Shades Of Reality

Chapter Six

As we mentioned in chapter two, the human mind has the ability to form generalized concepts from specific occurrences. For example we can look at two different oak trees and, even though they may be of different sizes and have completely different shapes in their branches, we still recognize and categorize them by using the word, "tree." Even a completely different type of tree, such as an evergreen tree, is still recognized as being in the category called "tree." Words such as "tree" therefore represent a kind of shorthand notation for summarizing all of the separate occurrences of such similar plants.

But if you plant an acorn, is it immediately a "tree?" When it first begins to sprout is it then a "tree?" Which things are properly described by the word, "tree," and which things are not? Words, like "tree," have an imprecision about them. There is no clear-cut distinction between "tree" and "non-tree." Therefore, words like "tree" represent fuzzy sets.

But nouns aren't the only words that are fuzzy. Adjectives too can have vagueness to them. For example we can speak of a "big" tree, or a "red" apple, or a "good" person. But where is the boundary between "big" and "little", between "red" and some other color, or between "good" and "bad?" Just how bad does a person have to be before they can no longer be referred to as being a "good" person? (As the old saying goes, "There's a little bit of bad in the best of us, and a little bit of good in the worst of us.") The answer is, there isn't any threshold line. Instead a person's, "goodness" is measured by his or her membership value in the fuzzy set called "good persons."

However not all words are fuzzy. Consider for example the adjective, "five," as in "five trees." While there may be a degree of vagueness associated with the word, "trees," there is nothing vague about the word, "five." (I don't mean "four," and I don't mean "six.") Since words are merely a shorthand way of expressing concepts, if a concept (such as "five-ness") is crisp, then so too will be the word corresponding to the concept.

Let's briefly take a look at some commonly used fuzzy words and phrases and see how they are generally given incorrect crisp interpretations.

As has been pointed out many times in these first several chapters, there is a human propensity to try to collapse the real world into perfectly straight lines and points. For example on a road atlas, most of the cities and towns are represented as single black dots, not because the towns are too small to be seen as identifiable shapes on the map, but simply because we think in terms of points.

Look at the three "objects" (dots) shown in Figure 6.1. Which of the two end objects (a or b) is nearer to the center object, c? Which one is farther away from c?

(Figure 6.1)

The answer of course is: a is closer and b is farther away. (In fact a is almost twice as close to c as b is.)

But now look at the three objects shown in Figure 6.2. Which of the two objects, a or b, is closer to the object, c? Which of the two objects is farther away from c?

(Figure 6.2)

Now you would probably answer that b is closer and that a is farther away, even though the center of each object is identical to the "centers" of the objects shown in Figure 6.1.

When objects have non-zero size (i.e., when they are bigger than points), there seems to be an unwritten convention mandating that distances be determined by using only the very closest two points on each of the objects. Why these two somewhat arbitrary points should be the sole determiners of distance is a mystery to me.

In fact one can argue that object b is both nearer and farther away than is object a! Because there exist points (on the left side of object b) that are closer to point c than are any of the points on object a. And at the same time there also exist points (on the right side of object b) that are farther away from c than are any of the points on object a!

This leads to one of those amusing contradictions that fuzzy logic is so famous for:

b < a

while at the same time,

b > a.

Of course these inequalities are only true to some degree. And the degree to which they are true is determined by the fraction of the objects' areas for which the inequalities are valid.

One of the activities I enjoy doing the most is traveling. In 1989 I bought a little motorhome, and I eventually drove it through every Canadian province and every state in the U.S. except for Hawaii. (I've been to Hawaii several times but never in the motorhome -- too damp trying to get there!) So am I now qualified to claim that "I have been in every state in the United States?"

Let's pretend that you are a native Californian and that you've never been outside of the state. But someday you would like to be able to make the claim of having "been in every state in the country." So you set out on your first trip -- an excursion along what is left of old historic Route 66, the famous highway that used to connect Los Angeles (or more precisely, Santa Monica) with Chicago.

Starting at Ocean Avenue in Santa Monica you head east along Santa Monica Boulevard, carefully following your Route 66 maps and guidebooks. (In many areas the old road no longer exists, and so you must take detours to follow the approximate route.) After driving about 200 miles through Los Angeles, San Bernardino, Barstow, and Needles, you eventually cross over the Colorado River to reach your first "new" state, Arizona.

