How many hairs does baldness need?

If someone says to you: Zhang San loses another hair, he will become bald; If he loses a hair, he is not bald. You must think this is nonsense.

In view of this problem, Oberlid, the representative of the ancient Greek Megara School, put forward two famous sophistries, which made people of insight in past dynasties think hard for more than 2000 years.

These two sophistries are: how many grains can make up a grain pile? One millet can't be piled, two can't be piled, and one can't be piled ... Similarly, if two are not many, three are not many ... and 10 is not many, when will it be more?

The sophistry of "baldness" is similar to the sophistry of "grain pile": if you lose one hair, two hairs, three hairs, etc. It won't make people bald. How many hairs does it take to get bald?

In traditional logic, the errors contained in these two sophistry, called the scattered use of nouns, are usually attributed to the errors of collective use. If you think about millet in a decentralized way, then of course they can't form a grain pile, but this doesn't mean that many millet as a whole can't form a grain pile. Some logical historians believe that the root of these two sophistry lies in the fact that the problem excludes the dialectical transformation from quantity to quality in advance.

It is true that these two philosophical explanations of sophistry are correct, but they are too general, and philosophical explanations cannot replace logical explanations.

Can traditional logic explain these two sophistries? No, traditional logic is based on the study of precise concepts and propositions. At the same time, a proposition has only two values: either true or false. In the words of western scholars, traditional logic is based on dividing sentences by swords. They don't study ambiguous and ambiguous sentences in critical state. Fuzzy and ambiguous sentences are excluded from the object of logical research. However, the propositions contained in the above two sophistry are vague and ambiguous. Losing one hair is certainly not baldness, and losing two or three hairs is certainly not. So how much is it? It is impossible to cut an exact number with a knife here. The answer is vague. Traditional logic is helpless when it encounters such a vague object.

In fact, there are countless vague phenomena in nature, human society and thinking. The leaves of the same tree are roughly the same, but it is impossible to find two identical leaves; It is impossible for the same person to write the same word exactly. Huang Zongying published a poem dedicated to young and middle-aged scientific and technological workers in Guangming Daily on March 30, 1980. The first paragraph reads:

Middle-aged people and young people, how to divide, how to calculate?

How many parts of life have been removed, or half of time has passed?

It means that "middle age" and "youth" are two vague concepts, and it is difficult to define them accurately. For example, the concepts of "height", "fat", "speed" and "weight" are inaccurate.

There is a gradual evolution from thin to fat. Can you tell exactly when he became fat? Of course not.

History shows that at what age Newton discovered the law of universal gravitation is a rather vague fact. If we say that creative ideas are generated, it is between 1665- 1666, but it is completed between 1685- 1686.

Does the same Faraday's law of electrolysis belong to physics? Or chemistry? Nor can it be divided by the sword.

Traditional logic can't deal with fuzzy objects, but in real life, people still have the ability to identify and judge. Ambiguity can be properly grasped by a famous doctor who feels the pulse, a skilled steelworker who adjusts the furnace temperature and a senior chef who knows the temperature. These experts not only have the mechanical and accurate rigorous logical reasoning ability, but also have the ability to deal with fuzzy objects flexibly, have the overall and parallel thinking ability, and have the ability of generalization, abstraction, intuition and creativity.

In order to reduce blindness and improve science, it is necessary to quantitatively describe the fuzziness of things. For the research of large-scale systems, such as aerospace systems, human brain systems and intelligent systems, which involve complex relationships and a large number of ambiguous objects, and for the development of machines that simulate human advanced wisdom, not to mention traditional logic, even modern mathematical logic is far from enough. So a kind of applied logic-fuzzy logic (translated as fuzzy logic by some people) came into being.

Chad, an American cybernetic scholar, first put forward the concept of fuzzy sets. A fuzzy set is a set of fuzzy concepts. For example, the concept of "baldness" is vague, and there is no line cut with a knife between people who are regarded as baldness and those who are not bald.

It turns out that in set theory, the basic concept is membership. There is at least one attribute between any set and its constituent elements, that is, the specified element either belongs to this set or does not belong to this set. Mathematically, this property is represented by a characteristic function whose binary values are 1 and 0, respectively, corresponding to true binary values and false binary values in logic. But this binary property can only describe and deal with accurate objects. Chad further quantified the relationship of "belonging", so that an element does not belong to a certain set or not, but can belong to a certain set to varying degrees, so he introduced the concept of membership.

