuous experiences, how can one explain experiences of continuous colors?

2. The non-transitivity of indiscriminability

An alternative answer starts with a rather awkward fact of phenomenology, namely that indiscriminability is a non-transitive relation. One may find that stimulus objects x and y look the same, or match in appearance. Strengthen this to the claim that x and y are indiscriminable: that one cannot reliably identify which is x and which is y in pairwise presentations. Technically, x and y are 'pairwise indiscriminable' if over a sample of trials of presentations of x and y, the distribution of one's choices of which one is x and which is y is not significantly different from a random distribution. Similarly, objects y and z may match or (more strongly) be indiscriminable. Nevertheless, we may find that objects x and z do not match--they can be discriminated. So even though x matches y, and y matches z, one cannot be assured that x matches z.

Matching would seem to be a paradigm example of an equivalence relation, and its surprising non-transitivity can be explained as follows. Suppose the differences between items x and y are not of sufficient magnitude to allow one to distinguish the two. Similarly, the differences between y and z may engage no discriminative response. Nevertheless, the sum of the differences, from x to z, may be sufficient to allow one to distinguish that pair. Indiscriminability is a failure to discriminate, and here the sum of two failures may lead to success.

`Matching' and 'indiscriminability' are terms that will be applied only to external stimuli--typically, physical objects impinging on receptor surfaces. The sensations or sense impressions caused by such things are internal states of the perceiver, and as such can neither literally match nor fail to match one another. They are likewise not in the domain to which the term 'discriminable' can be applied. Instead there is a different but analogous relation that will be applied to the sensations or sense impressions of the perceiver: such internal states are in varying degrees 'qualitatively' or 'phenomenally' similar to one another (see Shoemaker 1975, pp. 293-294; Shoemaker 1984). While sensations can neither match nor fail to match one another, we can say that the sensation of something red is qualitatively similar to the sensation of something reddish-orange, and much less similar to the sensation of something green. The limit of qualitative similarity is presumably qualitative identity. Sensations are qualitatively identical if having one of them is exactly like having the other.

The non-transitivity of matching is demonstrated whenever one views a sunset. If one focuses on a sufficiently small region of the sky (a narrow visual angle), one cannot detect any difference in color within the region; it appears all to be of the same color. If the night is clear enough, this may be true of all the (sufficiently small) regions of the sky. But the sum of many such regions yields a spectrum whose end points are clearly discriminable. The counter-intuitive consequence of these observations is that there must be qualitative differences between the sensations engendered by indiscriminable items. As Hume put it

For, if this shou'd be deny'd, 'tis possible, by the continual gradation of shades, to run a colour insensibly into what is most remote from it; and if you will not allow any of the means to be different, you cannot without absurdity deny the extremes to be the same (Hume 1739, Book I, Part I, Section I).

Suppose we have a match between x and y, and y and z, but not between x and z. If the sensation of y were qualitatively identical with that of x, then the indiscriminability of y and z would establish that x and z must also be indiscriminable. But x and z are discriminable. Hence the sensations engendered by x and y must be qualitatively distinct, although x and y are (pairwise) indiscriminable. For the sake of an abbreviation, such a difference will be called an 'indiscriminable qualitative difference': a qualitative difference between the sensations engendered by indiscriminable things. Qualitative differences exist below the thresholds of discriminability. Matching, or even indiscriminability, is too weak a relation to define qualitative identity; x and y must match, not only one another, but also the same sets of items z (see Goodman 1977, pp. 196, 209).

It is important to emphasize that the non-transitivity of matching can be established purely on phenomenal grounds, from 'within' the manifest image. Its grounds are quite distinct from those motivating any materialist identifications. Nevertheless, once one admits that qualitative differences can exist below the threshold of discriminability, one has the wherewithal to explain how a digital system could have experiences of continuous qualities.

3. How to Construct Continuous Experiences

Suppose an observer is examining a photo mural of the sky at sunset, and we label six contiguous patches of the photo with the letters 'a' through 'f'. Endnote 5. Suppose further that the mural is produced in the lab in such a way that the observer finds that pairs of patches match as long as the two members of the pair are within any contiguous span of four. (All pairs in {a,b,c,d} match, all pairs in {b,c,d,e} match, all pairs in {c,d,e,f} match, and so on.) Any two patches within the span of four 'look the same' upon inspection, and there is no noticeable difference between their appearances. Nevertheless, it may be possible to learn how to discriminate patches separated by lesser spans. Discrimination is indicated by a distribution of identifications that differs from a random distribution, and one can sometimes manage such statistically reliable discriminations among a collection of objects all of which 'look the same'. Phenomenologically this corresponds to an ability to sort stimuli into classes that are statistically distinct from random even though one lacks any awareness of any distinguishing characteristics of the classes. Suppose all the contiguous pairs of patches are indiscriminable, and those separated by a greater span are discriminable. So, for example, c and d are indiscriminable, as are d and e, but c and e are discriminable. The differences between the relations can then be diagrammed as follows:

      a       b        c        d        e       f 


            |<--------- matching --------->| 

Notice that one might find that b matches e, and e matches f, but b fails to match f. Even though b matches e, the sensation engendered by e must be qualitatively distinct from that of b, as e matches a patch that b fails to match. If one employs the more powerful relationship of indiscriminability, the same sort of result can be demonstrated for the sensations engendered by patches c, d, and e.

