The Barber's Paradox

The aim of every science is to discover the laws that could explain one or another phenomenon. Once these laws are discovered, then science proceed to study the other phenomena, which in the nature are of an infinite set. It is interesting to note that in the process of discovering a law, for example in physics, people make thousands of experiments, build proves, among them some experience, or evidence - useful for understanding a certain phenomenon, but other experiments or evidence proved fruitless. But it is found only with hindsight, when the law is already discovered. Therefore, with the discovering of the law it is enough to show 2 - 3 experiments or prove to verify its correctness. All other experiments were the ways of study and there is no need to repeat them, to understand how the law works. In the exact sciences, it is understood, and therefore the students studies only the information that is necessary to understand specific phenomena. By no means is the case with the study of formal logic.

Formal logic, as opposed to other sciences: physics, chemistry, mathematics, biology and so on, studies not an infinite number of phenomena in nature, but only one how a man thinks, how he learns the world surrounding us, and how people understand each other. In other words, what laws govern the logic of our thinking, i.e., our reasoning and judgments in any science or in everyday life.
By the beginning of XVIII century four laws of logic were formulated : the law of identity, Law of Contradiction, the Law of Excluded Middle and the Law of Sufficient Ground. The first three laws were formulated by Aristotle in the 4 th century BC, as the 4th law was introduced by Leibniz at the beginning of 18 century. So far for more than 300 years, none of the philosophers discovered neither the 5th or 6 th law of formal logic. During this time, all the "discovery" of formal logic were limited only to it’s the distortion and confusion.

If to consider a formal logic of Aristotle from the point of view of its essence , then its center of gravity is its Laws, that were discovered by Aristotle, based on analysis of the different types of syllogism, which Aristotle classified to track down those Laws. In his research, the syllogisms played the same role as the experiments in physics or chemistry for the discovery of regularities, to explain the process of certain events. Once these logical laws of thinking have been discovered, the syllogisms have fulfilled their role. And it would be foolish to assume that our knowledge in any science, is built only on Aristotle's syllogisms or others discovered later. Whatever syllogisms would not have been discovered since Aristotle, none of them had added something new in the laws of formal logic revealed by Aristotle and Leibniz. But philosophers still continue to analyze Aristotle's syllogisms, a historic mission of which ended more than 2000 years ago. Moreover, after the discovering of 4th of law of formal logic, the law of sufficient ground , the legality of any syllogism is easy checked from the viewpoint of the four laws of formal logic, because all our judgments and inferences must be obeyed to these laws, to be true.
Bertrand Russell is the one who belongs to this category of the philosophers, who in his book "History of Western Philosophy", examining the formal logic of Aristotle, has continued to pick weaknesses in his syllogisms, rather than focus his attention on the importance of the laws of formal logic in the human knowledge and to point out to the incompleteness of their definitions.

Here he writes about the formal logic: "Aristotle's most important work in logic is the doctrine of the syllogism... Apart from such inferences as the above, Aristotle and his followers thought that all deductive inference, when strictly stated, is syllogistic. By setting forth all the valid kinds of syllogism, and setting out any suggested argument in syllogistic form, it should therefore be possible to avoid fallacies.This system was the beginning of formal logic, and , as such, was both important and admirable. But considered as the end, not the beginning, of formal logic, it is open to three kinds of criticism:(1) Formal defects within the system itself. (2) Over-estimation of the syllogism, as compared to other forms of deductive argument. (3) Over-estimation of deduct5ion as a form of argument." (Bertrand Russell, "A History of Western Philosophy", p196-197, published by Simon&Schuster)

As we see, in his chapter "Aristotle's Logic" he did not mention at all about the importance of three laws of formal logic discovered by Aristotle. As I said earlier, the power of formal logic, its common to all sciences, based on its four laws, rather than on different types of syllogism. Russell's misunderstanding of this fact led him to an underestimation and distortion of formal logic.

