Judgments of Time, Aesthetics, and Complexity as a Function of The

Fractal Dimension of Images formed by Chaotic Attractors


Frederick David Abraham[1],[2], Julien Clinton Sprott[3], Olga Mitina[4]
Maureen Osorio1, Elvie Ann Dequito1, and Jeanne Marie Pinili1




In this experiment, we obtained judgments of aesthetic value, complexity, and duration of presentation of images of chaotic attractors as a function of their fractal dimension (F). We used four levels of fractal dimension (four stimuli at each level, mean Fs = .59, 1.07, 1.54, 2.27), with small samples of six students from each of three populations (elementary school students, graduate students, and special education students from ethnic minority groups in residence on our campus). In addition to replicating earlier studies of aesthetic judgments, we additionally asked for judgments of complexity to see if they were also a nonlinear function of the fractal dimensionality of stimuli (they were, increasing to a maximum at F = 1.54, and falling off at F = 2.27), suggesting the possibility that aesthetics and complexity judgments were at least in part mediated by perceived complexity. Perhaps this non-monotonic result was due to a loss of detail of contrast and detail within the attractors at the highest dimensional complexity. Judgments of duration did not yield a significant F-ratio between groups, but t-tests showed the lowest dimensionality yielded shorter time estimates than those for the higher dimensionalities. If complexity is a determinant of subjective duration of these stimuli, it is saturated at fairly low levels of the fractal dimensionality of the stimuli. Three-way ANOVAs within participants, showed only the fractal dimension as a significant source of variation in aesthetic and complexity judgments, and not academic level, gender, or cultural differences, although there were interesting individual and some cultural differences.


KEY WORDS:  psychophysics, time, complexity, aesthetics, chaos, fractal dimension





Studies of both the perception of aesthetics and of time have centered on cognitive and biological factors (Eisler, Eisler, & Montgomery, 1996; Anderson & Mandell, 1996). Many factors of stimuli, cognition, and biology have been investigated. One of the features of stimuli which make both biological and cognitive demands for time estimation is the amount of perceptual/cognitive effort involved, such as can vary with complexity of stimuli (Cupchik & Gebotys, 1988; Stoyanova & Yakimoff et al., 1987). The same can be said for aesthetic judgments (Aks & Sprott, 1996; Sprott, 1993, 2003).

Perceptual/neural organizational features of attending complex stimuli may affect both the estimation of time and aesthetics, but also, there is the possibility that aesthetics could affect the perception of time. There is likely a nonlinear interaction between the complexity of stimuli and experiencing time and aesthetics. Such nonlinearities have been at the foundation of perceptual psychophysical theories and research of Gregson (1995).

To investigate the possibility of a relationship between complexity and judgments of time and aesthetics it was decided to use abstract stimuli. Chaotic attractors were used for this purpose as they are easy to generate along with objective measurements of their complexity (Aks & Sprott, 1996; Sprott, 1993a,b, 2003) and relatively free of prior associations. Chaotic attractors as stimuli can be created by integration of three coupled differential equations that produce abstract computer images in two dimensions, the third being color coded which adds to their aesthetic potential. To check and see if mathematical complexity has any relationship to perceived complexity, it was decided to add an estimate of complexity to the ratings obtained from the participants.

While complexity of stimuli is the principal independent variable, demographic factors as well, namely age and culture (Eisler & Eisler, 1994; H. Eisler, , were also varied. Therefore, both children and adults, and both urbanized students of Silliman University and cultural minority visitors (Ati and Sulod) to specialized education program to our campus served as participants.

And to enrich the dependent variable side of the experiment, in addition to scales of aesthetics and complexity, and estimates of stimulus duration, some narrative explorations of the features upon which the participants might have been basing their judgments were also attempted.



Participants were from three populations:

(1)                                       Students (six, 1 female, 5 males, from grades 4-5) Silliman University Elementary School were recruited from a special computer learning project under the direction of the senior author. Their participation in this experiment had direct relevance to their goals of the integration of art, science, mathematics.

(2)                                       Graduate Students (six, 4 females, 2 males), Department of Psychology, Silliman University were recruited from one of the graduate seminars.

(3)                                       Ati and Sulod (six, 4 females, 2 males) were recruited from a residential special learning program for the current school year hosted by the University’s Department of Education. They were visiting from several Ati and Sulod communities (Negros Oriental Province, where the University is also located).



