In the 6th century BC, the philosopher epimenides, a Crete, said, "All Cretes are lying, and one of the poets also said so." This is the origin of this famous paradox.

It is mentioned in the Bible: "A local prophet of Park Yung-soo said,' The Celts often lie, but they are evil beasts, greedy and lazy'" (Titus 1). It can be seen that this paradox is famous, but Paul is not interested in its logical solution.

People will ask: Is Epiminides lying? The simplest form of this paradox is:

1-2 "I'm lying"

If he is lying, then "I am lying" is a lie, so he is telling the truth; But if this is true, he is lying again. Contradictions are inevitable. A copy of it:

1-3 "This sentence is incorrect"

A standard form of this paradox is: if event A occurs, non-A is deduced; if non-A occurs, non-A is deduced, which is a self-contradictory infinite logic cycle. One-sided body in topology is the expression of image.

Barber paradox

In Saville village, the barber put up a sign: "I only cut the hair of those people in the village who don't cut their own hair." Someone asked him, "Do you cut your hair?" The barber was speechless at once.

This is a paradox: a barber who doesn't cut his hair belongs to the kind of person on the signboard. As promised, he should give himself a haircut. On the other hand, if the barber cuts his own hair, according to the brand, he only cuts the hair of people in the village who don't cut their own hair, and he can't cut it himself.

So no matter how the barber answers, he can't rule out the internal contradictions. This paradox was put forward by Russell in 1902, so it is also called "Russell paradox". This is a popular and story-telling expression of the paradox of set theory. Obviously, there is also an unavoidable problem of "self-reference".

Paradox of set theory

"R is the set of all sets that do not contain themselves."

People will also ask: "Does R include R itself?" If not, according to the definition of R, R should belong to R. If R contains itself, R does not belong to R..

Kurt G?del (Czech Republic, 193 1) put forward an "incomplete theorem", which broke the ideal that mathematicians thought that all mathematical systems could be deduced by logic at the end of19th century. This theorem points out that any postulate system is incomplete, and there must be propositions that can neither be affirmed nor denied. For example, the negation of the "axiom of parallel lines" in Euclidean geometry has produced several non-Euclidean geometries; Russell's paradox also shows that the axiomatic system of set theory is incomplete.

Bibliographic paradox

A library compiled a dictionary of titles, which listed all the books in the library without their own titles. So will it list its own title?

This paradox is basically consistent with Barber's paradox.

Socrates paradox

Socrates (470-399 BC), an Athenian, is known as "Confucius in the West" and a great philosopher in ancient Greece. He was once opposed to the famous sophists Prut Golas and Gogis. He established a "definition" to deal with the confusing rhetoric of sophists, thus finding out hundreds of miscellaneous theories. But his moral concept was not accepted by the Greeks, and he was regarded as the representative of sophistry when he was seventy years old. Twelve years after expelling Prut Golas and burning books, Socrates was also sentenced to death, but his theory was inherited by Plato and Aristotle.

Socrates famously said, "I only know one thing, and that is nothing."

"Words are full of contradictions"

This is what Zhuangzi said in Zhuangzi's Theory of Everything. Later Mohism retorted: If "everything is against the truth", isn't Zhuangzi's statement against the truth? We often say:

1-7 "There is no absolute truth in the world"

We don't know whether this sentence itself is "absolute truth".

1-8 "absurd truth"

Some dictionaries define paradox as "absurd truth", and this contradiction modification is also a kind of "compressed paradox". Paradox comes from the Greek word "para+dokein", which means "think more".

All these examples show that logically, they can't get rid of the vicious circle brought by the concept of self-reference. Is there a further solution? We will continue our discussion in the last part of the next section.

Dichotomy paradox

This is also a paradox put forward by Zhi Nuo: when an object travels a certain distance to reach D, it must first reach half of the distance D, then a quarter, an eighth and a sixteenth, so that it can be divided indefinitely. Therefore, this object will never reach D.

These conclusions do not exist in practice, but they are logically impeccable.

Zhi Nuo even thought: "There can be no movement from one place to another, because if there is such movement, there will be' perfect infinity', which is impossible." If Achilles really catches up with the tortoise at T, then, "this is an illogical phenomenon, so it is by no means the truth, but just a scam". This means that the senses are unreliable and there is no logical reliability.

He believes that "it is absolutely impossible to be endless." According to this theory of motion, Zhi Nuo also proposed a similar paradox of motion:

2-3 "The arrow does not move"

In Zhi Nuo's view, because the flying arrow has an instantaneous position at every moment of flight, it is no different from being still in this position. So, is the sum of infinite static positions equal to motion? Or the infinite repetition of static motion? There was a similar saying in ancient China, such as:

2-4 "The scenery of birds has not moved"

This is the proposition of Hui Shi, a famous Chinese artist, which is the same as "flying an arrow without moving". This is a conflict between irresistible reasoning and inevitable facts.