After driving across Arizona for about 350 miles (of mostly desert), you eventually cross the state line into New Mexico (where you get to see another 350 miles of mostly desert!)

Next you reach the Texas panhandle, and after driving about another 175 miles (and passing through Amarillo) you arrive in your fourth new state, Oklahoma.

Once again you have another 350 miles of driving to do (taking you through Oklahoma City and Tulsa) before you reach your next new state, Kansas.

Now, Kansas is kind of unique in respect to Route 66 -- there are only about 12 miles of the road there. The route just barely passes through the southeast corner of the state before reaching the Missouri state line. (Once in Missouri, you will again have to travel hundreds of miles before you reach the final state for Route 66, Illinois.) So, having traveled through these 12 miles of Kansas, is it really fair to say that you have now "been in Kansas?"

If you had simply reached across the Kansas state line with your little finger and then turned around and went back home, would that too have counted as "being in Kansas?" If after returning from such a trip you were to tell one of your friends, "I was in Kansas," and you offered no further words of explanation, would your claim match your friend's perception of what really happened?

If a statement conveys a high degree of misinformation, then that statement must have a high degree of falseness associated with it. Therefore the expression, "I've been to somewhere," is true only to some degree, and false to some degree. And the degree of falseness is determined by how much of the "somewhere" that you haven't been to. (When viewed in this perspective, nobody has ever been much of anywhere!)

Even the claim that you traveled on old Route 66 is only partly true. Since most of the actual asphalt for the road no longer exists (or has been paved over with new asphalt), what exactly does the expression, "being on Route 66," mean? Does it count if you're merely in the general vacinity of the former road? And if so, then just how far away can you get before you are no longer on Route 66? Or you might even turn the question around slightly and ask, "How much of your trip was not spent on Route 66?" And even more to the point, "How much of the entire Route 66 did you miss traveling on" (either because you took side excursions, or simply because parts of the original road no longer exist)?

All claims of having "been" somewhere are fuzzy. For example, have you ever "been" to your own home town? There are many places even there that you have never been (for example, a restaurant in which you've never eaten, or an alley that you've never walked through, or the bathroom in a stranger's house on the other side of town).

Imagine an ordinary electric lightbulb connected to a simple on/off switch. When the switch is in the "on" position, the lightbulb glows at full brilliance, and we say that the lightbulb is "on." When the switch is in the "off" position, no electricity flows, and so we say that the lightbulb is "off."

But now imagine the same lightbulb connected to a dimmer switch. When the switch is set to its maximum setting, the lightbulb once again glows at its full brilliance. When the switch is set to its minimum setting, no electricity flows. When the switch is set to any intermediate setting, the lightbulb glows with an intermediate level of brightness that is linearly proportional to the setting.

Assume that the dimmer switch initially starts out at its minimum setting. And then, over a period of one minute, the switch slowly and uniformly moves to its maximum setting. When does the lightbulb go on?

Whenever I ask anybody this question, the answers they give are almost always the same: "The lightbulb goes on the instant the switch starts moving. Because the bulb has to be considered as being "on" if even so much as one electron starts flowing through it."

Then I reverse the question. I start the dimmer switch at the maximum setting and let it move toward the minimum setting. When I ask: "When does the lightbulb go off?" the answers are again unanimous: "The lightbulb doesn't go off until the switch reaches the minimum position and all of the electricity stops flowing."

But there is a fundamental inconsistancy to their reasoning. If the term "going on" is to be interpreted as making the transition from an extreme state (no electricity flowing) to a non-extreme state (a small amount of electricity beginning to flow), then by symmetrical reasoning, the term "going off" should mean making the transition from an extreme state (maximum electricity flowing) to a non-extreme state (a small amount of electricity ceasing its flow). In other words if "going on" means leaving the minimum setting, then "going off" should mean leaving the maximum setting.

For example instead of having a dimmer switch that changes the brightness of a lightbulb, imagine one that somehow changes its color. Let's say that when the switch is at its minimum setting the bulb is red. But as the switch moves to its maximum setting, the color of the bulb smoothly changes to blue. Therefore, red corresponds to "off" in our previous discussion, and blue corresponds to "on." If I now ask the completely equivalent question: "When does the bulb go blue?" would the answer still be that it happens at the instant the switch leaves its minimum position because "the presence of even the slightest amount of blueness makes the bulb blue?" And if so, then how about the second question: "When does the bulb go red?" Would the answer still be that the bulb doesn't go red until the switch is all the way back to its minimum position and all of the redness is gone? It seems to me that if the presence of even a small amount of blueness justifies calling the bulb blue, then the same reasoning should be valid for red as well.

