Lecture
There is no exhaustive list of logical paradoxes, and it is impossible.
The considered paradoxes are only a part of all found by now. It is likely that in the future many other paradoxes will be discovered, and even completely new types of them. The concept of paradox itself is not so specific that it was possible to compile a list of at least already known paradoxes.
“The set-theoretic paradoxes are a very serious problem, not for mathematics, however, but rather for logic and the theory of knowledge,” writes the Austrian mathematician and logician K. Godel. “The logic is consistent. There are no logical paradoxes, ”says mathematician D. Bochvar. This kind of discrepancy is sometimes significant, sometimes verbal. The point is largely what is meant by the logical paradox.
A necessary feature of logical paradoxes is a logical dictionary. Paradoxes attributable to logical should be formulated in logical terms. However, in logic there are no clear criteria for dividing terms into logical and non-logical. The logic involved in the correctness of reasoning tends to reduce the concepts on which the correctness of the practically applied conclusions depends, to a minimum. But this minimum is not predetermined unambiguously. In addition, in logical terms, non-logical statements can be formulated. Whether a particular paradox uses only purely logical premises is not always possible to determine unambiguously.
Logical paradoxes are not rigidly separated from all other paradoxes, just as the latter are not clearly distinguished from all that is non-paradoxical and consistent with the dominant concepts.
At the beginning of the study of logical paradoxes, it seemed that they could be distinguished by violating a certain, not yet studied, position or rule of logic. The principle of the vicious circle introduced by B. Russell was especially actively claiming the role of such a rule. This principle states that a collection of objects cannot contain members that can only be defined by the same collection.
Weight paradoxes have one common property - self-applicability, or circularity. In each of them, the object in question is characterized by a certain set of objects to which it itself belongs. If we single out, for example, the most cunning person, we do it with the help of the aggregate of people to which this person belongs. And if we say: “This statement is false,” we characterize the statement we are interested in by referring to the totality of all false statements that include it.
In all the paradoxes, the self-applicability of concepts takes place, which means that there is a kind of circular movement, leading eventually to the starting point. In an effort to characterize the object of interest to us, we turn to the totality of objects that includes it. However, it turns out that she herself, for her certainty, needs the object in question and cannot clearly be trapped without it. In this circle, perhaps, lies the source of paradoxes.
The situation is complicated, however, by the fact that such a circle exists in many completely non-paradoxical arguments. Circular is a huge variety of the most common, harmless and at the same time convenient ways of expression. Examples such as “the largest of all cities”, “the smallest of all natural numbers”, “one of the electrons of the iron atom”, etc., show that not every case of self-applicability leads to a contradiction and that it is not only important in ordinary language, but also in the language of science.
A simple reference to the use of self-applied concepts is not enough, therefore, to discredit the paradoxes. Some additional criterion is still needed, separating self-applicability leading to paradox from all its other cases.
There were many proposals on this subject, but the circularity was never successfully clarified. It was not possible to characterize circularity in such a way that each circular reasoning led to a paradox, and each paradox was the result of some circular reasoning.
An attempt to find some specific principle of logic, the violation of which would be a distinctive feature of all logical paradoxes, did not lead to anything definite.
Undoubtedly, some kind of classification of paradoxes would be useful, dividing them into types and types, grouping some paradoxes and opposing them to others. However, in this matter nothing sustainable has been achieved.
The English logician F. Ramsay, who died in 1930, when he was not even twenty-seven years old, proposed to separate all the paradoxes into syntactic and semantic. The former include, for example, the Russell paradox, the latter the “Liar”, Grelling, etc. paradoxes.
According to Russell, the paradoxes of the first group contain only concepts belonging to logic or mathematics. The latter include such concepts as “truth”, “definability”, “naming”, “language”, which are not strictly mathematical, but rather related to linguistics or even to the theory of knowledge. Semantic paradoxes are obliged, as it seems, by their appearance not to some kind of error in logic, but to the vagueness or ambiguity of some non-logical concepts; therefore, the problems they bring relate to language and must be solved by linguistics.
It seemed to Ramsay that mathematicians and logicians need not be interested in semantic paradoxes. Later it turned out, however, that some of the most significant results of modern logic were obtained precisely in connection with a deeper study of precisely these non-logical paradoxes.
The paradoxes division proposed by Ramsey was widely used at first, and retains a certain value even now. At the same time, it is becoming increasingly clear that this division is rather vague and relies primarily on examples, and not on in-depth comparative analysis of the two groups of paradoxes. Semantic concepts have now received precise definitions, and it is difficult not to recognize that these concepts really relate to logic. With the development of semantics, which defines its basic concepts in terms of set theory, the distinction made by Ramsey is more and more erased.
What conclusions for logic follow from the existence of paradoxes?
First of all, the presence of a large number of paradoxes speaks of the power of logic as a science, and not of its weakness, as it may seem. The discovery of paradoxes coincidentally coincided with a period of the most intensive development of modern logic and its greatest successes.
The first paradoxes were discovered even before the emergence of logic as a special science. Many paradoxes were discovered in the Middle Ages. Later, however, they were forgotten and were reopened in the 20th century.