How do qualitative differences between the sensations engendered by indiscriminable items allow one to explain the continuity of experience? All of the features explained by continuous experiences can be explained as well on the following hypothesis:

The Hypothesis of Digitized Experience: The sense impression of an object is instantiated in a digital array of elements. Neighboring elements in the array encode the qualities of neighboring points on the object. Those object points are spatially distinct, but cannot be discriminated as such. Likewise, the properties encoded by neighboring elements in the array are qualitatively distinct, but indiscriminably so.

Each element encodes the properties of some portion of the scene, and the elements are organized as an array by the relative locations of the portions they encode. Endnote 6. Neighboring elements encode portions of objects that are spatially distinct, but that cannot be discriminated as being distinct. Between neighboring elements of the array we suppose there is an indiscriminably small change in the position of the point encoded. Spatial discriminations are non-transitive just as are other kinds. Since the sum of two indiscriminably small shifts in position can result in a discriminable shift in position, we must allow that indiscriminably close positions can engender qualitatively distinct sensations.

The system is digital: it is instantiated with a finite set of discrete elements, and those elements have neighbors. The hypothesized array is an internal state, and is not itself perceived. The point of the argument is to show that its properties suffice to explain all the features of perception that might be explained by genuinely continuous experiences. The key to such an explanation is that the differences encoded by neighboring elements in the digital array are all below the threshold of discriminability. The locations encoded by neighboring elements are indiscriminably close, and the properties encoded are indiscriminably distinct. Crudely: the experience is instantiated in qualitatively discrete parts, which however encode properties too 'small' to apprehend.

Recall that the existence of indiscriminable qualitative differences was established phenomenologically, and is not simply ad hoc. We know that there are such indiscriminable differences. The hypothesis makes use of a known quantity.

Consider first the sunset. Suppose there is a digital array of elements in your visual cortex, neighbors in which encode color differences between portions of the sky--differences that are, however, indiscriminable. Endnote 7. When you examine the sky, you would then find no discriminable edges within it, since all changes in color occur beneath the threshold of discriminability. Nevertheless the sum of such changes across a meridian is clearly discriminable, and even noticeable. One explanation for the appearance would be that between any two sensuously apprehended colors there is a sense impression of a third color, and hence that experience is genuinely continuous. Another is that there are only a finite number of distinct impressions of color, but that neighbors encode color differences that are indiscriminable. Both would present a continuous appearance, but one is, at base, digital.

Consider next the homogeneous pink ice cube. It seems that every pink region of the cube is composed of a collection of pink regions. Since there is no element in a digital array between neighboring elements, we seem unable to explain homogeneous pinkness by appeal to any finite collection of parts. The array is gappy, while the pink expanse is not.

However, the 'gaps' between locations encoded by neighboring elements may be so small as to be indiscriminable. Just as one can show that there are indiscriminable qualitative differences in color sensations, so one can show that there are indiscriminable differences in the sensed places in the visual field. The simplest such test is the 'minimum separable', which tests the smallest visual angle at which two lines in a grating can be seen as two. As the lines in a grating are made smaller and closer together, eventually one can no longer detect distinct lines, but sees instead a homogeneous gray. The digital proposal is in effect that every 'homogeneous' gray is instantiated as a gridwork of distinct lines, in which neighboring elements encode visual angles below the minimum separable.

Between any two discriminable pink points of the homogeneous pink ice cube there will seem to be a third (and pink) point. One can identify no non-pink regions of the cube. However, both findings are consistent with the idea that between discriminably different points there are indiscriminably different ones. A finite number of such, each with some finite set of values, yields the appearance of a continuous color expanse.

These ideas can be expressed in terms of signal detection theory. In a decision task there is some criterion threshold below which items are judged to be the same, and above which they are judged to be different. Such a criterion is an inevitable feature of decisions concerning variable items. Items judged the same may be qualitatively identical, or they may have differences that fall below threshold. The hypothesis of digital experience is simply that encodings possess fidelity that exceeds the thresholds for pairwise difference detection. One mistake in classic theories was to suppose that the criterion for differences in sensations can be set at such fidelity that they are judged to differ just in case they do differ. (This is another version of the incorrigibility of the mental.) If one allows for a loss of information between encoding and difference judgments, then one must accept the existence of indiscriminable qualitative differences.

A photograph can be digitized by measuring each tiny region (or 'pixel') of the photograph, and representing its light intensity by some one of a finite number of distinct intensity values. The digital character of such pictures is quite easy to recognize if one employs a small number of pixels or of distinct intensity values. However, as the number of pixels and the number of intensity values increases, digitized pictures come closer and closer to approximating the original photograph. Indeed, at some point the digitized picture will be indiscriminable from the original. The digital hypothesis is that visual representation could, for all we know, be similar to such a digitized picture: one which is so well digitized that it is indistinguishable from the original, apparently continuous one. We know that there can be qualitative differences between sensations even though their corresponding stimuli are indiscriminable. Perhaps visual experience is digitized too, but at such a fine level of resolution that 'neighbors' encode indiscriminable differences. The overall result would be a visual experience that would be qualitatively similar--and perhaps even qualitatively identical--to our own.

