It appears that the proposition «the yellow ball is green» is contradictory. It is believed that yellow is not green. If yellow is not green, then the yellow ball is not green, either. Given this, to claim that «the yellow ball is green» would entail that «the yellow ball is not green, and, at the same time, is green.» In this paper, however, I want to show that this proposition is intelligible. In other words, I propose to argue that the yellow ball is, in a very important sense, green.
In physical science, there are basically two kinds of signals: (1) the analog signals, and (2) the digital signals. An analog signal is a physical quantity (or quality) which conveys information in a smooth and continuous form. For example, f (t) = sin (t) can be described as analog signal. In contrast to analog signal, a digital signal is an ordered sequence of information selected from a finite number of discrete quantities (or values). An easy and popular example is the dichotomy signal. We can use high/low, on/off, or 1/0...etc. to express the quantity (or value) of the signal. This makes the digital signals easier to handle. In short, we can manipulate digital signals with greater accuracy and efficiency than analog models would allow. Fortunately, an analog signal can be transformed into a digital one.
Likewise, in expressing (or describing) the objects or the states of affairs, there are basically two kinds of concepts: (1) digital concepts, and (2) analog concepts. The digital concepts are the concepts that generally cover (or include) merely a finite number of discrete categories (or paradigms). «Husband», «wife», «parent», and «sibling» are all among the digital concepts. Indeed, a person P is either a husband or not a husband. There is no such thing as being a little of a husband.Foot note 1 Of course, «ball», «stone», and «cat' are also among the digital concepts. On the other hand, the analog concepts are continuous concepts. Therefore, in theory, they can cover (or include) an infinite number of categories (or paradigms). «Color», «speed», «height», for example, are all among them. Given the characteristic mentioned above, we seldom (or never) apply our analog concepts directly. Conversely, we are inclined to apply them digitally. As Ronald de Sousa correctly says:
Whatever we know, we must categorize. From the continuum of our experience, we need to extract a finite and relatively fixed number of categories...Foot note 2
Yet, to efficiently apply our analog concepts, we should digitalize them first. Indeed, digitalization can provide the needed convenience of categories that we consider as discrete.
We can sometimes see a beautiful rainbow in the sky after it rains. How many colors does the rainbow have? To this question, some might answer, «Of course, five!» The others may insist, «Seven!» In fact, «color» is a continuous (or analog) concept, so, in theory, there can be an infinite number of colors in rainbow.Foot note 3 In practice, we assign color names to roughly mark out areas within a continuous spectrum of hues. But this practice may be different among different peoples or cultures. For example, it is possible for a group of people (the S tribe) living near the polar circle to respectively assign several color names to the corresponding areas within a spectrum of hues which people living in the USA call «white». Put loosely, different whites may have different meanings for the S tribe--danger, safety, predatory moments...and so forth. How precisely should we digitalize our analog concepts? Or how many parts should we divide an analog object or state of affairs (e.g., «color») into? In a sense, the digitalization is quite arbitrary. Indeed, we can divide «color» into any number of parts in terms of our common needs. In practical exercise, we merely assign a finite number of color names to mark out areas within a continuous spectrum of hues. For normally people are not able to sharply discriminate quite a lot of colors (say, 50 colors).
The digital concepts conform to the principle of non-contradiction; whereas the analog concepts do not. The claim that «this is a ball and this is not a ball in the room» is contradictory. So is the claim that «P is Q's father and P is not Q's father.» On the other hand, given that the speed of my bike ranges from 0 km/hr to 100 km/hr, it is true that the speed of my bike is 50 km/hr. It is also true that the speed of my bike is 90km/hr. Accordingly, it is not contradictory to claim that the speed of my bike is both 50 km/hr and 90 km/hr.Foot note 4
In any case, my bike has an infinity of speeds within the range of 0 km/hr and 100 km/hr. Now the question here is, «Would the digitalized (analog) concepts (e.g., quick/slow, yellow/green, big/small) conform to the principle of non-contradiction?» It appears that they conform to this principle. Indeed, as shown above, the digital concepts do conform to the principle of non-contradiction. And the digitalized (analog) concepts are a certain kind of digital concepts. One might argue for this point as follows.