In the era of Greek tragedy, the German philosopher Nietzsche called Zeno's paradox "the paradox of denying feelings" in his philosophy. Although it is completely true that Achilles caught up with the tortoise who started first in the race, why is it "illogical"? Because Zhi Nuo used the concept of infinity, which is a logical assumption, there can be no infinity in the real world, so the assumption and reality are contradictory.

"Father died before mother"

This is a self-evident proverb. There are also four explanations: first, "the father is there, and the mother dies first"; Second, "the father died first"; Third, if parents are alive, it can be interpreted as the future; Fourth, even if both parents die, it can be interpreted as "when the father is around, the mother died." Or "father died before mother", which is a two-way street.

In logical order, the above two examples are just the opposite. Both positive and negative propositions can be quibbled according to the so-called objective reasons, forming self-proof or cross-examination. So Gracian said in The Book of Wisdom: An Eternal Classic of Life: "The world is a kind of deception. At first glance, this is quite reasonable and frightening because it is exciting and novel, but when its disguise is exposed, it will bring shame to itself. "

On Deng's paradox of saving the corpse.

"Lu's Spring and Autumn Annals" recorded a story that a member of Zheng Fujia was drowned in a flood in the river. The body was salvaged by others, and the rich family demanded redemption. But the price of the person who found the body is too high, and the family members of the rich family are unwilling to accept it. They came to Deng to analyze their ideas. Deng Xi said, "Don't worry, who else will he sell it to besides you?" The man who found the body was very anxious. He went to Deng for advice. Deng Xi replied, "Don't worry, if he doesn't buy from you, who else can he buy from?"

Deng was born in the late Spring and Autumn Period. Contemporary with Laozi and Confucius, he was the originator of famous scholars in the Warring States period and also a famous litigator. His work has been lost.

With the same facts, Deng came to two diametrically opposite conclusions, each of which sounds logical, but when put together, it is absurd. Does Deng hope that after a period of stalemate, the two sides can find an acceptable price balance point? We can only guess.

"White horse is not a horse"

During the Warring States Period, Gong Sunlong, a native of Zhao, wrote Gong Sunlong Zi, which was well received by Ping Yuanjun. His famous propositions are "White Horse is not a horse" and "Debate on Similarities and Differences".

It is said that Gong Sunlong once rode through the customs, and the gatekeeper said to him, "According to the law, horses are not allowed to cross." Gong Sunlong replied: "I am riding a white horse, and a white horse is not a horse. This is two different things. " We don't know whether Gong Sunlong's "White Horse" has passed the test. From the point of view of ordinary people, 80% of the soldiers guarding the pass think that Gong Sunlong is sophistry. This is also a logical example of "inability to refute", which cannot be established in reality.

In On White Horse, Feng Youlan thinks that "a white horse is not a horse" is Gong Sunzilong:

First, emphasize the different connotations of "horse", "white" and "white horse". The connotation of "horse" is animal, the connotation of "white" is color, and the connotation of "white horse" is animal plus color. The three have different connotations, so a white horse is not a horse.

The second is to emphasize the difference between "horse" and "white horse" in extension. The extension of "horse" includes all horses, regardless of color differences; The extension of "white horse" only includes white horse, which has color difference. The extension is different, so a white horse is not a horse.

The third is to emphasize the difference between the * * phase of "horse" and the * * phase of "white horse". The appearance of a horse is the essential attribute of all horses. Not including color, only including "horse is a horse". * * * Different genders, "when a horse is a horse" and "when a white horse is a white horse" are different. So a white horse is not a horse.

As mentioned earlier, dialectics is developed in the process of dealing with sophistry. Hegel said in Little Logic: "Dialectics must not be confused with simple sophistry. The essence of sophistry is to look at things in isolation and regard their one-sided abstract provisions as reliable. " ("Further Provisions on Logical Concepts and Division of Departments")

From the perspective of dialectics, "white horse is not a horse" cuts off the relationship between the individual and the general. White horse belongs to personality, especially white horse; Horses are common and have the characteristics of horses of various colors. Gong Sunlong distinguished the difference between the two, but he also made it absolute. Although the color of the white horse is different from other horses, such as the yellow horse and the dark horse mentioned by Gong Sunlong, it is still a horse. As a sex, "horse" as a personality resides in "white horse". As a general category, "horse" includes horses of all colors, and Gong Sunlong's white horse is no exception.

Killing thieves is not killing people. "

This proposition is similar to "a white horse is not a horse", although the method and purpose of argument are different. Xunzi classified Mo's statement of "killing thieves but not killing people" as a sophistry of "confusing names". Xunzi believes that the category of "person" includes the category of "thief" in extension. So when we say "thief", that is to say, he is also a "person"; Killing a thief is also killing.