In accordance with standard practice for undergraduate research, the locus of informed consent was formally placed in the hands of administers of the educational units involved. Therefore permission to solicit participants was obtained from Ms. Iyoyo, Principal of the Silliman University Elementary School, Dr. Margaret Udarbe-Alvarez, Chair of the Psychology Department, and Dr. Betsy Joy Tan, Dean of the School of Education and administrator for the Ati Educational Project. Individual participants participated by verbal agreement, but that contained the usual elements of informed consent.


The elementary school students were not randomly conscripted from the school, and could be considered as among the high achieving students, being the whole of the members of the experimental enrichment program. The same could be said of the graduate students, who represent a high achieving profile. This achieving profile could work against our finding an age difference, due to the sophistication of both populations. The Ati, also are not a random sample of Ati, being leaders in their communities interested in bringing their Silliman education back to their communities, and from among those, the more adventuresome were among those who volunteered to participate. While these sophistications could have minimized differences in the results, their differences in age, education, life styles, and environment could possibly have influenced the way they responded to the abstract visual stimuli.



Sprott (1993) developed a program for the generation of computer images of chaotic attractors from systems of three nonlinear coupled difference equations which were adapted for use in this and similar experiments (Mintina and Abraham, in press). The equations are:

xn+1 = a + bxn + cxn2 + dxnyn +exnzn +fyn + gyn2 + hynzn + izn + jzn2

yn+1 = xn

zn+1 = yn                    where  x,y,z are variables and a–j are parameters.

Solutions (trajectories) are displayed in two dimensions (x,y, the plane of the screen of the computer) with the third dimension (z) represented by color coding. These stimuli vary in complexity depending on the parameters used in each integration, and are selected by the program at each integration. These programs develop the trajectory (attractors, plotted by point, not as a continuous solid line)) by iteration of the equations, although they evolve so fast as to be essentially perceived as filling space rapidly over time, but not as a trajectory growing over time. Thus of the measures of complexity produced for each image by the program, the fractal dimension of the attractor (which measures complexity in space) is a more realistic measure of the complexity of the trajectory than the largest Liapunov exponent (which measures aspects of the evolution of the trajectory in time). The program was used to generate 168 of these stimuli. A statistics program (STATISTICA) was used to analyze these two measures of these images. Aks and Sprott (1986) used both measures and found aesthetic responses to vary with both measures for some participants. However we chose our stimuli on the basis of having the Liapunov exponents well correlated with the fractal dimension, so it would be irrelevant to try to separate the two aspects of complexity. The fractal dimension, F, employed here is the correlation dimension, usually designated as D2 (introduced to dynamics by Grassberger & Procaccia, 1983; and reviewed in many books and articles, including Sprott, 2003, p. 307-311, and Abraham, 1997, pp. 17-18). Stimuli were chosen with Fs in four ranges, .5–.85, .86–1.4–1.6, and 2.2–2.4 (mean Fs, 0.59, 1.07, 1.54, 2.27). Twenty stimuli, five from each of these four ranges were selected. Four were to be used at the beginning of each run in a fixed order to give a warm-up for the participant, and the remaining 16 were presented in a randomized sequence for the remainder of the run, as described below. These 16 stimuli were used in analyzing the results. These images can be viewed at http://blueberry-brain.org/silliman/jemstim.htm.

Computer Implementation and Instructions

A session on the computer started with a set-up phase by the experimenter in which the instruction set and presentation method were selected, and the participant’s identification and data file name were entered. Next, the participant sat at the computer, and the experimenter gave the participant verbal instructions on how the experiment was to be run. Then the stimuli were presented in succession, going through the sequence twice. The participant responded to each stimulus with three judgments as follows.

The following comprised the two runs through the stimuli, the first run to obtain the participant’s ratings of aesthetics and complexity, and the second run to obtain estimation of the duration of presentation of the images. The first run was completed as follows:

(1)  When the experimenter finished setting up the computer to run the experiment, the participant pressed any key to begin.  Then the first stimulus appeared on the screen. For 1.04 seconds, the stimulus remained on, and then the first question appeared at the top of the screen:

How beautiful was the image [1 least - 9 (most)]?

(2)   Then the experimenter explained the rating scale with the first stimulus.

(3)   The participant responded on the keyboard, with a number and followed with an entering key-press.