Suppose the speed of my bike ranges from 0 km/hr to 150 km/hr. We can, therefore, digitalize its speed-range into three zones: the low-speed zone (0 km/hr - 50 km/hr); the middle-speed zone (50 km/hr - 100 km/hr); and the high-speed zone (100 km/hr - 150 km/hr). The high-speed zone is also called «the 125 km/hr-speed zone» because all speeds within this zone are assigned as 125 km/hr. Likewise, the middle-speed zone is also called «the 75 km/hr-speed zone»; and the low-speed zone called «the 25 km/hr-speed zone». Now speed 100 km/hr is both in the middle-speed zone and in the high-speed zone. However, «high-speed» is not «middle-speed». Thus, to say that «speed 100 km/hr is in the middle-speed zone and in the high-speed zone» would seem to entail that «speed 100 km/hr is in the middle-speed zone and not in the middle-speed zone.» This appears contradictory. However, I suggest that, in fact, it is not. Remember that these digitalized concepts are in reality analog. Let me explicate this point.
In the 75 km/hr-speed zone (or the middle-speed zone), only speed 75 km/hr can precisely be assigned as 75 km/hr. As for speed 80 km/hr, it is said that it is very close to 75 km/hr, and, therefore, is also treated as 75 km/hr. Similarly, the speed 100 km/hr is assigned as 75 km/hr. On the other hand, in the 125 km/hr-speed zone (or the high-speed zone), only speed 125 km/hr can be precisely assigned as 125 km/hr. As to speed 110 km/hr, precisely speaking, it is very close to 125 km/hr and, thus, also can be treated as 125 km/hr. Similarly, the speed 100 km/hr is assigned as 125 km/hr. Given this, speed 100 km/hr is in a sense in the 75 km/hr-speed zone (or the middle-speed zone), and in another sense in the 125 km/hr-speed zone (or the high-speed zone). In other words, 100 km/hr is in a sense assigned as 75 km/hr, and in another sense assigned as 125 km/hr. Therefore, there is no contradiction here.
At this point, I would like to further argue that «high-speed» can also in a sense be «middle-speed». As we have shown, since speed 100 km/hr is close to 75 km/hr, we assign it as 75 km/hr. What about speed 101 km/hr? Isn't it close to 75 km/hr, as well? If so, we can also, in this sense, assign it as 75 km/hr (or call it «middle-speed»).Foot note 5 Since 101 km/hr is in the high-speed zone, and since it can also be assigned as «middle-speed» as shown above, «high-speed» is «middle-speed» in this very sense. We are now in a good position to justify that the yellow ball is green.
The lights we can see are the ones with frequencies ranging from about 4.3 x 1014 Hz to about 7.5 x 10 14 Hz. Within this frequency range, a light, in an appropriate condition, with a certain frequency would cause us to perceive a certain color. There are an infinite number of frequency values (or quantities) within this range. Put in another way, this frequency-range can be infinitely divided. Thus, in theory, there are an infinity of colors within this frequency-range.Foot note 6 In order to handle it well, we should digitalize it. We customarily use the term «yellow» to refer to those colors caused by lights with frequencies ranging from about 5.0 x 10 14 Hz to about 5.5 x 10 14 Hz. The term «yellow» in fact indicates many (or an infinity of) colors within a certain area of spectrum of hues. Next to yellow is green. The term «green» is customarily used to call those colors whose frequencies range from about 5.5 x 10 14 Hz to about 6.0 x 10 14 Hz.Foot note 7 Here, the exact yellow color can be stipulated as the color which is caused by the light with frequency at 5.25 x 10 14 Hz. Call it color-525.Foot note 8 As to color-530, precisely speaking, it is very close to the exact yellow color. In fact, in a strictly philosophical sense, we cannot say that it is yellow. However, in practical exercise, both color-525 and color-530 are called «yellow». Or both are assigned to the same category--«yellow». Likewise, the exact green color can be stipulated as color-575. If so, color-570 can only be said to be very close to the exact green color. Since it is very close to the exact green color, it can be assigned as «green». Here, we can actually locate a color just at the border of yellow and green. Suppose this is color-550. Since color-550 is close to color-525, it is called (or assigned as) «yellow». On the other hand, it is close to color-575 too, it is also assigned as «green». Therefore, color-550 is yellow, and, at the same time, green. There is no contradiction here.
