Shades Of Reality
Chapter Five
Fallacy and Paradox
Fallacy
In studying high school algebra you may have come across "proofs" of mathematical absurdities, such as: 2 = 1. Such a "proof" may have proceeded as follows:
a) Let x = 1
Squaring both sides of equation (a) yields -
b) x2 = 12 = 1
Since x = 1 and x2 = 1 we have -
c) x2 = x
Subtracting one from both sides of equation (c) gives -
d) x2 -1 = x - 1
Factoring x2 - 1 into (x - 1) (x + 1) yields -
e) (x - 1) (x + 1) = (x - 1)
Cancelling out the (x - 1) term from both sides of the equation gives -
f) (x + 1) = 1
Substituting x = 1 from equation (a) produces -
g) 2 = 1 Q.E.D.
Of course we know that 2 doesn't really equal 1, even though each step of our reasoning seems valid. And indeed upon further investigation we see that in going from equation (e) to equation (f) we discover that we cannot cancel out the (x - 1) term. Since x = 1, then the term (x - 1) is equal to zero. Therefore, if we try to cancel the (x - 1) terms, we accidentally divided both sides of the equation by zero (which is a mathematical "no-no").
The above "proof" that 1 equals 2 is an example of what is known as a fallacy. In proving a fallacy you must commit one or more hidden mistakes (such as dividing by zero). When the mistake is finally discovered, you have a good chuckle at your misstep and you move on, secure in your understanding that the rules of mathematics are indeed consistent after all. If there had not been a misstep -- if you had indeed discovered an actual inconsistency in the rules of arithmetic -- your proof would have shaken the very foundations of mathematics! Such is the power and importance of consistency. (Recall our discussions about consistency, which were presented in Chapter Zero.)
Paradox
A logical paradox, like a fallacy, starts with a known fact and then proceeds, through a logical sequence of step-by-step reasoning, to arrive at a contradiction. But a paradox differs from a fallacy in at least one very important aspect -- a paradox has no hidden missteps. Each step in the argument is valid. Logical paradoxes, therefore, are indications that there is something fundamentally wrong with the foundations of traditional logic!
Sorites Paradox
We have already encountered and resolved one simple example of a paradox in our previous discussions -- the sorites paradox of when does a heap of sand become a non-heap. Starting with a heap of sand you remove one grain and then ask the bivalent question, "is it still a heap?" If so, you continue to remove one grain of sand at a time, each time asking for a 100 percent "yes" answer or a 100 percent "no" answer to the question "is it still a heap?"
Fuzzy logic realizes that the correct answer to the question at each step is not an all-or-nothing "yes" or an all-or-nothing "no," but a degree of "yes" and "no." As each grain of sand is removed, the degree of "yes" decreases slightly and the degree of "no" increases slightly. Therefore there is no grain of sand whose removal causes the answer to instantaneously jump from 100 percent "yes" to 100 percent "no."
An equivalent way of looking at the problem is to realize that the term "heap" is merely the name of a fuzzy set. Every pile of sand is (to some degree) a member of this set that we call "heap." As each grain of sand is removed, the resulting pile's membership value (in the "heap" set) gradually becomes less and less.
Who Shaves the Barber?
The mathematician Bertrand Russell showed that the assumptions of crisp set theory could lead to contradiction and paradox. To illustrate, he imagined a small village in which a barber had a shop. On the window of his shop was the sign: "I shave all, and only, those men in the village who don't shave themselves." If the sign is true, then who shaves the barber? If he shaves himself, then his sign says that he doesn't. And if he doesn't shave himself, then his sign says that he does! [Footnote: An amusing "resolution" of this paradox is presented by Paul Sloane in his book, Test Your Lateral Thinking IQ. Mr. Sloane suggests that the barber is a woman!]
The Barber paradox is just one of many such paradoxes that can result when one tries to apply the all-or-nothing Aristotelian reasoning to statements of self-reference. And as amusing and trivial as they may seem, they actually reflect serious problems that arise in set theory when discussing sets that contain themselves. Specifically, if a set contains all (and only) those sets that don't contain themselves, does it contain itself? It can't -- and yet, it must!
Epimenides Paradox
Another example of self-reference is the famous Epimenides Paradox. Epimenides, who came from the island of Crete, supposedly once made the statement, "All Cretans are liars." If his statement was true, then it was false. And if it was false, then it was true.
As a slightly different version of the Epimenides Paradox, consider the following two statements:
a) Statement (b) is true.
b) Statement (a) is false.
Once again, if we assume that statement (a) is true then we are immediately led to the conclusion that statement (a) is false. And if we assume that statement (a) is false, then statement (a) is true. But if we do not limit the scope of our assumptions to only extreme all-or-nothing truth -- if we allow for the existence of partial truth -- then we can conclude that both statements (a) and (b) are each half true and half false. (If we assume that statement (a) is 50 percent true, then statement (b) is only 50 percent true. This in turn makes statement (a) only 50 percent false, which is in complete agreement with our initial assumption.)
As an aside, what if we were to change either one of the statements to its opposite? For example, suppose that we were to leave statement (a) unchanged, but change statement (b):
a) Statement (b) is true.
b) Statement (a) is true.