(4)   The second question then replaced the first question:
How complex was the image [1 (least) - 9 (most)]?

(5)   To insure that the participant used a wide range of ratings of complexity, the experimenter suggested numerical values for this and the following three warm-up stimuli.

(6)   After that, the participants self-paced themselves for the remainder of the run through the remaining 16 of the 20 images.

(7)   The experimenter then reset the experiment for the second run that was used to get the time estimates.

The second run to obtain the ratings of perceived duration (time) was completed as follows:

(1)   For the estimation of duration of presentation of the images, the same images were used, with the 5th-20th being in a new random order.

(2)   The second run was started with the experimenter preparing the run as in the first run, choosing the new presentation format. Then the participant, as before, sat in front of the computer and initiated the run.

(3)   The stimulus was presented for 10.05 seconds, turned off, and the following question presented:
How many seconds was the image on?
(The participant had to hide the face of a watch if worn, and asked not to count.)

Finally, after the runs are completed, the participants were then asked to describe which features of the images determined their aesthetic and complexity judgments.

Each run took about 5-8 minutes, the whole session took 20-30 minutes. Some participants were run singly, and some were run simultaneously, using up to the 6 computers in the psychology computer laboratory where the experiment was run. When six participants were run simultaneously, it was found that each of the three experimenters could supervise two participants at a time without difficulty.


Data Analyses

The computer program used to run the experiment recorded the participant’s responses automatically. These were then transferred to an Excel spreadsheet program as follows. The responses to the test stimuli for each of the three measures at each of the stimulus complexity levels were averaged for each participant. Also the Pearson product moment correlations were computed for all six pair-wise combinations of the three response measures and the fractal dimension for each participant. Thus a case-by-variables data table was assembled with 24 variables per case: group, gender, the fractal dimension (F) and the judgments of aesthetics, complexity, and duration at each of the four complexity levels, and the six correlation coefficients.

This table was imported by the STATISTICA data analysis program, where descriptive statistics and tables were completed (box & whisker plots, Figures 1, 2, & 3,). Inferential statistics were also computed, namely 3-way ANOVAs for a within-participant design for each of the three measures (with the complexity level being the repeated variable, and group and gender being the independent between-group variables. T-tests for paired measurements were also used to determine the differences for each of the main measures (complexity, aesthetic, and duration judgments) between each pair of adjacent levels of F (low vs low mid, low-mid vs high mid, and high-mid vs high). An alpha level of .05 was used for all statistical tests.

Statistical Results

Judgments of Complexity

The box & whisker plots (Figure 1) show the means, standard error of the mean, and the 95% confidence limits of the complexity judgments. The y-axis represents the means of the judgments of the complexity for each level of complexity of the images. The x-axis represents the four levels of the complexity of the images. The levels of that variable, going from left to right are: low (F = .59, .52–.65), low-mid (F = 1.07, 1.05–1.09), high-mid (F = 1.54, 1.46–1.55), and high (F = 2.27, 2.21–2.36). F is the mean of the fractal dimension for the four stimuli in that level of the fractal dimension; the range is given after the mean.


                                                                                    Insert Figure 1 about here (click)


This figure shows judgments of complexity to be a non-monotonic increasing function for most of the range of F as might be expected, but decreasing at the highest values. That is increasing complexity is perceived as such, until the images at the highest F values which are reported as less complex. Even within the increasing portion of the curve, the result is decidedly non-Fechnerian: increases in F are not met with proportionately increased magnitude of the judgments of complexity. The ANOVA showed significant results for F, but not for gender or group, nor for any interactions. The t-tests showed all adjacent means to differ significantly from each other. All probabilities were way below the alpha level. Tables can be found at http://blueberry-brain.org/silliman/jemanovacmplx.htm.

Judgments of Aesthetics

The box & whisker plots (Figure 2) show similar results for the aesthetics judgments. This plot also shows a non-monotonic form nearly identical to that for the complexity judgments, with maximum aesthetics judgments also occurring at the high-mid level of the fractal dimension of the images. That means that for the three lower levels of the fractal dimension, the participants scaled the aesthetics as increasing with increased complexity, but for the highest level of mathematical complexity (F), they tended to judge the aesthetics as less, quite comparable to the low-middle level.  Again, that result is non-Fechnerian even within the monotonic increasing phase of the curve.