One can pairwise discriminate some finite number of distinct locations in one's visual field. According to the digital hypothesis, there are more pixels in the array instantiating your visual experience than there are pairwise discriminable locations in your visual field. Endnote 8. Any discriminable location is therefore instantiated as a collection of indiscriminably close pixels. Pairwise spatial discrimination is too crude to identify all the qualitative differences between sensations of location. Endnote 9. Furthermore, each pixel has some one of a discrete set of intensity values, but properties encoded by successive values cannot be discriminated from one another. So perhaps the qualitative differences between neighbors are discrete, but encode differences below the resolving power of discriminability. The result is a visual field that seems continuous.

These problems are not new, and indeed many of these ideas are strongly suggested by certain passages in Berkeley's Essay Towards a New Theory of Vision (1709). Following Berkeley, let us call the visual angle subtended by the smallest visible point a minimum visibile. He writes:

Of these visible points we see at all times an equal number. It is every whit as great when our view is contracted and bounded by near objects as when it is extended to larger and remoter ones (Berkeley 1709, Section 82).

In fact the number of such points must be some finite integer. Furthermore, "the minimum visibile is never greater or lesser, but in all cases constantly the same" (Berkeley 1709, Section 86). That is, since sensitivity to visual angles is unaltered by the items one is examining, the array of minima visibilia is just the same when one is viewing distant hills as when one is in one's study, or peering through a microscope.

The main difference between Berkeley's account and a modern version is that the former denies while the latter accepts the principle that minima visibilia have 'parts'--e.g., that analysis can isolate visually significant elements with a finer resolution than those of the minimum visibile. Berkeley argues that

...the minimum visibile having (in like manner as all other the proper and immediate objects of sight) been shewn not to have any existence without the mind of him who sees it, it follows that there cannot be any part of it that is not actually perceived and therefore visible. Now, for any object to contain several distinct visible parts, and at the same time to be a minimum visibile is a manifest contradiction (Berkeley 1709, Section 81).

For Berkeley, analysis of visual qualities must stop at the minimum visibile, since anything smaller cannot be seen. All differences between neighboring visibilia must be immediately apparent to the senses. If one allows indiscriminable qualitative differences, however, one can reject this principle. Although minima visibilia are the smallest visible elements, analysis of visual qualities must proceed to a finer level of detail, simply because two elements that both match some third one may fail to match one another.

On the digital hypothesis, a minimum visibile is composed of distinct parts (pixels), which can encode indiscriminable differences in visual qualities. This hypothesis denies the Berkelian principle that all differences between two sensations can be (directly) sensed.

4. Other Examples

It should be stressed that the hypothesis of digitized experience is purely hypothetical. Its intent is to show how one could explain experiences of what appear to be continuous qualities without invoking either continuous experiences or some new class of scientific particulars that have some counterpart property to continuity. It will be useful to consider how the digital hypothesis might explain some other examples.

Altered Illumination. Evidence for the digital and perhaps quantal nature of visual sensation is provided by vision under optimal conditions of sensitivity, which are those after prolonged dark adaptation. Adaptation can increase one's sensitivity to light by a factor of at least 100,000. At maximal sensitivity, random noise in visual channels may occasionally summate to produce brief pinpoint flashes in the visual field. You may experience these yourself after a sufficiently long adaptation in perfectly dark room. In terms of the hypothesis, under dark adaptation the discrimination of brightness differences is so heightened that activity in just a few pixels can sum to a discriminable point. Indeed, quantal activity sufficient to excite some ten receptors can, under optimal conditions, yield a reliable discrimination (Cornsweet 1970, p. 25).

What happens to our homogeneous pink ice cube as the lights dim? Certain light intensities are optimal for color discrimination. In daylight the cube may readily be discriminated from one composed of red and white speckles. As the lights dim, however, color discriminations worsen, and a greater number of pixels are recruited to sum to a discriminable difference. At some point one will be unable to discriminate the pink cube from the red and white one. At that point, what seems to be a homogeneous pink cube may in fact be a digitally speckled red and white one: the speckles are simply below the threshold of discriminability. Why cannot the same obtain for experiences in daylight? It does not suffice to point out that illumination and discrimination thresholds are different in daylight. One simply supposes smaller speckles for the correspondingly lower thresholds.

Flicker Fusion. Visual experience has an apparent temporal continuity as well as spatial continuity. If one perceives a constant stimulus, there seems to be no moment in which the stimulus is absent. So one assumes that between any two sensations of the presence of the stimulus, there is a sensation of the presence of the stimulus.

At appropriate intensities, frequencies, and amplitudes inconstant or flickering lights can be perceived to flicker; that is, one perceives periods of absence of the light. As the frequency increases, however, it becomes more and more difficult to perceive the flickering, until at some point (the 'critical flicker fusion frequency') the flicker will be undetectable, and the stimulus will appear to have some constant intensity. Thousands of experiments have been done determining the psychometrics of the fusion frequency and its dependence on intensity, amplitude, and other properties of the stimulus.

Flicker fusion has philosophical significance. It shows that the experience of something that appears continuous does not entail the existence of continuous experience. Neural firings are, after all, temporally discrete, and one may crudely wonder how some sequence of such discrete events could instantiate the experience of a continuous stimulus. But there is no contradiction to be resolved. Experience of a continuous stimulus may be a sequence of discrete events whose temporal discreteness is below the level of discriminability. A sensation of visual continuity may just be one whose encodings are discrete, but occur at such a frequency that the events encoded are above the critical flicker fusion frequency.