By now we should realize that a lightbulb on a dimmer switch doesn't just go "on" or "off" (or blue or red) at a specified point. Instead, at any setting of the switch, the lightbulb is always "on" to some degree, and also "off" to some degree. Therefore my question, "When does the lightbulb go on?" was intentionally misworded (since some degree of "on"-ness exists at every position of the dimmer switch). But our discussion about the common interpretation of the word "on" illustrates the traditional notion that if any part of an entity exists (such as "on"-ness), then the entity totally exists. (This same notion was also manifested in our former example of claiming to have "been in Kansas" simply by virtue of having been in any part of Kansas.) I refer to this mind-set as: drawing the line at zero. We will see more examples of this peculiar mind-set, particularly in Part Two of this book.

Place a small chunk of dry ice on an empty table inside of a warm room, and then leave for several hours. Chances are when you return to the room, the dry ice will have completely sublimed, and the table will once again be empty.

This question might seem like nothing more than a warmed-over version of the sorites paradox of the heap. But there is at least one important, albeit subtle, distinction. The heap paradox looked for the point of transition between two existing entities (i.e., a cluster of sand grains called a "heap" vs. a cluster of sand grains not called a "heap"). The question about the dry ice takes the matter even further and raises the question about the boundary line between existence and non-existence.

You might want to argue that the chunk of dry ice never did cease to exist. It's molecules are still there, but now they exist as a gas instead of a solid. But then a similar objection could also be raised about the heap paradox -- each of the removed particles of sand is still there too, only in a different location.

Next, you might be tempted to claim that the chunk of dry ice ceased to exist when it became too small to be seen. But too small to be seen how? With the naked eye? With a magnifying glass? With a microscope? Where do you "draw the line" between visible and not visible? Of course you could just choose some arbitrary size, like maybe the wavelength of light, and declare that to be the limit of seeability. But to what avail? Only an ostrich would think that something is really gone just because it can't be seen.

And so you eventually "draw the line at zero" and claim that chunk of dry ice continued to exist until its size reached "zero" (i.e., the exact instant at which its very last molecule of carbon dioxide sublimated). At that precise moment in time it instantly changed from the category called "existing," into the category called "not existing."

But we already know from the Smoothness Principle that no such exact time can be determined. Besides, there were probably many carbon dioxide molecules in the air touching the surface of the table, even before you brought the chunk of dry ice into the room. Would you call each of those molecules "chunks of dry ice" too?

The English language has many words for expressing comparitive gradations of certain concepts. For example, "colder," "cooler," "warmer," "hotter," etc. all describe degrees of temperature (no pun intended). And words like "tinier," "smaller" "larger," etc. all describe degrees of size. But there are no words like "iser" and "isn'ter" (i.e., more than or less than "is" and "isn't") for describing degrees of existence. Such words don't exist because the concept of degrees of existence doesn't exist (until now!). To our all-or-nothing mentalities, existence either is, or else it isn't. There is nothing in between those two possibilities. Something either exists -- or else it doesn't. Period!

When it comes to matters about existence we "draw the line at zero." On the "above zero" side of the line is total existence. On the other side is total non-existence. But if "everything is a matter of degree," as the fuzzy principle states, then what are the degrees of existence?

At one time or another in your life you've probably had somebody ask you the following riddle (or some version of it):

If it takes three men four hours to dig a hole, then
how long will it take one man to dig half of a hole?

Upon first hearing the riddle you probably did a little arithmetic -- the number of man-hours required was 3 times 4, which is 12. So it should take half that long (6 hours) for a single person to dig half of the hole.

"Six hours," was your answer.

"Wrong!" exclaimed the riddler. "There's no such thing as a half of a hole!"

Conventional logic dictates that holes exist all-or-nothing. The instant you remove the first shovelful of dirt, you immediately have a hole -- a whole hole.