Insert Figure 2 about here


The ANOVA results yielded results almost identical to those for the complexity variable (a table is at http://blueberry-brain.org/silliman/jemanovaaesth.htm). That is, the two between-group independent variables, group and gender, where not significant, while the within-participant main independent variable (levels of F), was also significant. The one difference compared to the results for the complexity judgments were that the 3-way interaction was also significant, a result difficult to interpret (see the figure at http://blueberry-brain.org/dynamics/jemint3xaesth.jpg). The t-tests for adjacent values of F also were similar to those for the complexity judgments (see again the table at http://blueberry-brain.org/silliman/jemanovaaesth.htm).

Judgments of Duration

The results of the judgments of the duration of the presentation of stimuli were not significant. The box & whisker plot (Figure 3) shows (1) no simple relationship to the mathematical complexity (F) of the images, (2) remarkable accuracy of the means of the time estimates to the actual time of 10 sec. despite (3) great variability in the responses (range: 2-16.25 sec). This accuracy is reflected in the means of the four levels, which were 9.31, 10.06, 9.69, and 10.31 sec. (left to right on the box & whiskers plot), with a grand mean of 9.84 sec.


Insert Figure 3 about here


The ANOVA (the table is at http://blueberry-brain.org/silliman/jemanovatime.htm) shows no significant main effects or interactions, which is not surprising considering the lack of trends and the great variability seen in the box & whisker plot. Despite that, the t-test for one of the comparisons, the lo-mid was significantly greater than the lo level of the mathematical complexity variability.

While the interaction terms were not significant, the plots of the means for the 3-way interaction (see table at http://blueberry-brain.org/silliman/jemint3xtime.jpg) showed consistently lower time estimates from females compared to males, and for the Ati/Sulod group compared to the elementary and graduate students.


Correlations: Overall

Pearson product-moment correlations (see the Table) mirrored the results of the box & whisker plots and the ANOVAs. The mathematical complexity (fractal dimension, F) was positively correlated to both the complexity ratings (r = .37) and the aesthetic ratings (r = .46), but not the time estimations (r = .03). The highest correlation was between the aesthetic and complexity ratings (r = .64). The remaining two correlations were remarkably unremarkable, the one between time estimation and complexity ratings (r = .07), and the one between time and aesthetics judgments (r = .05).  Thus, the three correlations involving relationships among fractal dimension, complexity judgments, and aesthetic judgments were strong, while the three involving time estimates were essentially zero.


Insert the Table about here


Correlations: Group Comparisons

The students from the elementary school were similar to the overall means of the correlations, with the exception that their correlation between aesthetic and complexity ratings were higher, and in fact were the highest among the groups (r = .84).  The graduate students were similar to the overall means of the correlations except for being lower (lowest compared to the other groups) on the correlation between fractal dimension and the aesthetic ratings (r = .24). The students from the Ati and Sulod had the highest correlation compared to the other two groups for the relationship between the fractal dimension and aesthetic ratings (r = .67), and the lowest correlation compared to the other two groups for the relationship between the aesthetic and complexity ratings (r = .44).

Thus the highest correlations were those between the complexity and aesthetic judgments (r = .64), with the elementary students being the highest (r = .84), the Ati/Sulod-adult education students being lowest (r = .44), and the graduate students being in between, right at the mean (r = .63). Another interesting group difference was for the correlation of the fractal dimension to aesthetic ratings, which was highest for the Ati/Sulod students (r = .67), lowest for the graduate students (r = .24), and intermediate, right at the mean for the elementary students (r = .47). 

Correlations: Individual Differences

Half the participants showed the typical pattern of high correlations among the fractal dimension and aesthetic and complexity ratings and low correlations involving time judgments (participants: at, fd, lh, rn, mr, jq, jy, ly, am, although one of these—jy— had a somewhat lower correlation between aesthetic ratings and fractal dimension). One (ly) had identical correlations of F to aesthetic and complexity ratings, and a perfect correlation (+1) between aesthetic and complexity ratings, a clear case of response bias or generalization. Two other participants (en, ml) showed this response generalization reflected in high correlations between aesthetic and complexity ratings while the remaining five correlations were negligible, including no relationships to the fractal dimension. Participant (jn) was similar to these two participants, except that the aesthetic to complexity correlation was smaller.