Flicker fusion underlies perception of motion pictures. The scenes pictured by the sequence of discrete frames are perceived as continuous. One does not perceive any moment during which the scene is absent, so one naturally supposes that there must at every moment be a perception of its presence. But again the inference fails. There may be a sequence of internal 'frames' (encodings) whose frequency is such that successive frames are temporally indiscriminable. Perhaps the experience of a continuous stimulus is at base digital, but the frequency of encodings is such that the events encoded fuse.

Apparent Motion. When the movie actor swings a golf club, the club seems to describe a continuous path. It appears to traverse every point in the swing, and there is no point that it appears not to traverse.

Again one assumes that the apparently continuous swing must be encoded by a dense series of sensations. Such an assumption is not mandatory. There is a threshold for flicker fusion, and a threshold for spatial location, and so there is one as well for spatial change (measured in change of visual angle) per unit time. For a movement to be detectable it must exceed a certain visual angle per unit time. As long as the change in position per unit time is less than that threshold, the change in position is not discriminable. Perhaps the experience of continuous motion is a sequence of discrete encodings of position, in which the change per unit time between neighbors is below that threshold. Then at no time would one perceive a discrete jump in position, yet position over time would change.

The phenomena of apparent motion are here quite suggestive (see Goodman 1978, Chapter 5). We arrange two light sources near one another so that we can flick them on and off separately, but precisely control the interval between flashes. Below a certain interval, the two flashes are experienced as being simultaneous. Above a certain interval, two temporally distinct flashes are seen. Between those limits, one has the experience of a single light, moving from one source across to the other. The light seems to traverse all the points between the two sources. Clearly the visual system has no encoding of the light passing through those various points, since in fact the light does not move through them. Hence distinct encodings could, within the appropriate thresholds of change in visual angle over time, give rise to the experience of continuous motion. Perhaps all our experience of continuous motion employs the mechanism revealed in apparent motion.

Other Modalities and Qualities. The existence of an apparently continuous color spectrum, or of any of the other apparently continuous qualities (such as sounds, tastes, or smells) can be explained on the same principles. The retina contains three different kinds of color receptors, which differ merely in the frequency of light to which each is maximally responsive. Each neuron in the optic nerve fires some discrete finite number of times per second. It may seem impossible to explain the structure of our color experience in terms of such neural firings. How can colors seem continuous, so that between any two there is an intervening series at no point in which one can perceive a discrete change?

Since any neuron has a finite maximum rate of firing, there are only a finite (but large) number of distinct ratios of excitation of three distinct color systems. This suggests that ultimately the 'color solid' is a discrete structure of points. Between distinct points there is no intervening series, but a quantal jump in color. But perhaps the neighboring points are so 'close' that their differences cannot be perceived, and so the color solid appears continuous. In traversing such a solid there is no place at which one perceives a distinct change in color, but one can reach an endpoint with a color distinct from the beginning.

For other modalities the principles of the explanation are the same. If there is some feature of perception which manifests apparently continuous gradations, that feature can be used as a 'dimension' to order encodings; and we hypothesize that neighbors in that order encode indiscriminable differences. Any apparently continuous gradation is then explained as a discrete sequence of such indiscriminably small steps. Many modalities are spatially topographic (e.g., vision, touch, hearing, somesthesia) and the original hypothesis was stated in terms of arrays of elements organized by the spatial reference of encodings. However, the same explanation can succeed for modalities in which perceptions are not organized spatially. For example, smells may make a smooth transition from musky to acrid, even though odors are not spatially localized. To account for such modalities, we suppose that olfactory encodings are ordered, not in terms of spatial reference, but rather along whatever dimension corresponds to the musky-acrid axis. The digital hypothesis is then that neighbors in that order encode indiscriminably distinct odors, and that a sequence of such signals yields phenomenally gapless olfaction.

5. Two Objections

If the hypothesis of digitized experience succeeds in explaining how a digital system could instantiate experiences of what appear to be continuous qualities, then it shows there is no inconsistency in ascribing an 'ultimately homogeneous' sense impression to a system of particles. But two related objections concerning the success of such explanations are likely to be raised.

The first is that the hypothesis does not genuinely explain the apparent continuity of experience, but merely shifts the burden onto the back of some ghostly homunculus. One can invoke the homunculus by asking: who has the experiences of what seem to be continuous qualities? While the visual system may employ digitized arrays, temporally discrete encodings, a discrete color solid, and so on, what explains apparent continuity in each case is the existence of qualitative differences below the threshold of discriminability; and who is doing the discriminating? The explanations all seem to posit an internal movie, viewed by a ghostly homunculus; and it is the homunculus who has the experiences of apparent continuity, not the digital system.

This objection misportrays the intent of the hypothesis offered above, but in a rather subtle way; and it is easiest to explain the problem with an example. Thomas Young was the first (in 1801) to demonstrate that any color in the spectrum can be matched by suitably mixing different proportions of three 'primary' colors. From this demonstration he concluded that there were just three different kinds of 'sensitive particles' in the retina, and that all colors are formed by the different ratios of 'undulation' of sensitive filaments from the three.

