Shades Of Reality

Chapter Four

When we make a statement, we assert a description of some aspect of reality. The extent to which the statement conforms to fact is a measure of the truth of the statement.

Philosophers distinguish logical truth from descriptive truth. Statements that indicate logical or mathematical relationships (such as: 3 + 2 = 5) are referred to as logical truths, while statements that specify facts about physical reality (such as: "The Earth is round") are referred to as descriptive truths.

The Polish logician, Alfred Tarski, expressed the relationship between descriptive statements and their truth. His statement formula of truth says:

"STATEMENT" is true if and only if STATEMENT.

The left-most term, "STATEMENT" (in quotes), indicates an assertion. The right-most term, STATEMENT, describes a fact. (In other words, "STATEMENT" are the actual words we use to express the fact.) For example, the statement "The sky is blue" is true if and only if the sky is blue. And "Der Himmel ist blau" is true if and only if the sky is blue.

Fuzzy logic (or "continuum logic") views truth as accuracy, and accuracy is obviously a matter of degree. Statements of logical truths are always either 100 percent accurate or 0 percent accurate. But statements about physical reality have accuracy values that lie in between these extremes.

Words represent fuzzy sets. For example a term like "blue" does not have one and only one precise meaning. The word "blue" includes a continuum of possible colors: light blue, baby blue, azure blue, powder blue, etc., each having a slightly different membership value in the set called "blue." And so a statement like "The sky is blue" is accurate only to the extent that the sky's actual color is a member of the "blue" set. (We will return to this concept in Chapter Six.)

The more accurately a statement describes reality, the more truthful the statement is. And to whatever extent the statement inaccurately describes reality, the more false the statement is. All statements are therefore both true to some degree, and false to some degree. Even statements of logical truth (such as "2 + 2 = 4") are both true to some degree and false to some degree -- their degree of truth is 100 percent, and their degree of falseness is 0 percent. (Of course an incorrect statement, such as "2 + 2 = 3," is 0 percent true and 100 percent false.)

Other statements whose degree of truth can be 100 percent are statements of establishment (or definition). For example the statement, "The city of Madison is the capital of Wisconsin," is a statement whose degree of truth is 100 percent, simply because Madison was established (or defined) as being the capital of Wisconsin. These kinds of statements reflecting such "man-made" truths are therefore tautologically true. However, as we pointed out in Chapter Zero, man-made "truths" are (in general) only as true as they are consistent.

In summary, statements of truth can be classified into three general types:

Let us return now to the discussion we started in Chapter One about what point in time the sky gets dark. The terms "day" and "night" represent fuzzy sets. Therefore, any point in time has some degree of membership in both sets. And since the "night" set is the complement of the "day" set the sum of the membership values in the two sets for any point must be 100 percent.

If at some point in time, someone were to make the statement: "The sky is dark." That statement would have a degree of truth somewhere between 0 percent and 100 percent, depending on the time's membership value in the "night" set. Similarly, the statement would have a degree of falseness (equal to 100 perecnt minus the degree of truth). For example if the statement were to be 83 percent true, then it would also be 17 percent false. So at that particular point in time, the question, "Is it dark yet?" would have an answer. The answer would be: 83 percent yes, and 17 percent no!

Also along these same lines, in the previous chapter I asked the question: "Is coin f (in Figure 3.3) inside of the circle?" By exactly the same reasoning, the answer to that question would be 25 percent yes, and 75 percent no.

It is very important that you fully understand this concept of partially "yes" and partially "no" (or "yes" to some degree, and "no" to some degree) as being a valid answer to a question. Such an answer does not mean that I can't make up my mind about which choice is the correct answer. Nor does it mean that I'm being wishy-washy by trying to avoid answering the question. An answer of "yes" to some degree and "no" to some degree is the answer to the question.

Most people find it difficult to comprehend this apparent contradiction. They've been brought up to think that all questions should always have answers that can ultimately simplify to a "bottom line" result of either completely "yes," or else completely "no." They feel that it's got to end up being one way or the other. But the answer of "yes" to some specified degree and "no" to some specified degree is the "bottom line" result.

Of course you can always pretend that it's OK to somehow round off this "bottom line" result to the nearest "yes" or "no." But if you don't want to accept the correct answer as it stands, then you can pretend anything you like! (It's always possible to find "easy" answers to any question, if you don't mind having wrong answers!)