Now if we assume that statement (a) is true, we conclude that statement (a) is true. But what if we assume that statement (a) is false? Then, from statement (b), statement (a) is indeed false! So once again we that:
Statements of self reference leads to only half truths.
The Paradox of the Unexpected
As a final example of how fuzzy logic makes mince meat out of paradox, consider the famous "paradox of the unexpected." There are many slightly different scenarios that have been used to express this paradox. One tells of a teacher who announces that there will be an unexpected exam sometime during the course of the semester. Another tells of an egg that has been hidden away inside one of ten numbered boxes, and if you open the boxes in numerical order you will not know in advance which box contains the egg. In each case the reasoning is exactly the same. So I will present:
Before I present the resolution of this paradox, let me first point out that the king partially contradicted himself when he told the prisoner that the execution day would come as a complete surprise. It clearly cannot come as a complete surprise if the prisoner already knows that the execution will definitely occur on one of only seven upcoming days. The prisoner therefore "knew" the day of his execution to degree 1/7. And so the king was being only 6/7 honest when he told the prisoner that there would be no way for him to determine the day. (Yes, even honesty occurs in degrees.) With each passing day, the degree to which the prisoner would know the day of his execution would steadily increase. On the second day he would know it to degree 1/6, on the third day to degree 1/5, ... , on the seventh day to degree 1 (i.e., to a certainty).
To understand the solution to the paradox let's imagine that this episode was not just a onetime event. Let's suppose that it was a standard policy of the king to make this very same offer to every condemned prisoner. Surely the king, himself, could have come to the conclusion that day 7 would have to be avoided as this would result in a certainty of having to free any prisoner who lasted that long. But if he always avoided day 7 without exception, then that fact would soon become common knowledge among the prisoners, and they would therefore be correct in eliminating day 7 as a possible execution day. In that case, the king would be "right back where he started" so to speak. Now he would have to avoid day six without exception, and this too would soon become common knowledge, etc. The downward trickle would continue until eventually there would be no more days for executions, and all prisoners would always be set free! This would make the king become a total liar, because he had explicitly stated that each execution would occur.
The king would therefore be caught between a rock and a hard place. No matter what he did, there would be no way for him to maintain the complete integrity of his statements to the prisoners. He would have to choose between being either a total liar or, at best, being only a partial liar. To maximize the degree of truthfullness of his statements, the king would not be able to exclude any day. Instead, he would have to randomly distribute the executions evenly over all of the days, including day 7. Of course this would mean that, on the average, 1/7 of the prisoners would always go free. But then we already knew that any prisoner could have just "taken a guess" at his execution day, and he would have had one chance in seven of being correct.
In summary, the paradox of the unexpected execution assumes that all of the king's statements are 100 percent true, and that the prisoner's knowledge of the execution day is 0 percent. These all-or-nothing assumptions are clearly incorrect. Instead, the king's statements about the unknowability of the execution day are only 6/7 true, and the prisoner's degree of knowledge of the execution day is 1/7. If the king had chosen to be completely honest, and to admit to the prisoner that he (the prisoner) does have this degree of knowledge, then there would no longer have been a paradox.
Aristotelian Logic and Integers
In conclusion, a logical paradox can arise when all-or-nothing (i.e., integer) assumptions are made about an inherently fuzzy (i.e., non-integer) system. In this regard we can make the observation that:
Aristotelian logic is to integers,
as fuzzy logic is to fractions.
Suppose for a moment that mathematicians had developed modern day arithmetic with the same all-or-nothing guidelines that Aristotelian logic uses. (In fact ancient arithmetic was that way.) Such an "Aristotelian arithmetic" would therefore recognize the existence of only integer numbers. Concepts like fractions and irrational numbers would simply be unheard of (or at least not acknowledged as being anything more than maybe "the cocaine of mathematics").
Just think about the kinds of mathematical "paradoxes" that could arise inside such a framework! Values resulting from divisions would have to be rounded off produce integer results. For example 5 divided by 3 would be perceived as being exactly equal to 2. And the number 1 divided by any number greater than zero could be perceived as being either 0 or 1. (The value could be perceived to be 1 since, as is often argued, "if any part of an entity exists then the whole entity exists." We will talk about this erroneous notion further when we discuss the concept of "drawing the line at zero" in the next chapter.)
In Aristotelian mathematics one would have, for example, the "paradox" that 6 equals 5:
5 / 3 = 5 / 3
Multiplying both sides by 3 -
3 (5 / 3) = 5
Since 5 / 3 is perceived to be identically equal to 2, we have -
3 (2) = 5
or -
6 = 5 Q.E.D.
It is important to understand that the above proof would represent a paradox, and not a fallacy. There were no hidden missteps in any part of its derivation. Instead, the problem lies in the foundations of such an Aristotelian mathematics. The paradox resulted from the failure of the Aristotelian system to recognize the existence of any kind of numerical values except for whole numbers. However, when we allow for "fuzzy integers" (like fractions), then the paradox of 6 = 5 no longer exists.