The membership degree of an element to a fuzzy set can take any value between 0 and less than or equal to 1. Chad extended ordinary set theory to fuzzy set theory. It not only takes the binary value of (0, 1), but also takes the continuous infinite value in the interval of (0, 1).

For example, with the discovery of gravity, a distribution function with different membership degrees was developed between 1665- 1686, or in Newton's life, there was a distribution with fuzzy membership degrees between the ages of 23 and 43. Another example is Faraday's law, which is 0.6 in physics and 0.3 in chemistry.

Chad said: "Perhaps the simplest way to describe fuzzy logic is to say that it is an approximate reasoning logic." Inference based on imprecise propositions is specious, and its conclusion is vague and not unique. The validity of its reasoning rules is also approximate rather than accurate.

Now let's go back to the two sophistries at the beginning of this article.

Assuming that someone has a lot of hair, he is definitely not bald. Then, there is a person who has only one hair less than the person who is definitely not bald. We asked: Is this man who has lost a hair bald? Obviously, he is not bald. If the person who has lost (less) one hair is not bald, is the person who has lost two hairs bald? Obviously not considered bald. By analogy, if people who have lost n hairs are not bald, then people who have lost n+ 1 hairs are not bald. The general reasoning is as follows:

If the person who loses 0 hairs (not one) is not bald, then the person who loses 1 hair is not bald.

People who lose 0 hairs are not bald,

Therefore, people who lose 1 hair are not bald. ( 1)

If the person who loses 1 hair is not bald, then the person who loses 2 hairs is not bald.

People who lose 1 hair are not bald.

Therefore, people who lose two hairs are not bald. (2)

If the person who loses n hairs is not bald, then lose 1+ 1 hair.

There are no bald people,

People who lose n hairs are not bald,

Therefore, people who lose n+ 1 hair are not bald. (noun)

Finally, the following conclusion will be drawn: for any n hairs, people who lose n hairs are not bald. Suppose n is all the hair of someone, all the hair has fallen out, and he is not bald yet. Obviously, this conclusion is absurd.

We can see that the sophistry of "baldness" contains a series of paradoxes of reasoning.

Let's check whether this reasoning is valid. This series of reasoning is the positive premise of using sufficient conditions to assume reasoning, and all of them are correct in form.

The second premise of reasoning (1) "People who lose 0 hairs are not bald" is obviously true. The first premise "If the person who loses 0 hairs is not bald, then the person who loses 1 hair is not bald" also holds. It can be seen that the reasoning (1) is effective. Similarly, inferences (2) and (3) hold. When the value of n reaches a certain level, does the first premise (n) of reasoning hold? It's also true. If the premise of (n) holds, that is, "If the person who loses n hairs is not bald, then the person who loses n+ 1 hairs is bald", it is difficult to understand intuitively. It is hard for people to accept the view that if a person loses one hair, he is not bald, but if he loses one more hair, he will become bald. It is obviously not in line with the usual view to distinguish whether a person is bald by the difference of a hair. So the negation of the first premise (n) is false, and the first premise is still true.

Because n can take any value and reasoning (n) is effective, the conclusion of reasoning n is inevitable, that is, true. This conclusion contains this meaning; People who have lost their hair are not bald. It doesn't match the reality, so the conclusion is false. From truth to falsehood, contradiction!

The key is that we use binary logic and its law of excluded middle. Binary logic can only take true and false binary values, and law of excluded middle requires that a proposition and its negation must have a truth value. So we can only choose between the first premise of each reasoning and its negation, and the paradox arises.

Restricted by binary logic, the paradox of chain reasoning is difficult to explain, and fuzzy logic makes a reasonable analysis of it.

A bald man's set is a fuzzy set. A person's membership degree to this set can be not only 0 and 1, but also greater than 0 and less than 1. Therefore, people who lose n hairs are not completely equal to people who lose n+ 1 hairs, and people who lose n+ 1 hairs have slightly higher membership than people who lose n hairs. When the value of n reaches a certain level, the membership degree of most people to "balding people" will change from 0 to 1, and those greater than this number will be taken as 1. The above continuous reasoning can be transformed into approximate reasoning of fuzzy logic, and the conclusions obtained by each step of reasoning are approximate, and the truth value of the conclusion is a little more than the truth value of the premise. The true value of the conclusion changes gradually from 0 to 1, and thus a false conclusion is drawn.