Has no relation. Truth has nothing to do with logic. The only story that has nothing to do with the car. Almost none

Sometimes Prayer Doesn't Seem to Be Related to the Actual Outcome The third reason why prayer requires us to exert ourselves is that we often feel as if prayer is not related to the actual outcome. So that you don't blame

Autograph | "Modern Logic"

Vladislav Preobrazhensky

Everyone considers himself to one degree or another to be proficient in logic. However, as with language, theoretical knowledge opens up much more possibilities than intuitive use. Moreover, the theory of logic, contrary to Kant’s premature statement, does not stand still and continues to develop. To find out more about this, PostNauki editor Vladislav Preobrazhensky spoke with the author of the book “Modern Logic” Alexander Arkhipovich Ivin.

How does modern logic differ from classical Aristotelian logic?

Aristotelian logic existed in the 4th century BC and ended with Aristotle, because he did nothing but syllogistics. I wrote two thick books about this - good books, but this was only the beginning, the first logical theory. Aristotle was also the first to formulate the basic principle of logic: the correctness of reasoning depends only on its logical form. Then logic developed, which is usually called traditional: Diodorus Cronus, Philo, who introduced the central concept of “material implication”, etc. A more fundamental propositional logic was invented. In the Middle Ages it had already taken shape: modus ponens, modus tollens, from where disjunctive syllogisms, conjunctive ones, etc. come. In modern times, development completely died out, and classical logic as a theory was finally formed only at the end of the 19th century, so it should not, of course, be called “Aristotelian.”

Where did the next stage begin?

Even 300 years ago, Leibniz suggested that logic and mathematics are almost identical: a logical proof is a calculation, but we do not know the rules of this calculation, although we use “and”, “or”, “if, then”, logical consequence, denial, etc. If we describe these rules, we will get a formalized language - one in which there are only symbols and not a single thing that we call a word. Leibniz tried to do this, but it didn’t work out. In the 19th century, mathematicians took up this task independently, and the first was George Boole. He constructed an algebra without degrees, and it turned out that his algebra fits exactly with logic. This is where the second stage of development began.
The first non-classical implication was introduced in 1912 by the American logician K.I. Lewis, he improved it until the 20s. Based on his theory of “strict implication,” Lewis built modal logic (“necessary,” “possible,” “impossible”). Around the same time, Polish logician Jan Łukasiewicz constructed his own version of modal logic.

That is, modern logic appeared as a result of a merger with mathematics?

Yes, it arose at the intersection of very different sciences. From the very beginning it was called mathematical logic because it was similar to mathematics. The idea is very simple: proofs are calculations. It is also called symbolic logic, because it builds artificial languages ​​in which there is not a single word of natural language, and this allows you to speak strictly, juicily and unambiguously. And now the name “modern logic” has become established.

After Boole, the German Gottlob Frege took up logic, and he had the idea that mathematics could be reduced to logic. But for this, again, it is necessary to formalize logic, axiomatize it and bring mathematical axioms to logical ones (at that time Peano had already proposed how to axiomatize mathematics). Frege was already finishing his book, and at that time the young Englishman Russell discovered a paradox in his reasoning.

The one about the set of all sets?

Yes, that's absolutely right. The paradox of the set of all sets that are not their own elements. Is such a set its own element or not? The only thing Frege was able to do was write in a note that he received such a letter, but he did not know the solution to the paradox. This struck him so deeply that, after living for another 20 years, he did not write a single line of logic. He was deeply worried because he believed that he was close to a tremendous success - reducing all mathematics to logic. But it turned out that no, it does not reduce.

And then they found a solution to this paradox?

One solution or another has been proposed for each paradox, but how acceptable is it? As a rule, different authors disagree on whether to make a decision or not. They proposed dividing the language into the one in which we talk about the world, and the one in which we talk about the language, that is, a metalanguage. But a radical solution, i.e. There is still no single generally accepted one. The paradox remains.

You write that there are an infinite number of laws of logic. Can we say that these laws exist in nature and we discovered them, or are they the fruit of the human mind?

You are asking one of the most difficult questions. It is clear that no one discovered any laws. They seemed to exist on their own. Let's say, the law of contradictions - you cannot say “yes” and “no”, you cannot affirm and deny at the same time. Did it exist or not? Yes, human thinking has always been guided by this principle. Did someone open it? Aristotle was the first to describe it, and before him Plato spoke about this law, and even earlier Socrates.
There are infinitely many logics. They are partially opening up. Let's say, Post, an American, built many-valued logics in the 20s, but he simply designed them, they were such artificial systems. There are many values ​​between zero and one, and all those numbers in between are truth values. Zero is a lie, one is true, and everything in between is something between a lie and the truth. And he built, formally speaking, a completely correct system. At the same time, Lukasiewicz was working - for some reason he loved three-valued logic. True, in 1954 he built a four-digit logic, which he said was the best. But it coincided with double digits.

If there are an infinite number of different logics, then what are the criteria for whether a logic is correct in its reasoning or incorrect?

This cannot be said. There are no criteria for determining which of the infinite number of logics is correct. There is an infinite number of many-valued logics. There is an infinite number of numbers and, accordingly, you can choose your own logic for each set of numbers. This is not difficult to do, but which one is true and which one is false? And is multi-valued logic necessary?

