Outer Limits of Reason (11 page)

One must also make a distinction between vague statements and
relative statements
. “Jack Baxter is smart” might be true or not depending on who he is being compared to. If you are comparing Jack to the other people in his class, then he might very well be considered smart; however, the class might not be the smartest class. The truth of a relative statement can be determined by looking at the context of the statement. Who are we talking about? One can imagine the salutatorian at a Harvard University graduation legitimately being called stupid . . . by the valedictorian.

In both ambiguous cases and relative cases there is a lack of specificity. In other words, there is missing information. Usually, if one adds more information, then the statements can be clearly understood. If one identifies the subject of an ambiguous statement or the context of a relative statement, then we can determine if the statement is true or false. In contrast, vague statements usually cannot be tweaked by adding more information. There is no more information to add. When is a person considered bald? The answer is “blowin' in the wind.” There is no real answer.

Vagueness is not necessarily a bad thing. Sometimes vagueness is a necessity. Biologists use vague characteristics to describe different species.
¹⁷
Many lawyers are employed to work with vagueness (and to obfuscate the truth). Diplomats are vague when they make treaties with foreign countries so that they are not caught by their own words. When a woman asks if a certain dress makes her look fat, it might be wise to be vague in your response.

Philosophers are usually split as to why there is vagueness. Some philosophers promote
ontological vagueness
—that is, the reason some terms do not have an exact meaning is that an exact meaning of these terms really does not exist. While there is an exact definition of “above six feet,” there is no exact definition of “tall.” In contrast, other philosophers promote
epistemic vagueness.
They believe there is an exact definition of vague terms but we simply do not know what it is.

Which is it? Ontological vagueness or epistemic vagueness? While everyone has an opinion, no one has the decidedly knockdown argument. Unfortunately it is an unanswerable metaphysical question. I humbly lean toward ontological vagueness.
¹⁸
For reasons elaborated in
section 3.1
, it is hard to believe that exact definitions of
tall
,
bald
, or
red
exist. Who determines these exact definitions? Are they to be found in Plato's attic? Is there some exact height that is considered tall? Is there an exact number of equally distributed hairs that make a person hirsute (not bald)? Is there an exact wavelength associated with red and not with cherry? I highly doubt it. Since we discounted an exact definition of the ship of Theseus, it stands to reason that we discount an exact definition of tallness, baldness, or redness.

One problem with vague terms is that the usual tools of logic and mathematics that we use to understand the world do not work for such terms. For example, one of the main rules of logic is that for any proposition P, it is always a fact that either P or not-P is true. So for example, “It is either colder than 32°F or it is not colder than 32°F now” is always true (and hence devoid of any content). This rule is called the
law of excluded middle
. That means that either a proposition is true or false but nothing in the middle. However, for vague predicates the law of excluded middle does not work. We all know people who are neither tall nor not tall. They are simply somewhere in the middle. There are men who are neither bald nor not bald . . . they are, like many men, heading toward baldness.

One of the main tools of logic is the law called
modus ponens
. This law says that if a statement P is true and the statement “P implies Q” is true, we can then derive that the statement Q is true. In symbols, we write this as

P

P → Q

Q

For example, from the fact that “it is raining,” and “if it is raining, then there are clouds in the sky,” we can derive that “there are clouds in the sky.” This basic law of logic is at the root of all reasoning. However, this law fails when we deal with vague terms. In the next few paragraphs I describe certain strange logical deductions that come about from the failure of modus ponens.

If a man does not have any hair on his head he is definitely bald. What if he has one hair on his head? Most people would say that a man with exactly one hair on his head is still considered bald. How about two hairs on his head? It is hard to believe that if a person with one hair is considered bald, one more little hair is now going to make him hairy. He must be considered bald. How about three hairs? There must be a rule that says that

If a man with 3 hairs is bald, then with 4 hairs he is also bald.

Again, we are only adding one little hair so this rule must be true. In fact we can generalize this rule to the following rule for all positive whole numbers
n
:

If a man with
n
hairs is bald, then with
n
+
1 hairs he is also bald.

Pressing on with our analysis, we can come to the conclusion that a man with 100,000 hairs or even 10 million hairs is still bald. But this is simply not true. A man with that much hair is not bald. This is called the
bald-man paradox.

Such a paradox is an example of a type of argument that goes back to ancient Greek times and is called a
sorites paradox
(from the Greek word
soros
for “heap”). Eubulides of Miletus (fourth century BC) is usually credited with being the first to formulate this puzzle.
¹⁹
He asked how many grains of wheat form a heap. Is one grain of wheat considered a heap? Obviously not. How about adding one grain to it? Are two grains considered a heap? Still not. After all, we only added one grain. We can formulate the following law:

If
n
grains are not a heap, then
n
+
1 grains are also not a heap.

Following a similar analysis of the bald man, we come to the obviously wrong conclusion that no amount of grains form a heap. What went wrong?

Let us carefully analyze the argument given. We start with the obvious statement:

1 grain is not a heap.

We also use the
n
-grain rule for
n
=
1 to get

If 1 grain is not a heap, then 2 grains are also not a heap.

Combining these two rules using modus ponens, we get

2 grains are not a heap.

Furthermore, combining this with

If 2 grains are not a heap, then 3 grains are also not a heap.

gives us:

3 grains are not a heap.

Continuing on with this shows us that for any
n
, no matter how large,

n
grains are not a heap.

This is obviously false.

We can also go the other way. Consider a heap with 10,000 grains of wheat. If we take off one little grain are we to come to the conclusion that 9,999 grains are not a heap? Obviously they are still a heap. A rule can be formulated:

If
n
grains are a heap, then
n
− 1 grains are also a heap.