One can interpret Young's theory so as to invoke a homunculus. We imagine three inner projectors throwing lights up on a ghostly screen, and a homunculus watching the inner movie. After all, nerve undulations have no apparent color, and so we suppose the appearances must be constituted at some later stage, inside a homunculus. But one need not endorse this interpretation, and it misinterprets Young's result. What Young's demonstration shows is that since any spectral color can be matched by mixing just three colors, all of the information available to the nervous system concerning color can be captured in just three independent variables. Young's color 'particles' instantiate those variables. They are not themselves seen; instead they are internal states of a mechanism, intended to explain seeing. They encode sufficient information to produce any color match, and hence could yield experiences qualitatively the same as those occasioned by any arbitrary color.

Similarly, I have argued that any of the stimuli providing appearances that are considered continuous (e.g., of sunset, pink ice cubes, apparent motion, and so on) could be matched by a digitized version, as long as differences between stimuli encoded by neighboring elements are indiscriminable. This would show that the visual system could function with no more information than is available in the digitized versions, and hence that all the feats described by 'continuity of experience' could be achieved with discrete means. Such demonstrations of indiscriminability do not establish that the nervous system only employs discrete encodings, just as Young's did not establish that it uses only three kinds of color receptor. It could of course employ continuous encodings, or more than three kinds of receptor. But such demonstrations show that digitized experience (or trichromatic color vision) is possible; that no more than that is required.

Finally, just as with Young's theory, there is here no need to claim that the hypothesized internal states are themselves perceived. The digital encoding is not something that can be examined in the same way as a picture. You do not literally see it; you see objects. Nor do any thresholds of discriminability apply to it; they too apply only to objects. The digital encoding is an internal state that helps explain some features of perception. Since a digitized stimulus can in each case be indiscriminable from one giving rise to experiences of continuity, all of the variables required to explain experiences of continuity are available to a digital system.

This leads to a second objection. One may complain that homogeneous sense experience has not been explained, so much as explained away. Instead of dissolving the dilemma posed by Sellars, does not this proposal merely accept one side of it?--namely that, as Sellars puts it

the neurophysiological image is complete and the ultimate homogeneity of the sense qualities (and, hence, the sense qualities, themselves) is mere appearance in the very radical sense of not existing in the spatio-temporal world at all (Sellars 1963b, p. 36).

The proposed account seems to adopt the view that while experiences seem homogeneous, in fact they are not homogeneous (but ultimately digital); hence that the apparent homogeneity of experience is only an appearance. Instead of accepting the continuity of experience, and attempting to explain how it could be manifest in a particulate system, have we not simply denied that experiences are continuous?

In one sense of course the digital hypothesis does deny that experiences are continuous, since its point is precisely to show that we do not need to invoke new scientific entities that are continuous or that have some counterpart to continuity in order to explain experiences of continuity. But it should be noted that there is a second sense in which the digital hypothesis does not deny that experiences are continuous, but rather explains how it is that they could have that character.

Here is an ambiguous sentence: The experience of the pink ice cube is continuous if between any two pink points of the ice cube there seems to be another pink point. A simple shift in the scope of the intensional context gives us the two different senses of the claim that experiences are continuous. One is that the experience of the ice cube is such that it seems that between any two pink points there is a third. The second is that between two experienced pink points, there is an experience of a third pink point. The former I have called 'experience of continuity'; the latter 'continuous experience'. My claim is that the former (which yields gracefully to digitization) is all that can be meant by the 'ultimate homogeneity' of sensory experience, and that the latter is in fact not a phenomenal feature of experience at all. It is not a feature which experience can appear to have.

In what sense does the pink ice cube present itself as a pink continuum, as pink in all its regions no matter how small? There is a certain oddity in the claim that something appears as a continuum. Suppose we come across an ice cube whose appearance is described as follows:

(C) pink in every region, no matter how small.

Call this a 'continuously pink' appearance (adverbially, we sense such a cube continuously-pinkly). We come across a second ice cube whose appearance we describe as:

(D) pink in every visible region, no matter how small; colorless in any region subtending an angle less than the minimum separable.

Call this a 'digitally pink' appearance. Regions subtending angles less than the minimum separable have no visible width or height, so in a sense are not visible regions at all, and so have no apparent color. The problem (as argued above) is just that cube (C) might visually match (D), or the two might even be indiscriminable. If so, in what sense is (C) a property that a cube can appear to have?

Cubes (C) and (D) purportedly differ in the appearance presented by regions subtending angles less than the minimum separable. (C) implies those regions are pink, while (D) implies that they are colorless. These appearance claims are incompatible: if a region appears pink, it does not appear to be colorless, and conversely. But the two cubes match. For this reason, insofar as (C) is distinguished from (D), it cannot be a purely phenomenal description of an appearance; it must make use of what has been called the 'epistemic' or 'propositional' sense of 'appears' statements as well.

In the purely phenomenal or comparative sense of 'looks', if x looks P and y looks just the same as x, then y looks P as well. Certainly if y looks non-P, then it cannot look the same as x. However, there is also an epistemic or propositional use of "appears" and similar verbs, distinguished by an embedded "that" clause, in which "it appears that it is P" does not classify an experience but rather has the force of endorsing the proposition that that thing there is P--that the object has that property.

