Let's take a look at an interesting mind game first.
In order to test the analytical reasoning ability of four students, A, B, C, and D, the teacher showed them five hats of the same style, and emphasized: "There are two white hats here, a red hat, a yellow hat, and a blue hat. Then he sat the four men on the four steps in sequence, told them to close their eyes, and put a hat on each of them. Finally, he asked the students to open their eyes and judge what color the hat was wearing on their heads.
The results were unexpected. Although the person sitting in the back could see the color of the hat worn by the person in front, the three people A, B, and C looked at it and thought about it, and they all shook their heads and said that they couldn't guess.
Ding sat at the front, he couldn't see the color of the other person's hat, but now he spoke, saying that he had guessed the color of the hat he was wearing. How did Ding determine the color of his hat? You've probably figured out the game's mystery. In fact, Ding's judgment is not difficult, he thinks like this:
"A is blessed to sit on the tallest, and he can see the hats of the other 3 people, why can't he guess it? Surely he saw someone in front of him wearing a white hat. Because if the people in front of him were wearing variegated hats, then he would be able to guess that he was wearing a white hat. Besides, B is a smart person, and she naturally knows what A thinks. So why did she say she couldn't guess? She must have seen someone in front of her wearing a white hat. Otherwise, she would judge that she was wearing a white hat based on A's attitude and the color of other people's hats. Finally, C, her IQ is by no means lower than B, but why can't she guess it! There can only be one reason, and that is that she saw me wearing a white hat on my head. ”
In this way, Ding affirmed the color of his hat from the negation of everyone!
The above game can be generalized to multiple people, but the variegated hat is one less than the number of people, and the white hat has at least two. The method of reasoning is the same. It's just that no matter whether the conclusion is positive or negative, thinking must conform to certain laws.
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What are the basic laws of logical thinking? In general, there are the following 3 points:
(1) The law of identity: that is, the thinking should be unified from beginning to end.
(2) The law of contradiction: that is, two opposite or incompatible judgments in the mind cannot be true.
(3) The law of exclusion: In the process of thinking, a logical judgment is either affirmed or negatived, and it is either false or true. The above three laws put forward requirements for the consistency, certainty, and non-contradiction of human correct thinking from different angles.
It should be pointed out that many people think that sentences consisting of "yes" and "no" must be opposite judgments. If one of them is true, then the other must be incorrect.
In fact, this view is not necessarily true. The following "Achebe Puzzle" may surprise you!
Achebe loved the study of formal logic, and once he came across the following two sentences:
"×× is ○○○"
"×× is not ○○○"
In these two sentences, the "× ×" at the beginning of each sentence also indicates the same word, and the "○○○" at the end of each sentence also indicates the same word. They differ only in the middle of the "yes" vs. "no".
However, both sentences are true! It may be strange to some readers that this is due to the fact that there is too much formal logic in the mind that "A is not equal to non-A".
However, if the subject of two sentences is the same but the content is different, then there may not be a logical contradiction even if the expression is the same. For example:
"This sentence is a six-word sentence."
"This sentence is not a six-word sentence."
This is one of the answers to the Achebe conundrum. In the two sentences, the subject of the former sentence and the latter sentence, "this sentence", contains different contents.
The following story will help you become more familiar with the laws of logical thinking.
The tiger occupies the mountain as the king and commands the beasts.
One day, the tiger was hungry and wanted to change his tricks and make some animals to eat. So they called sika deer, foxes, rabbits, and monkeys, and asked everyone to talk about the smell in their mouths to test their loyalty.
The sika deer was the first to be assigned to answer, and it reported that the tiger had bad breath and was killed on charges of "defamation".
When the fox saw that the situation was not good, he immediately slapped his horse. Unexpectedly, the tiger did not buy this account. The fox was also killed.
The rabbit was frightened and bleeding from both eyes. It learned from the lessons of the past, and sincerely reported with trepidation: "It is difficult to say whether Your Majesty's mouth stinks or not. The tiger was furious, saying that those who rode the wall and compromised would never be allowed to survive in the world!
Finally, it was the monkey's turn, and the monkey scratched the back of his head, and respectfully walked up to the tiger and said, "King, I have a little cold lately, and my nose is blocked, if you can let me go back to recuperate for a few days, and when my nose is clear, I will be able to accurately tell the smell of the king's mouth." "The tiger was short of words, so he had to let the monkey go. Naturally, the monkey took the opportunity to escape.
At this end of the story, please analyze the reader from a logical point of view, why did the sika deer, the fox and the rabbit not escape the doom, but only the monkey was able to turn the corner? Did the monkey's words violate the law of exclusion?
Sometimes, people start from some seemingly correct and acceptable conventions, and after concise and correct reasoning, they come to contradictory conclusions. Such arguments are called paradoxes. "Paradox" means chaos and conflict.
For example, given a proposition A, there would be:
A→B
A→B'
Here B and B' are true at the same time, which is contrary to the laws of logic.
Paradoxes are not uncommon in everyday life. For the convenience of readers, a library compiles a "catalog" of books in its collection. Now ask: Is this "catalogue" itself cataloged? Such questions can be very difficult for you.
Ancient Greece was a country full of myths. There is a legend that a crocodile snatched a child from a mother. The crocodile wanted to eat the child, and wanted to be justified, so he said to the mother wisely:
"Am I going to eat your child? If you answer this question correctly, I will return the child to you without harm. ”
The mother thought for a moment and replied, "You are going to eat my child." ”
At this time, the crocodile had a problem: if the child's mother answered incorrectly, then I could eat her child, but she clearly said that I was going to eat her child, wouldn't that be a pair again? If her answer is correct, it means that I am going to eat her baby, but I must return the child to her without harm!
The clumsy crocodile was confused, and in order to show respect for the promise, he had to return the child to the witty mother.
The paradox stems from a fairly long period of time. The famous "liar" paradox appeared in the 6th century BC. It was to the effect that Mr. E of Crete said, "The people of Crete are liars. "No matter how you interpret it, there will be a contradiction.
The most influential in modern mathematics is the so-called "Russell's paradox". In 1902, the British mathematician Bertrand Russell (1872-1970) put forward the following famous problem in response to the imperfection of the basic theory in the early days of set theory:
"Divide all sets into two categories, the sets in the first category have elements in themselves, and the sets in the second category do not have elements in themselves. Suppose that the set of the first type of set is P, and the set of the second type of set is Q, then there is
P={A|A∈A}
Q={A|A∉A}
Q: Does the set Q belong to the first type of set P? Or does it belong to the second type of set Q? ”
Logically, the answer to this question can only be "Q∈P" or "Q∈Q", and one of them must be the other. However, either answer leads to the opposite.
Russell
The paradox arises logically against the basic laws that human beings should follow in their correct thinking. For mathematics, which is known for its rigor, paradoxes are naturally not permanent. But it can prompt mathematicians to think seriously and look for the causes of paradoxes, so as to create a scientific theory that is at least logically perfectly coordinated, and unassailable.
Source: Origin Reading
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