Another participant (am) also showed very high correlations of F to aesthetic and complexity ratings, but their correlation between the two rating was somewhat lower than the mean (r = .35 compared to the mean of .64) showing independence of aesthetic and complexity judgments while both sets of judgments were strongly related to the fractal dimension. This participant showed the unusual result of fairly strong negative correlation of time estimation to F and complexity ratings.

Two participants (ao, ma) showed very strong correlations among F and aesthetic and complexity ratings and some fairly strong positive correlations involving the time estimates.

One graduate participant (ms) a collegiate art instructor as well as a graduate student in psychology, expressed a preference for open space in art, but his response revealed this preference in a negative relationship between aesthetic ratings and F, perhaps small due to the nonlinear relationship between area covered and F. His low correlations indicate of a strong independence of complexity from F as well, and the slight positive correlation of complexity to F, and slightly negative correlation between aesthetic and complexity indicate an independence of these judgments from each other, while somewhat related to F. The zero correlations indicate a constant estimation of time. He appears to be a person with strong independent convictions.




The individual correlations between complexity judgments and aesthetic judgments were the highest of all, with 12 of them being greater than .8, and one was a perfect 1. This is a key finding. Why is it not so for all individuals? Can there be different reasons for different individuals to have high correlations? Can one unravel some of the individual differences by examining the pattern of responses across all measures for particular individuals? The reader is invited to try, before we look at a few in the discussion section later. For example, S15  had identical scores for the two ratings at each of the level of mathematical complexity. Thus (1) correlations of aesthetic ratings were higher to complexity judgments than to actual mathematical complexity (functionally because both were similarly nonlinearly related to mathematical complexity), (2) These were degraded for a few individuals into lower correlations, because some people had aesthetic preferences for lower dimensional stimuli. One of these, S10, a graduate student and art teacher, stated such a preference in his interview, but this preference did not make for large negative correlations which his claim would indicated, but certainly did degrade not only this correlation to near zero (-.066), but all of his correlations. And (3), the only hint of cultural (and gender) differences that can be found in the data are swamped by these individual differences, and the relationships to mathematical complexity are so strong as to survive the individual differences.


Of the remaining three correlations (all for the time judgments), only a few were strong, mainly for participants 2, 7, 14, 15, 16, 17. Thus, there was greater individual variability in the Ati/Sulud group (4 Participants compared to 1 each in the other two groups). Of these, two were strongly negative (Participants 7 & 17), and four (Participants 2, 14, 15, & 16) strongly positive. Three were strongest for the correlations to the mathematical (fractal) complexity (Participants 2, 7, 15), and three were strongest for the correlations to the complexity judgments (Participants 14, 16, & 17), all three of these in the Ati/Sulud group.


There were no large negative correlations of other variables to aesthetic ratings. The exceptions could be with the time variable, where two of the Ati/Sulod group had correlations of .604 and .434. The single large positive correlation between time and aesthetic ratings also came from a participant in this group.




Since the judged complexity is not a simple increasing monotonic function of complexity as measure by F (D2), falling off when F> 2, the question arises as to why that should be? Visual inspection of the images gives one answer. For stimuli where the mean F = .59, the space is quite empty. When going to the next higher F (mean F = 1.07), there are considerably plumper images filling the space more completely (both the viewing space as a whole and within the attractor itself) and having greater intuitive complexity. Going to the next higher (mean F = .154), the increase in the filling of space and the apparent boundaries within the attractor (the third, colored dimension helps a lot with that) is increased only a little, thus the judgments likewise are not much increased. At the highest level (mean F = 2.27), the attractors occupy about the same amount of the viewing space as a whole, but the increase in density of points within the attractor fill the attractor more completely and the attractors appear more like clouds, with less interior definition and boundaries, they are more homogeneous, leading probably to their being judged less complex.

Why does the F (Grassberger-Procaccia D2) measure not see all that the eye sees. There could be other measures of the mathematical complexity of the images, but F sees only the filling of space and does not enjoy the advantages of the eye, brain, and cognition in giving weight to the boundary effects, especially those from the coloring. And unlike other measures of complexity, such as D (the Hausdorff, capactity, box-counting dimension which weights a hypercube in the space equally no matter the amount of occupancy), D2 weights the hypercubes by the extent to which they are each occupied, and thus the more dense images are given greater dimension. We shall return to some of these alternative properties when considering other judgments as well.