If the cubes are indiscriminable, any characterization of differences in their appearances must employ the epistemic or propositional sense--not the phenomenal sense--of "appears." The point is illustrated by an old Vermont joke:

Two Vermonters are passing by a freshly painted barn. One says "It looks as if Hebb painted his barn." The other responds "On this side, anyway."

How can the barn appear as if its currently invisible regions are freshly painted? (And of course, a purely phenomenal use of "appears" would eliminate reference to painting as well.) Similarly, to the extent that "appearing as a continuum" is an appearance, our digitally pink ice cube can share that appearance. The ice cube appears homogeneous in the same sense in which the entire barn appears freshly painted. To distinguish the appearance of a continuum from the appearance of a non-continuum, one must employ the epistemic sense of "appears".

A pink ice cube appears continuous in the sense that it matches the appearance that would be presented by a pink continuum. A digitally pink cube could do the same. Since the two might be phenomenally indistinguishable, it seems the digital hypothesis can in principle explain all the phenomenal features of experiences of continuity.

6. Ultimate homogeneity and simple qualities

While the argument from apparent continuity is of interest in its own right, I said above that it is fairly clear that it is not Sellars' argument. For Sellars the 'ultimate homogeneity' of the pink ice cube is not the continuity of the series of pink locations, but is rather related to the reducibility of properties. Roughly, his "principle of the strong reducibility of properties" is that "every property of a system of objects consists of properties of, and relations between, its constituents" (Sellars 1963b, p. 27). The color of a compound object (such a brick wall) may consist of the colors of its various constituents. However, if the constituents of a system of objects have no color, it would seem that the system they constitute must also have no color. This is what Sellars means by the 'ultimate homogeneity' of color: the color of a system of objects does not consist of properties and relations among non-colored constituents.

Applied to my example it says that the pinkness of a whole (the pink ice cube) does not consist in a relationship of non-pink parts. ... the being colored of colored objects (in the naive realist sense) does not consist in a relationship of non-colored parts (Sellars 1971, pp. 408-409).

Similarly, sense impressions (or sensings) are also ultimately homogeneous, since "a sensing can include other sensings, as when we sense a-red-circle-in-a-green-square, but it cannot consist of non-sensings" (Sellars 1971, p. 409). The claim that colors (or the analogous properties of sense impressions) are ultimately homogeneous is in effect one way to deny that the principle of strong reducibility applies to the color properties of systems with non-colored constituents (see Sellars 1981a, pp. 68-70).

Since according to Sellars it is absurd to say that molecules are red (or that they sense redly) the implication of strong reducibility conjoined with the ultimate homogeneity of color is that it is false to ascribe color to a system of molecules. Furthermore, one must deny strict identities between manifest objects and systems of molecules, as well as between sensings and physical2 states of the brain. Sellars states explicitly that both denials rest on the principle of reducibility:

Without it, I cannot conclude from the "ultimate homogeneity" of color that colored objects do not consist of non-colored constituents (micro-physical particles). Nor could I conclude from the ultimate homogeneity of sense impressions that persons do not consist of particles which severally do not sense (Sellars 1971, p. 408).

Objects and sense impressions are conceived in the 'manifest image' to have properties which do not consist of properties and relations of micro-physical particles, and so manifest objects and sense impressions cannot be identified with systems or states of such particles. Ultimate homogeneity bars the reducibility of color properties, and so bars the identification of manifest objects or impressions with systems of scientific objects.

One somewhat ambiguous way to describe ultimate homogeneity is to say that all the parts of a pink expanse, no matter how small, are pink. This may seem to imply (as the continuity analysis of homogeneity does imply) that every pink region of the expanse consists of smaller pink regions. But the Sellarsian analysis of homogeneity in terms of strong reducibility can avoid such an implication. Simply enough, one can claim that when the parts of the pink expanse become so small as to be invisible, they are no longer properly 'parts' of the pink expanse. Unfortunately the term 'pink expanse' is ambiguous, and can mean either the pink area of an object that is sensed or the sensation of a pink area. It is the latter sense that is critical here, and the claim is properly that every part of a sensation of a color expanse is a sensation of some visible expanse. I will call this view the principle of the universal visibility of the parts of a sensed visual expanse. It is recognizable as a variant of Berkeley's claim that the minimum visibile cannot have any parts or properties that are unperceived.

In one sense of 'homogeneous' colors, pixelized vision allows homogeneous colors within a particulate system. Can it also help us to see how the pinkness of an expanse may consist of properties and relations of non-pink parts?

One immediate implication of the non-transitivity of matching is that the universal visibility principle must be rejected. Instead one must admit that there are elements of the sensation of a color expanse that do not encode visible expanses. This admission is forced upon us by any of the series of objects x, y, and z in which x and z both match y but fail to match one another. In such a series one must admit a qualitative difference in the sense impressions of x and of y, even though one senses no difference between x and y. That qualitative difference is not sensed, but is rather inferred from matching data. Furthermore, non-transitivity of matching will apply to places within the expanse as well. The sum of the indiscriminable differences in location from x to y and from y to z yields a discriminable difference in location from x to z. It seems that there are parts and properties of the sense impression of a surface that encode places so close together as to be indiscriminable. What seems to be a single sensed 'place' in the phenomenal field can be a composite of qualitatively distinct sensations. The locations they encode differ, but indiscriminably. Berkeley's minimum visibile turns out to be a sum of distinct pixels whose separate contributions are imperceptible.