Consider an airplane pilot who wanted to fly from Miami, Florida, to Seattle, Washington. He looked on a map and asked himself the question: "Is Seattle north of Miami, or is Seattle west of Miami? The correct answer, of course, is that Seattle is both north to some degree, and west to some degree. But our pilot didn't like having to deal with reality and its confusing fractions and percentages. So he decided to round off the correct answer to the nearest "north" or "west," and then he chose that direction in which to fly. (He's still looking for Seattle, but at least he has an easier course to follow!)

Truth is a matter of degree, and this degree must always be acknowledged somewhere. Therefore, any question involving a truth must either:

The two options are equivalent and the choice of which one to use is entirely up to you. Therefore, if you feel uneasy with fuzzy answers (ones that are "partially yes" and "partially no"), then simply refrain from asking questions that require such answers. Instead, rephrase your questions in such a way that they explicitly request answers involving indications of degree. In other words, instead of expressing questions in the form:

Is "STATEMENT" true?

simply ask the equivalent question:

To what degree is "STATEMENT" true?

Since the question itself then acknowledges the concept of degree, the answer can now be crisp rather than fuzzy. For example instead of asking a question like, "Is it dark?" simply rephrase the question so that it now becomes: "To what degree is it dark?" or more simply: "How dark is it?" This will allow for a more traditional-sounding answer.

We all live in the same physical world, and we are all members of the same species (homo sapiens). And yet, we all have slightly different philosophies. Some of us are Liberals, and some of us are Conservatives. Some of us are Pro-life, and some of us are Pro-choice. Some of us are Republicans, and some of us are Democrats. Some of us are in favor of having strict gun control laws, and some of us oppose the idea.

And we even have our own private disputes. The other driver claims that you ran into him, and you claim that he failed to yield to you. You say that your next-door neighbor is making too much noise, and he claims that he isn't. The cop claims that you were driving at night without your headlights on, and you claim that it was not yet dark.

But if we're all involved in the very same physical reality, then how come there are such disagreements? If there is only one true reality, then shouldn't one side or the other ultimately end up being proven to be the correct side?

There's an old adage which says: "There are two sides to every argument." Of course the traditional interpretation of this saying is simply: don't try to determine who's right and who's wrong until both sides have had a chance to present their cases. However, since truth is a matter of degree, a more realistic interpretation of the adage presents itself -- an interpretation that does not require the eventual choosing of a correct side: Both sides of any argument are usually right -- to some degree, that is -- and, of course, both sides are usually wrong, also to some degree.

In almost every argument, both sides feel that their claims are legitimate -- and they usually are. But each side sees only its own "rightness" and only the other side's "wrongness." Therefore, it's easy for each side to claim that its own view is totally right while that of the opposition is totally wrong. Moreover, each side generally seems to think that by merely being able to successfully defend his own position, that he will have therefore somehow "disproved" the opposing view. And so each side usually tries to "win" the argument by focusing exclusively on his or her own merits, while totally ignoring the valid counterarguments of the other side.

Most arguments can be resolved fairly and accurately, not by looking for an absolute "winner," but by acknowledging the degrees of rightness and wrongness of both sides of the argument. (But trying to get both factions to accept this fair and accurate resolution may pose a problem!)

In a court of law, witnesses are instructed to "tell the truth, the whole truth, and nothing but the truth." This means that they are not allowed to omit portions of the truth (they must tell the whole truth), nor are they allowed to add false testimony to otherwise true statements (they must tell nothing but the truth). In other words, their testimony must be both accurate and complete.

But as we have already seen, truth is a matter of degree. Therefore, every statement (except for logically true statements and statements of definition) contains at least some small degree of falseness. And so making any statement (even supposedly "true" ones) would not comply with the directive to tell "nothing but the truth!" And yet not to make any statement would violate the directive to tell "the whole truth!" (The dilemma is similar to the one in Chapter Two where you were told to make an accurate and complete list of all your favorite foods.) And so it is generally impossible to comply with a court's request for such black and white testimony.