A final note could be made as to what one might investigate further concerning judging complexity. Our instructions were quite nonspecific, giving little guidance to what meaning they might wish to construct or assume for it, and there very likely was a great range in both what they considered complex, and the extent to which they may have been aware of features of the images to which they were responding. We have not made a formal analysis of the narratives, but many corroborated the kinds of details mentioned above. The similarity of the three groups suggests that all were rather sophisticated in observing these details, and we shall return to a group difference later after discussing the other judgments. The possibility of interest is to see if people can learn to discriminate different types dimensionality by viewing images and training with them. Can the mathematical definitions of images be learned through experience judging them?

[will check out Atteneve (1957) who used polygons to study judgements of complexity; search for follow up work; derived from his interest in information theory, which is mathenmatically related to nonlinear measures of complexity.]


The history of the psychological study of aesthetics is long and large. Since this feature was not part of the motivation for the present experiment, we will review only the part of it that relates to issues related to complexity and order. While much of the literature deals with simple properties, e.g., those on the well known Golden Mean, surprisingly the empirical work on complexity goes back to Gustav Fechner (1801-1887) and also surprising is that some of the conjectures about complexity apply to some simple forms as well as complex ones (Berlyne, 1960). Berlyne concludes that “The work on the experimental aesthetics of simple visual forms that began with Fechner’s Vorschule der Äesthtik (1876) tends to confirm the view that some intermediate degree of complexity produces the most pleasing effect and the extremes of simplicity or complexity are distasteful (p. 237).” Fechner mentioned the veining in marble. Also, interestingly enough, two of the early contributors to this literature included pioneers of dynamics, the mathematician George David Birkhoff (1884-1944), and the mathematical biological scientist, Nicholas Rashevsky (1899-1972) (see Abraham & Shaw, 1982, 1992), although their analysis of aesthetics apparently did not involve dynamics, but did look at complexity.

The early history of ideas of unity and diversity in aesthetics (summarized in Berlyne, 1960; Gilbert & Kuhn, 1953) was addressed mathematically by Birkhoff (1933) with his formulation that aesthetic value (M) was a function of complexity C of the image (diversity or numerosity) upon which attention and tension depended, and order (unity, due to properties such as symmetry) of the image, upon which  (as updated by Graves, 1951) resolution of the tension depended. Birkoff’s formula was M = O/C. That is, aesthetic value was proportional to order, and inversely proportional to complexity. Very shortly, there were attempts to test this theory (Eysenck, 1941; Davis, 1936) which found, similar to our result, that there was a maximum of aesthetic judgment at intermediate values of Birkhoff’s M, or our F. Our measure of complexity, F, treats complexity as a single dimension stretching between order and complexity, rather than assuming a composite function, although much dynamical thinking looks at oppositional forces along dimensions of the state space, and our generative equations utilize a complex combination of three variables. The curvilinear relationship of all our judgments suggests that these judgments are not unidimensional.

Rashevsky (1938) suggested a possibility based on mathematical assumptions of excitation and inhibition in cortical neurons which gave a “measure of total excitatory effect, which is identified with aesthetic value. Rashevsky’s measure sounds very much like a measure of complexity, but it actually bears a curvilinear relation to Birkhoff’s M, reaching a sharply delineated maximum when M is at an intermediate value (quoted in Berlyne, 1960, p. 239).


Summary & Conclusions





We thank Debbie Aks for sharing her expertise in conducting psychophysical experiments and Anna Lourd Villaneuva, a friend and graduate student, for assisting at making her friends among the Ati and Sulod feel comfortable participating in our experiment. We also thank Ms. Iyoyo, Principal of the Silliman University Elementary School, Dr. Margaret Udarbe-Alvarez, Chair of the Department of Psychology, and Dr. Betsy Joy Tan, Dean of the School of Education, for their support, and Dr. Christian Schales for setting up our computer laboratory, and Dr. Agustin Pulido, President of Silliman University, for establishing that laboratory for the Department.




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 This view was also reflected by Külpe (1895), “The more complicated the object of affective appreciation, the less possible, of course, is it to decide a priori what aspect of it will be effective, and consequently what will be the nature of the resultant feeling.”

[1] Depatment of Psychology, Silliman University, Dumaguete City, Philippines 6200.

[2] Correspondence should be sent to abraham@sover.net

[3] Department of Physics, University of Wisconsin, Madison, WI 53706

[4] Department of Psychology, Moscow State University, Moscow, Russia