In effect the sensation of the color expanse has elements whose identity criteria are not directly phenomenal, but which require a finer level of differentiation. I shall call them sub-phenomenal. Equivalence classes with respect to matching (or even indiscriminability) may contain several distinct sub-phenomenal elements. It would be inaccurate to say simply that these elements are non-phenomenal, since ultimately their individuation does rest entirely on matching data; furthermore, sums of sub-phenomenal differences can yield a phenomenal difference. For this reason too sub-phenomenal elements of the sensation of the expanse are genuinely elements of the sensation. After all, if in our sensation of the sunset or ice cube there was not an element y that matched both of the non-matching elements x and z, then one would have a sensation of an abrupt change; and one does not have any such sensation. It is best to think of a sub-phenomenal element as one that is qualitatively distinct from its neighbors, but whose distinctness cannot be revealed by direct discriminations (or sensings).

The location encoded by a single pixel is sub-phenomenal: it is indiscriminable from locations encoded by neighboring pixels. That location is not pairwise distinguishable from those of its neighbors, although it can be distinguished from them by appeal to matching data. A 'single' place in the phenomenal field can be thought of as an equivalence class with respect to spatial matching; it contains several such sub-phenomenal (indiscriminably distinct) locations.

Can this pixelized vision offer any suggestion as to how the pinkness of an expanse might consist of properties of non-pink parts? There is certain irony in the choice of a pink ice cube, since its color must be obtained by a mixture of colors. Pink is a desaturated red; that is, a red mixed with white light. White is itself a mixture of wavelengths, and can be obtained by combining approximately equal proportions across the spectrum. Pink must be obtained by mixing lights; there is no single wavelength which is ever seen as pink. Nor do we possess receptors that respond specifically to 'pink' wavelength mixtures.

Once pixels are allowed to encode indiscriminably distinct locations and qualities, an obvious suggestion is that the pink expanse consists entirely of a mixture of red and white pixels, digitized at such a resolution that their distinct identities and the distinct colors they encode cannot be discriminated. A pointillist can, after all, paint a pink expanse using only red and white dots. Perhaps none of the sub-phenomenal parts of the sensation of the pink expanse have that pink qualitative character; instead they have the qualitative characters corresponding to red and to white. A pinkly sensing may be nothing but a well-stirred mixture of redly sensings and whitely sensings.

Current psychophysiology might be called upon to buttress this suggestion. All wavelength combinations are coded in terms of the extent to which they activate a red-green system, a yellow-blue system, and a brightness system (Hurvich 1981). Perhaps all the pixels involved in sensing any shade encode only varying intensities of red, green, yellow, and blue. White must be coded by the visual system in terms of equal levels of activation in the red-green and yellow-blue systems. One can imagine each pixel encoding just one opponent process system, with the resulting array somewhat similar to the differently colored dots of a color television. Its main difference from video is that the locations encoded by adjacent pixels in the perceptual array will be indiscriminably close.

This still does not quite get us to the final resting place for pink. One can admit that every pink expanse might consist of indiscriminably small red and white parts (or better: red, green, yellow, and blue parts), and nevertheless maintain that its pinkness is ultimately homogeneous. The point of claiming that colors are homogeneous is not to deny that colors sometimes consist of mixtures, but rather to deny that they consist of mixtures of colorless parts. Generically, "the being colored of colored objects ... does not consist in a relationship of non-colored parts" (Sellars 1971, p. 408). Although pink may consist of non-pink constituents, the important point is that they are still colored (red, green, yellow, and blue). Can one explain pink as consisting of properties and relations among constituents that have no color at all?

Pixelized vision perhaps yields a faint glimmer of an answer to this last question. Each individual element in the digital array encodes a sub-phenomenal location: in our digital array, the place encoded by y is indiscriminable from the places encoded by neighboring pixels x and z. Phenomenally, its place is the same as theirs. Point y certainly has no discriminable width or height. We are obliged to assign it a location which differs from that of the apparently identical x only because it matches a point that x fails to match.

Sub-phenomenal locations are rather counter-intuitive. The final stage of the argument is to suggest that, properly speaking, such locations do not have properties corresponding to sensed colors. Crudely put, they are too small to be seen, and anything too small to be seen has no sensed color. More precisely, the region of the color expanse encoded by a given pixel has no discriminable width or height, and indeed seems located in just the same place as the distinct region encoded by the neighboring pixel. If both of these characteristics obtain, how can such a location have a color? In a sense it is not a visible location at all. In principle it has no visible extension; there is no way in which the area encoded by a single pixel can ever be seen as an area. If things that in principle are too small to be seen have no color, then the location encoded by an individual pixel has no color.

This view suggests one way in which it might be plausible to claim that pixel x encodes some quality that cannot be seen, and pixel y encodes some quality that cannot be seen, but their aggregation yields a visible colored location. The qualities ascribed to a single pixel are not phenomenal qualities; at best they are sub-phenomenal ones, simply because none of them can be (directly) sensed. In combination with others they yield sensed color; yet they are not sensed, and so cannot be sensed color. Perhaps from unsensed qualities one can derive a sensed quality. In effect pixelized vision suggests the existence of sensory atoms, too small to be seen, whose patterns of combination suffice to generate the properties observed in those combinations.