Truths exist independently from us. If all of mankind were to disappear from the universe, the fact that 2 + 2 = 4 would still be true. But we do exist, and we are aware. When we become aware of a truth, we say that we know that truth. Knowledge is therefore awareness of a truth.

Let's play a little game. I'm thinking of a number. Can you tell me what the number is? Of course not. I could be thinking of any number. It could be five. Or it could be -7.38. I might even be thinking of one million times the square root of pi.

But now suppose I were to tell you that the number that I'm thinking of is an integer. Now can you tell me which number I'm thinking of? No, but you can eliminate a lot of possibilities. For example you now know that I'm not thinking of a fraction, and I'm not thinking of an irrational number or a number with a decimal point somewhere between two of its digits.

Next, suppose I tell you that the number is less than 100.

Suppose I tell you that it's greater than 20 but less than 30.

Suppose I tell you the number is more than 22 but less than 27.

Suppose I tell you the number is more than 24.9 but less than 25.1.

"Ah ha!" you say. "The number you're thinking of is 25."

"Correct!" I reply.

But now let me ask you one more question: When did you know that the number was 25? You would probably say that you knew it after I gave the last clue, because 25 is the only integer between 24.9 and 25.1. And prior to hearing that clue you had no idea what number I was thinking of.

We tend to use expressions like "I had no idea" in a rather cavalier way. While it may be true that you initially "had no idea" what the number was, you at least knew that I was thinking of a number. I was not thinking of a letter of the alphabet, nor was I thinking of a color in the rainbow, nor was I thinking of a city in the United States, etc. (If I had merely said that I was thinking of something, then you might truly have "had no idea" what I was thinking of.) As each new clue came, you were able to add more and more items to the list of things that I was not thinking of. In other words, your knowledge about what I was thinking of continued to increase. By the time that I was ready to give you the very last clue, you already had a pretty good approximation of the number, even though you may not have known it completely and precisely.

Still, you might want to insist on adopting the all-or-nothing stance that either you know the number completely, or else you don't know it at all. (Old habits and old ways of thinking die hard!) In response to your objection I'll ask you another question: "Do you know the value of pi?" The numerical value of pi has the approximate value, 3.1416. But pi is an irrational number with an infinite number of digits. Nobody could ever know what every single one of those digits is. And so, by your reasoning, nobody has the slightest idea of what the value of pi is!

If you're still not convinced that knowledge is a matter of degree then let's invoke a "smoothness principle" type of approach -- At what precise point in time did you know the correct answer was 25? After you had read half-way though the final clue? After you had read three-quarters of the way through the final clue? After you reached the period at the end of the clue's sentence?

Your brain didn't instantaneously create the correct answer of 25. It took at least several milliseconds to even comprehend the final clue, and then at least several more milliseconds to calculate the answer. You didn't just suddenly "see the light" in zero time. Instead, the light took a small but finite amount of time to gradually come on. Even this analysis shows that, at any instant in time, knowledge is present in degrees.

When you were in school and took a class in a certain subject, at what point in time did you completely "know" the course material? When you attended the first day of the class? Halfway through the semester? After the very last lecture? After you completed the final exam? In fact, did you ever completely "know" the course material?

And speaking of the final exam, if you missed one or more questions on that exam, was that an indication that you didn't know any of the course material? Instead of receiving a passing grade, should you have failed the course merely because you didn't remember or understand every single thing that was taught to you?

Knowledge is not something that you either have or don't have. Knowledge is not an absolute. All knowledge is a matter of degree.

Forgetting, therefore, represents a diminishing level of knowledge with time. And unless you have a brain disorder (such as amnesia) you don't generally forget everything all at once. Nor do you generally forget things in an all-or-nothing manner. Even when you think that you've totally forgotten a piece of information, hearing that information again will usually "ring a bell" because some degree of that knowledge still remains in your mind. (That is why some schools offer "refresher courses" in certain subjects.)

There are definite procedures that you can use to improve your ability to remember things, and there are a number of institutions that offer "memory classes." The fundamental idea taught in these classes is the relatively simple trick of associating specific pieces of information with visual images. In Rex Dante's memory classes he used a process that he called the "TAVE" method (an acronym for: Touch, Affect, Visualize, and Enjoy). "To have it, you must TAVE it," was the slogan he taught.