The law of contradiction is probably the first law to be discovered. What is paraconsistent logic and how final is the law of contradiction?

No law is final; these are the first steps on an endless path. And the law of contradictions was discovered even before Aristotle, and there were disputes about it. Somewhere in the 50s of the last century, paraconsistent logic appeared. In traditional logic, anything could be deduced from contradictions. If there is A and not-A, then B - any arbitrary statement - follows from this. Tokyo is a big city, Tokyo is a small city, therefore two times two equals five, two times two equals seven, the moon is made of green cheese.

But this is completely counterintuitive.

However, traditional logic still believes this, and it underlies all logic. And so the Brazilian Da Costa proposed that it was necessary to build a logic in which there would be no possibility of such transitions from contradictions to anything. From a contradiction you can move on to some of its parts, but not to anything, especially not connected in meaning. And they built a logic in which the law of contradiction did not operate, but at the same time it was real logic. Although it turned out to be extremely complex, there has long been no interest in it, and what it gave is unclear. She didn't give anything.

Is there any consensus among logicians on how to deal with the law of contradiction now?

There is no consensus, and there is no talk about this law. After all, every logical system is an infinite set of laws, and usually among this set there is also a law of contradictions. What to do with it? The law is like a law, ordinary, and until the 50s of the last century it did not arouse any interest, and even now it has ceased to arouse it. And, say, the law of the excluded middle - either it snows or there is no snow - mathematicians argued about it. The Dutchman Brouwer believed that it was applicable only for finite sets, and threw out the law of excluded middle from his logic.

You write that among the main problems that modern logic studies is the relationship between knowledge and values. How can a researcher approach this?

Knowledge is that which is true or false. And values ​​or assessments are something that does not have a true purpose: “it’s good that it’s raining,” “it’s bad that it’s raining,” or, say, “Alexander the Great is a wonderful person.” These are neither true nor false, and these are called values.
There is a logic of assessments, there is a logic of norms created after the 50s of the last century. I was the first to construct the logic of absolute assessments: “good”, “bad”, “indifferent”, how they relate to each other. Since 1948, economists began to build the logic of comparative assessments: “better”, “worse”, “equal”. So all this has been sufficiently studied, but to what extent all this is applicable is difficult to say. My book “Foundations of Evaluation Logic” was translated into 20 languages, and the result? Where should I put it? And there is nowhere to put it.

By the way, which of your books do you think are the best?

Among scientific ones, I consider the first book where I built the logic of absolute assessments to be the most successful. It is stated there, as well as the logic of utilitarian or instrumental assessments associated with the first. It includes assessments of the type: “if A, then B and B is good, then A is good” (this is a special case).
And among the popular books - “According to the Laws of Logic”, published in 1983 in the “Young Guard” (I was then a teacher at Moscow State University). It is simply, cheerfully and clearly written. It was a good book, I still like it, and it is still on sale. She, however, caused some scandal. Firstly, the editor and I missed the phrase “some people from high stands say something that is not what they really think.” These are Soviet times! Then there was the chapter “With the Thought of the Meaningless.” And so all this was presented in such a way that supposedly Soviet life is nonsense, and the theory of communism is something that is designed to give the appearance of meaning, although the chapter was about something completely different. They tried to impose all sorts of penalties on me for this, but I am not a member of the editorial board and even a non-party member - I cannot be punished. But the editor - he was an elderly man - was immediately fired. The deputy editor-in-chief of Young Guard was reprimanded.

Before the interview, you said that you are quite critical of Modern Logic...

It is written competently, but if I had realized my original plan, I would have made a more lively and entertaining book. Another thing is that it contains a large number of new and interesting ideas, including those that seem important to me. But they didn’t seem like that to the editor.

What ideas are you talking about?

I think that modern logic makes a big mistake by connecting its reasoning directly with the concept of truth. Truth has nothing to do with logic. What is important in it is not truth, but correctness, and it does not depend on the content at all. We say that the correct reasoning is “if A, then VA. Therefore B.” We do not have A and B, these are arbitrary statements, but the reasoning is always correct in its form. The correctness of reasoning and the problem of truth are completely different things.

Doctor of Philosophy, chief researcher at the Institute of Philosophy of the Russian Academy of Sciences, specialist in the field of logic, theory of argumentation, philosophy of science

Remember once and for all what the main logician of the country says, despite the fact that he absorbed the dialogue with isthmath from the atmosphere of our society along with all of us loved:
“Truth has nothing to do with logic. What is important in it is not the truth, but the correctness [of the chain of statements - ], and it does not depend on the content at all.”
And logicians know better what is what than individual uninvited comrades from the back alley, declaring that truth is compliance with reality and its laws, revealed to him from popular books.
Reality, comrades, is not something that is, but, to begin with, it is something that appears to us in our psyche, smoked by all the fumes of the modern world, and on the basis of which we react to its images in our being...
And if this is not so, then study logic, finally discover the final truth and convey it, tell it to all those different comrades around the world who are stuck in age-old conflicts...
And then, and then!..
Yeah.

well and HERE, especially those who are dissatisfied with the Soviet authorities and other disreputable people, please do not look, but rather read my materials according to the tags in this entry. Pay special attention to tags true And conventionalism .
And you will be shushed!