Of course what I have called the 'faint glimmer' yielded by pixelized vision will not banish darkness everywhere; at best it suggests one possible way in which colors might consist of non-colored constituents. If the suggestion is consistent it will have fulfilled its purpose. Discriminations are poor in such murky illumination, however, and there is no intent to suggest that reality necessarily corresponds to the outline sketched here.

Ultimate homogeneity certainly seems to be a feature of colors, but the non-transitivity of indiscriminability suffices to refute simplistic formulations and to force revisions in others. It suggests a route by which colors might consist of properties and relations of constituents that are not exactly colored, but have qualities which when aggregated yield color. Perhaps a reconciliationist scientific realism is after all possible, in which the pinkness of that famous ice cube consists of properties and relations of non-colored constituents. Endnote 10.


1. For example, Hume and Berkeley seemed to countenance only finite sets of impressions, while Kant (in the First Critique, Anticipations of Perception, A 169/B 211 and elsewhere) argued that there were no finite limits to differences in the field of appearance. <Back>

2. See also Sellars 1971, pp. 406, 409, 422-423; and Sellars 1963b, pp. 36, 37. <Back>

3. See Russell 1903, p. 214. Russell writes "this definition gives not merely a criterion, but the very meaning of betweenness." <Back>

4. Sellars 1971, pp. 409-410, 416-417. See also Sellars 1963a, pp. 104-105; 1963b, p. 37; 1981a, pp. 85-88; 1981b, pp. 56-57, 59. <Back>

5. We use a photo--rather than the actual sky--purely as a matter of convenience. The photo makes it easier to test discriminations by presenting the same patch on multiple occasions. To do the same with the sky, one would need to capture the luminous flux from different regions, and somehow present that same flux--or at least an instance of the same type--on multiple occasions. <Back>

6. "Encoding" should not be taken in the sense of a linguistic or conventional system of representation, but rather in the pure information-theoretic sense of statistical-cum-causal regularities between two ensembles of classes of events. See Dretske 1981, Chapter 1. Such regularities are necessary and sufficient for an event in one ensemble to 'signal' or (in the sense intended) 'encode' an event in the other. <Back>

7. Neighboring elements in the array need not be instantiated in anatomically neighboring neurons, though as a matter of fact this happens surprisingly often. The neighborhood relations of elements in the array are defined purely by the relative locations of the stimulus points each encodes. Discontinuities in the somato-topographic mapping from receptors to cortex are hence allowed, and they too occur. <Back>

8. Each pixel is associated with a particular visual location, where 'location' is construed as viewer-centered, or, more precisely, eyeball-centered: a particular visual azimuth and altitude defined relative to the axis of one's visual fixation. Properties of the pixel presumably encode properties of whatever thing is encountered at that azimuth and altitude. The pixel should not be identified with the location or with the thing occupying that location, but is rather the element in the digital array which encodes that location. <Back>

9. Note that discriminations can be used to distinguish y from its matching neighbor x. The fact that y matches some z that does not match x licenses the inference that y is distinct from x. In that somewhat indirect sense discrimination data can be used to distinguish each of the phenomenally distinct elements. However, direct pairwise discriminations will not do so. These matching results create an interesting puzzle for some classic theories. They seem to show that there exist differences in the qualitative character of sense impressions that cannot be 'known by acquaintance' and cannot be revealed in 'immediate awareness'. Instead such qualitative differences can only be established by inferences from matching data. <Back>

10. I would like to thank Paul Brown, Hector-Neri Castaneda, Larry Hardin, Chris Hernandez, Leigh Kelley, Dick Lind, and Bill Lycan for their comments and criticisms. Errors remaining are all my own. <Back>


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

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

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

Goodman, N.: 1978, Ways of Worldmaking, Hackett Publishing Company, Indianapolis.

Hume, D.: 1739, A Treatise of Human Nature, edited by L. A. Selby-Bigge, reprinted 1888, Oxford University Press, Oxford.

Hurvich, L. M.: 1981, Color Vision, Sinauer Associates, Sunderland, Massachusetts.

Russell, B.: 1903, The Principles of Mathematics, W. W. Norton, New York.

Sellars, W.: 1963a, 'Phenomenalism' in Science, Perception and Reality, Routledge & Kegan Paul, London, 60-105.

Sellars, W.: 1963b, 'Philosophy and the Scientific Image of Man' in Science, Perception and Reality, Routledge & Kegan Paul, London, 1-40.

Sellars, W.: 1965, 'The Identity Approach to the Mind-Body Problem', The Review of Metaphysics, 18(3), 430-451.

Sellars, W.: 1968, Science & Metaphysics: Variations on Kantian Themes, Routledge & Kegan Paul, London.

Sellars, W.: 1971, 'Science, Sense Impressions, and Sensa: A Reply to Cornman', Review of Metaphysics, 23, 391-447.

Sellars, W.: 1981a, 'Is Consciousness Physical?', The Monist, 64 (1), 66-90.

Sellars, W.: 1981b, 'Naturalism and Process', The Monist, 64 (1), 37-65.

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

Shoemaker, S.: 1984, 'Phenomenal Similarity' in his Identity, Cause, and Mind, Cambridge University Press, Cambridge.

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