Let's say that you don't want to forget the name of a person who you've just met. (Let's suppose his name is William.) First create in your mind an image that you can easily associate with that person's name (For "William" you might visualize a sheet of paper labeled "Last Will & Testiment, or perhaps a duck's bill, or maybe a windowed envelope containing your telephone bill, etc.) Next identify some aspect of the person's physical appearance that stands out in your mind. (Let's say he has a tiny scar on his chin.) Then you might imagine, for example, a duck's bill poking (Touching) the scar until it starts to bleed (Affecting it). The blood can appear as either liquid gold (if you perceive William as being a nice person with a "heart of gold"), or green slime (if you don't like him). You must then concentrate on (Visualize) this mental scenario for a moment, and have fun Enjoying that image. If you carefully follow all of these steps, you will probably not only remember William's name whenever you see him again, but you will probably chuckle a little bit each time as well.

A technique that works well for memorizing numbers is to associate each digit with a different letter of the alphabet (or with a different spoken sound). By utilizing such a "phonetic alphabet" the meaningless digits in numbers can be converted into sequences of letters (or sounds) which can be associated with recognizable objects.

By utilizing tricks like the TAVE method along with the phonetic alphabet, it is possible to greatly improve your ability to recall almost any kind of information. (Since I've taken the memory class, my memory has become so good that I can't even recall the last time that I've ever forgotten anything!)

Pehaps we also need to make a distinction between forgetting and simply being absentminded. When a person becomes preoccupied with some activity, he/she may "forget" about previous commitments and obligations. This is not quite the same kind of forgetting as the "loss of recall" type of forgetting that we have been discussing (although absentmindedness also occurs in degrees). An absentminded person still remembers the information. But that information has simply been moved out of the "immediately active" area of his mind. (For example, do you remember your own telephone number? Of course you do. But I'll bet you weren't actively thinking about it until I brought the question up.)

Perhaps you are musically talented and know how to play a musical instrument. Almost everyone can manage to pick out at least one simple tune on a piano, even if they can only do so very clumsily and with just one finger. But would you say that such a clumsy one-finger player "knows" how to play the piano? Probably not, at least not in the sense that we usually mean (i.e., that they're skilled at playing it).

Skill is similar to knowledge in that it represents acquired information (e.g., which piano keys correspond to which musical notes), but different in that it requires degrees of physical and mental dexterity as well. When all three of these aspects are developed together, we say that the person is skilled.

But mental dexterity is essentially knowledge at the subconscious level. The brain is "taught" to make connections between neurons that previously were not connected. These connections constitute patterns of information in the brain. Therefore mental dexterity can also be regarded as a form of knowledge.

Let's return now to our one-finger piano player. She has just decided that it would be kind of fun to take some piano lessons. So after a couple of days of diligently practicing at the keyboard, she can eventually play a simple one-fingered tune almost all the way through without hardly making any mistakes at all. Does she now know how to play the piano?

Several more weeks pass and our piano student now knows a few simple chords. She even knows how to play some chords with her left hand while her right hand pounds out a simple melody. Would you say that she now knows how to play the piano?

How proficient must a person be before it can be said that he or she "knows" how to play the piano? How terribly must they play before they should be regarded as "not knowing" how to play. Is there anybody on earth who "completely knows" how to play the piano?

Clearly, skill is a matter of degree. Otherwise, everyone who "knows" how to play a musical instrument would be just as good as everyone else who "knows" how to play that instrument. But what exactly causes skill to be a matter of degree? If all knowledge were a bivalent all-or-nothing matter,then the only thing that would account for the existence of different degrees of skill would be the differences in degrees of physical dexterity.

Consider two virtually identical students. One student spends four years studying the piano. The other student is given absolutely no music lessons at all, but instead spends an equal amount of time exercising the very same finger, hand, and arm muscles that the first student is exercising by taking piano lessons.

After fourteen years the first student becomes a concert pianist. The only thing preventing the second student from being able to play the piano just as well should therefore be the missing knowledge of music. And so the second student could then become as equally good of a concert pianist as the first student, simply by reading music books to supply her with musical knowledge. Furthermore, if knowledge were all-or-nothing, there should be a well-defined instant in time at which she makes the transition from "not knowing" how to play the piano at all, to being a fully qualified concert pianist!