Using this rule and applying the modus ponens rule many times, we arrive at an obviously false conclusion that a collection of 1 grain is also a heap. A similar argument can show that a man with 1 hair, or even no hairs, is not bald.

Another sorites-type paradox is the
small-number paradox
(also called
Wang's paradox
). 0 is a small number. If
n
is a small number, then so is
n
+
1. We conclude with the apparent false fact that any number is considered a small number. There are many other types of sorites paradoxes. Is a person tall if we add another centimeter to their height? Does a person become heavy if they add one more pound? Similarly, for any other vague terms like
rich
,
poor
,
short
,
clever
, and so on, one has an associated sorites-type paradox.

How is one to understand such paradoxes? Some philosophers say that the sorites paradoxes show us that there is something wrong with the logical rule of modus ponens. By following modus ponens we came to a false conclusion, so modus ponens cannot be trusted. This seems a little too harsh. The modus ponens rule works so perfectly in most logic, math, and reasoning. Why should we abandon it? Other philosophers (who believe that all vagueness is epistemic—i.e., they believe exact boundaries exist that we are not aware of) assume that the rule

If
n
grains are not a heap, then
n
+
1 grains are also not a heap.

is simply false. For them, there is some
n
for which
n
grains do not form a heap but
n
+
1 grains do form a heap. We mortals are not aware of which
n
this is but it nevertheless exists. For such philosophers modus ponens is true, but this implication is simply not valid and so cannot be used in a modus ponens argument. As noted above, to us it seems that vagueness is not an epistemic but an ontological problem. There are no exact boundaries and the implication from
n
to
n
+
1 grains is, in fact, always true.

Rather than saying that there is something wrong with the obvious rule of modus ponens, I prefer to say that this amazing rule is perfect but cannot always be applied. In particular, one should not use modus ponens with vague terms. Although modus ponens seems to work with the first few applications of the rule (i.e., that 2, 3, and 4 grains do not make a heap), for many more applications of the rule we come to obvious false conclusions. We must restrict ourselves to using modus ponens only with exact terms. We will not be able to use modus ponens with vague terms because that will take us beyond the bounds of reason.

It makes sense that these logical and mathematical tools do not work with vague terms since these tools were formulated with exact terms in mind. One needs exact terms to do science, logic, and mathematics. When we leave the domain of exact definitions—that is, when we talk about baldness, tallness, and redness—we are necessarily leaving the boundaries where logic and math can help us. Vagueness is beyond the boundaries of reason. While we all freely live and communicate with such terms on a daily basis we must, nevertheless, be careful about crossing the outer limits of reason.

 

As shown above, when it comes to vague statements, mathematicians and logicians are somewhat at a loss. Their usual tools in their toolbox do not work. However, since these vague terms are ubiquitous, we simply cannot ignore them. Researchers have developed a number of different methods to make sense of the vague world. Here I will highlight several of them.

Logic usually deals with terms that are either true or false.
Fuzzy logic
is a branch of logic that deals with terms that can have any intermediate value between true and false. Say that true is 1 and that false is 0. Rather than dealing with the two-element set {0,1}, fuzzy logic deals with the infinite interval [0,1] of all real numbers between 0 and 1. With this we can give different values in different cases. Telly Savalas and Yul Brynner are both totally bald and hence would have the value 0. People with full heads of hair would get a 1. People in the middle will get middle values. 0.1 means almost bald, while 0.5 is halfway there. Someone might get the value of 0.7235. With these different values set up, researchers have gone on to develop different operations similar to AND and OR to work in this logic.

Similar to fuzzy logic is a related field of logic called
three-valued logic.
Rather than saying that a statement is either true or false, say that a statement is true, false, or indeterminate. These branches of logic are used extensively in the field of artificial intelligence, which tries to make computers act more like human beings. If we are going to have computers interacting with human beings, then they are going to have to deal with vague terms like humans beings. These multivalued logics have been very successful in dealing with vague predicates.

Another method used to deal with vague terms is to restrict logic. Consider a man who is halfway between being bald and being hairy. Rather than saying he is neither bald nor not bald, say that he is both bald and not bald. In classical logic if a statement and its negation are both true, we have a contradiction and the system is inconsistent. The major problem with such a system is that anything can be proved within such a system—that is, from a falsehood one can derive anything. While most logicians avoid such systems, others, like Graham Priest, work with them. They attempt to extend the realm of logic to the vague by permitting certain types of contradictions. The belief that there are types of contradictions that are true is called
dialetheism
. The logics that they deal with are called
paraconsistent logics
. Essentially what they do is restrict the logic so that not every statement is derivable from a contradiction. With these restrictions in place, one can derive meaningful statements about vague terms. This direction of research has also progressed over the past few years.
²⁰

3.4  Knowing about Knowing

Imagine being a contestant on the television game show
Let's Make a Deal
with its host, Monty Hall. Monty presents you with three doors and tells you that behind two of the doors are goats and behind one of the doors is a brand-new fancy car. You are allowed to keep whatever is behind the door you choose. After selecting one of the doors but before you open the door to see what you won, Monty stops you and knowingly opens another door and shows you a goat. He now offers you the choice of staying with your original selection or of switching to the third unopened door. What should you do?

Your immediate reaction is that you might as well stay with your original choice. After all, when you started each door had one-third of a chance of having a car. Now that one of the doors is open, there is a fifty-fifty chance that the car is behind your original choice. What is to be gained by going to the other door?