Such a perversion, and was introduced by the famous philosopher, B. Russell, in formal logic, as shown by his following explanation to the 3rd position above: " All the important inferences outside logic and pure mathematics are inductive, not deductive; the only exceptions are law and theology, each of which derives its first principles from an unquestionable text, viz, the statute books or the scriptures" (p. 199)
And in another place he writes:"Valid syllogisms, in fact, are only some among valid deductions, and have no logical priority over others. The attempt to give pre-eminence to the syllogism in deduction misled philosophers as to the nature of mathematical reasoning. Kant, who perceived that mathematics is not syllogistic, inferred that it uses extra-logical principles, which, however, he supposed to be as certain as those of logic. He, like his predecessors, though in a different way, was misled by respect for Aristotle" (p.199)

Rather than to say that even outside of logic and pure mathematics, four of law of formal logic, without any doubt remained valid, he emphasizes on the syllogisms that do not constitute the essence of formal logic and for this reason are not general to all science. His misunderstanding of role of the laws of formal logic in human thinking is confirmed by his paradox with the notion of set. Let us show how this paradox violates the basic laws of formal logic.

The paradox of Russell in the original form is linked to the notion of set or class. But the whole world knows it in another interpretation. Russell proposed the following popular version of discovered him paradox of set theory. Suppose there is a town with just one male barber; and that every man in the town keeps himself clean-shaven: some by shaving themselves, some by attending the barber. It seems reasonable to imagine that the barber obeys the following rule: He shaves all and only those men in town who do not shave themselves. Under this scenario, we can ask the following question: Does the barber shave himself ? Asking this, however, we discover that the situation presented is in fact impossible:

If the barber does not shave himself, he must abide by the rule and shave himself.
If he does shave himself, according to the rule he will not shave himself.

In this paradox, all the men in town are divided into two categories : those who shave themselves, and those who do not shave themselves. And barber is in one of these categories (one sufficient ground) , as he is a man from the town. But on the other hand, the man identified as a barber (another sufficient ground) with the functions that are contrary to the first two categories or sufficient ground:
he shaves all and only those men in town who do not shave themselves. Thus barber is determined in two ways for both categories:(1) as a man who shave himself and (2) as a barber, who shave all and only those men in town who do not shave themselves; or (1) as a man who does not shave himself and (1) as a barber, who shave all and only those men in town who do not shave themselves. The paradox violated the laws of formal logic: the law of identity and the law of sufficient ground. Violation of the Law of Identity takes place by introducing into the paradox of the two sufficient grounds: town men and town barber. And if our assumptions have violated the basic laws of formal logic, the conclusions would be incorrect.

For example, imagine a foreigner with an excellent memory, who memorized 5000 English words, but he absolutely does not know the rules of the English language. Rather than to say: "Today I read an interesting book", he said: "I book today read an interesting ." This sentence does not make sense to anybody because it has no meaning. It is well known that in order to learn new language one has to know besides the words, all the rules of language, another words to be familiar with the laws of the language which one tries to learn. The same thing is in any science, including formal logic. But it does not mean that the English language does not always work or it is not powerful enough to express any idea, because the foreigner breaks the rule of the language. But based on the same logic philosophers come to the conclusion that the formal logic is not perfect , when confronted with logical paradoxes that violating the laws of formal logic .
It is interesting to note that Russell is aware of the presence in the paradox of two sets of definitions, which poses a problem. Russell's own answer to the puzzle came in the form of a "theory of types." The problem in the paradox, he reasoned, is that we are confusing a description of sets of numbers with a description of sets of sets of numbers. " - but he does not associate it with a violation of the law of Identity , not to mention about the law of Sufficient ground, because in this case there would be nothing to talk about and there would be nothing new to be discovered, because this problem has been resolved by Aristotle over 2000 years ago.

His critique of Aristotelean formal logic remind me the story about the curious man who visited Zoo and tried to pay attention to everything including tiny insects, but after his visit of Zoo, he started to boast about his knowledge of animals that he saw, and one of his listeners asked him did he see an elephant in the Zoo and he said that he did not noticed one.
As I have said early, to understand the importance of the law of sufficient ground in its aggregation it is necessary to understand very well the relationship of formal and dialectical logic, and Russell denied the existence of the latter: "Even if (as I myself believe) almost all of the teachings of Hegel is false, it is still important, which not only belongs to history, as it best represents a certain kind of philosophy, which have other, less coordinated and less than comprehensive." After this statement about the dialectical logic, and with such understanding of formal logic, he can not be called not only as a great philosopher but even as a philosopher, because he did not introduce anything new to the development of formal logic of Aristotle, except of the distortion and perversion....

Ilya Stavinsky

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