For our purposes, a hole can be regarded as being an anti-heap (and likewise, a heap can be regarded as being an anti-hole.) And if you start digging the hole on level ground, you will create both simultaneously. Every grain of sand that comes out of the hole will go to form part of the heap. And when the time comes to fill in the hole, the heap and the hole will annihilate each other to once again produce level ground. In fact, if there were such a word as "isn'ter," you might say that a hole isn'ter a heap! (If a "non-existent heap" refers to almost level ground, then a hole would be a heap that has even less than non-existence!)

By symmetry, any arguments about heaps should also apply to their anti counterparts, holes. If a heap is a fuzzy concept, then a hole is a fuzzy concept.

Besides dry ice and holes, what other kinds of things might have degrees of existence? (Actually, everything has a degree of existence. But the degree is usually 100 percent!)

Let's return to our question about when the sky gets dark. At any point in time the question, "To what degree is the sky dark?" has an answer that lies in the range 0 percent to 100 percent, depending on how dark it is. But this simply means that, at any point in time, darkness exists to some degree. (And, at any point in time, daylight also exists to some degree.)

Next, consider the alternate form of the same question. "Is it dark yet?" can be answered by "yes" to some degree, and "no" to some degree. But this merely implies that "yes" partially exists in the answer, and "no" partially exists in the answer. In fact any question that requires a degree of "yes" and a degree of "no" in its answer implies partial existence of each of these two components.

As we have already seen, truth is a matter of degree. Therefore, any statement is only true to some degree. But once again, this simply means that truth only partially exists in the statement.

One of the best examples of partial existence comes from the field of atomic physics.

Around 1920 the Danish physicist, Niels Bohr, fostered the belief that the electrons of an atom were tiny negatively charged particles that revolved around the positively charged nucleus (in much the same manner that the planets of our solar system orbit the sun).

In 1923 the French physicist Louis de Broglie suggested that electrons (and all subatomic particles) had the properties of waves, and in 1927 experiments were performed that showed him to be correct.

Around the same time, the German physicist Erwin Schrodinger had developed a model of the atom in which all of its particles were viewed as consisting of tiny packets of waves.

Today the atom is no longer viewed as a nucleus surrounded by orbiting electrons, but as a nucleus surrounded by an electron cloud. Until quite recently this "cloud" (which is essentially the solution of Schrodinger's wave equation) was regarded as indicating the probability of finding the electron at each particular point in the atom. But the view currently coming into acceptance is that the electron cloud is not a probability distribution, but the actual distribution. Each electron therefore partially exists everywhere around the nucleus, but its degree of existence is a function of location.

As another example, consider the very concept of "shades of gray." "Gray" represents nothing more than the partial existence of "black" and the partial existence of "white." In fact this forms the basis for a computer graphics technique known as "anti-aliasing."

One of the problems with trying to draw a black and white line in low resolution computer graphics is that if the line has only a gentle slope, the resulting image becomes a series of noticeable "stair steps," (see Figure 6.3). This is because each pixel in the image is either a black pixel or a white pixel. (Either a dot of light exists on the computer screen, or else it doesn't exist on the screen.)

(Figure 6.3)

But if we allow for partial existence of pixels (i.e., pixels which are not just on or off, but shades of gray in between), then it is possible to produce a line in which the visual effect of the stair steps is reduced. An example of such an anti-aliased line is shown in Figure 6.4. (Stand back several feet and compare Figure 6.4 with Figure 6.3.)

(Figure 6.4)

At first, the concept of something only "partially existing" might have seemed like an impossibility. But when viewed in the proper perspective, it becomes quite understandable. In fact we can now see that the entire concept of existence can be viewed equivalently in a number of different ways. For example if a lightbulb (or a pixel, etc.) is "on" to some degree (let's say, 30 percent) then we can make the following equivalent statements:

In closing, let me just point out that intangibles (like "good," "evil," "pain," "happiness," etc.) can now be viewed in terms of having only partial existence (or existence to some degree) at any point in time. For example, saying that a person is experiencing "minor pain" is essentially equivalent to saying either that "partial pain" exists, or that pain "partially exists." For all practical purposes both are synonymous.

However, when dealing with tangibles (like an apple), the concept of partial existence becomes somewhat vague. A half of an apple is not quite the same thing as a whole apple that only half-exists. (Perhaps half-existence of a whole apple would result in some kind of "etherial" apple that you can't quite get your hands on.) But in another sense, a half of an apple is an apple that only partially exists. (After all, the whole apple isn't there.) It's kind of a matter of semantics. It's kind of fuzzy.