Suppose there is a ball whose color is color-545. As mentioned above, color-550 is close to the exact yellow color (i.e., color-525), it is called «yellow». Clearly, color-545 is closer to the exact yellow than color-550 (or put loosely, color-545 is yellower than color-550). Since color-550 is assigned as «yellow», color-545 should be assigned as «yellow» too. Therefore, this is a yellow ball.
On the other hand, as stated above, color-550 can also be assigned as «green» because it is close to color-575. Color-545 is very close to color-550, and, in a sense, also close to color-575.Foot note 9 Thus, it can also be assigned as green, in this sense. Given this, this (yellow) ball can also be green in a sense. There is no contradiction here. Indeed, in terms of «vagueness», V. J. McGill and W. T. Parry claim:
...[i]n any concrete continuum there is a stretch where something is both A and ~A...There is a sense in which the ranges of application of red and non-red [in so far as «red» is vague] overlap, and the law of non-contradiction does not hold.Foot note 10
Similarly, Dominic Hyde writes, «...a man growing a beard is at some stage both bearded and not bearded.»Foot note 11 He goes further, saying that:
When a vague predicate is applied to a borderline case we are confronted by a sentence which is neither determinately true...nor determinately false...but indeterminate, which now amounts to the claim that the sentence is both true and false. It is true since true on some admissible precisifications and false since false on some.Foot note 12
To sum up, the analog concepts do not conform to the principle of non-contradiction. To be handled well, they must be digitalized first. But note that these digitalized concepts are in reality analog. Because of this, and, on the basis of our above discussion, the following propositions are all plausible:
Center for General Education and Core Curriculum
Tamkang University, Taiwan
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[Foot Note 1]
Ronald de Sousa, «Love Undigitized», in Love Analyzed, ed. Roger E. Lamb (Boulder, Colo.: Westview Press, 1997), p. 196.
[Foot Note 2]
2. Ibid., p. 197.
[Foot Note 3]
3. Austen Clark also dresses this point. He says:
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.
See Austen Clark, «The Particulate instantiation of Homogeneous Pink», Synthese 80 (1989), p. 277.
[Foot Note 4]
4. Note that «the speed of my bike at t--85 km/hr» is a digital concept. Thus, it is contradictory to claim that «the speed of my bike is both 85 km/hr and 90 km/hr at t).
[Foot Note 5]
5. In fact, how precisely should we digitalize our analogous objects (or states or affairs) can be very arbitrary in terms of our common needs. We can also, for some reason, stipulate «middle-speed» as the speed-range between 50 km/hr and 120 km/hr, and «high-speed» as the speed-range between 120 km/hr and 150 km/hr.
[Foot Note 6]
6. In claiming that there are, in theory, an infinity of colors within this frequency-range, I do not rule out the possibility that, in practice, we perceive colors digitally.
[Foot Note 7]
7. See Vincent P. Coletta, College Physics (New York: McGraw-Hill, 1995), p. 633.
[Foot Note 8]
8. For the rest of this paper, I call a color caused by the light with its frequency at P x 10 12 Hz, «color-P».
[Foot Note 9]
9. There is no reason for us to insist that «green» should cover merely those colors ranging from color-550 to color-600. We can also stipulate that «green» covers those colors ranging from color-540 to color-610 in terms of our common needs.
[Foot Note 10]
V. J. McGill and W. T. Parry, «The Unity of Opposites: A Dialectical Principle», Science and Society 12 (1948), p. 428.
[Foot Note 11]
Dominic Hyde, «From Heaps and Gaps to Heaps of Gluts», Mind 106 (1997), p. 645.
[Foot Note 12]
12. Ibid., p. 649.
[Foot Note 13]
13. I am very grateful to Dr. Graham Priest (University of Melbourne) and Dr. Dominic Hyde (University of Queensland) for their comments on earlier drafts of this paper. Thanks also to Dr. Cheng-Hung Lin (Soochow University) for his help on the topic of paradox.