The concepts of "set" and "element of set" introduced into science only in the second half of the 19th century were not known to medieval logicians. But the flair for paradoxes was sharpened in the Middle Ages so much so that already at that ancient time certain concerns were expressed about self-applicable concepts. Their simplest example is the concept of “being an element of its own,” which appears in many of the current paradoxes.
However, such concerns, as well as all the warnings concerning paradoxes, were not properly systematic and definite before our century. They did not lead to any clear proposals for the revision of the usual ways of thinking and expression.
Only modern logic has learned from oblivion the very problem of paradoxes, has discovered or rediscovered most of the specific logical paradoxes. She further showed that the ways of thinking, traditionally investigated by logic, are completely inadequate to eliminate paradoxes, and pointed out fundamentally new methods of handling them.
Paradoxes pose an important question: what, in fact, do some of the usual methods of conceptualization and reasoning fail? After all, they seemed completely natural and convincing, until it was revealed that they were paradoxical.
Paradoxes undermine the belief that the usual methods of theoretical thinking themselves but without any particular control over them provide reliable progress towards the truth.
Demanding radical changes in an overly trusting approach to theorizing, paradoxes are a sharp criticism of logic in its naive, intuitive form. They play the role of a factor controlling and limiting the design of deductive systems of logic. And their role can be compared with the role of an experiment that tests the correctness of hypotheses in such sciences as physics and chemistry, and forces them to make changes in these hypotheses.
The paradox in theory says that the assumptions underlying it are incompatible. It acts as a timely detected symptom of the disease, without which it could be overlooked.
Of course, the disease manifests itself in various ways, and, in the end, it can be solved without such acute symptoms as paradoxes. For example, the foundations of the theory of sets would be analyzed and refined if even no paradoxes in this area were discovered. But there would not be the harshness and urgency with which the paradoxes found in it were posed by the problem of revising the theory of sets.
Paradoxes devoted extensive literature, offered a large number of their explanations. But none of these explanations of ns is generally recognized, and there is no complete agreement on the origin of paradoxes and how to get rid of them.
“Over the past sixty years, hundreds of books and articles have been devoted to the goal of resolving paradoxes, but the results are remarkably poor in comparison with the effort expended,” writes A. Frenkel. “It seems that,” concludes his analysis of the paradoxes of X. Curry, “that a complete reform of logic is required, and mathematical logic can become the main tool for carrying out this reform.”
Attention is drawn to one important difference. Elimination of paradoxes and their resolution is not the same thing at all. Eliminating the paradox from a certain theory means that it should be restructured so that the paradoxical statement turns out to be unprovable in it. Each paradox relies on a large number of definitions, assumptions and arguments. His conclusion in theory is a chain of reasoning. Formally speaking, one can cast doubt on any of its links, discard it and thereby break the chain and eliminate the paradox. In many works, and do so, and this is limited.
But this is not the resolution of the paradox. It is not enough to find a way to eliminate it, it is necessary to convincingly substantiate the proposed solution. Doubt itself in any step leading to a paradox should be well grounded.
First of all, the decision to abandon any logical means used in the derivation of a paradoxical statement should be linked to our general considerations regarding the nature of logical proof and other logical intuitions. If this is not the case, the elimination of the paradox is devoid of solid and stable foundations and degenerates into a primarily technical problem.
In addition, the rejection of any assumption, even if it is ensured by the elimination of a particular paradox, does not guarantee at all that all paradoxes will automatically be eliminated. This suggests that paradoxes should not be “hunted” one by one. The exclusion of one of them must be so justified that a definite guarantee appears that by the same step other paradoxes will be eliminated.
Every time a paradox is revealed, A. Tarsky writes, “we must subject our methods of thought to a thorough revision, reject any assumptions we believed in, and improve the methods of reasoning that we used. We do this by striving not only to get rid of antinomies, but also to prevent the emergence of new ones. ”
And finally, an ill-conceived and careless rejection of too many or too strong assumptions can simply lead to the result that, although not containing paradoxes, a significantly weaker theory, which is of particular interest only.
What could be the minimal, least radical set of measures to avoid the known paradoxes?
One way is to isolate, along with true and false sentences, also meaningless sentences. This path was adopted by B. Russell. Paradoxical reasoning was declared meaningless to them on the grounds that they violate the requirements of logical grammar. Not every sentence that does not violate the rules of ordinary grammar is meaningful - it must also satisfy the rules of a special, logical grammar.
Russell built a theory of logical types, a kind of logical grammar, the task of which was to eliminate all known antinomies. Later this theory was significantly simplified and received the name of a simple type theory.
The main idea of the theory of types is the selection of different types of objects in a logical sense, the introduction of a peculiar hierarchy or ladder of the objects in question. To the lower, or zero, type are individual objects that are not sets. The first type includes sets of objects of zero type, i.e. individuals; to the second - sets of sets of individuals, etc. In other words, a distinction is made between objects, properties of objects, properties of properties of objects, etc. At the same time, certain restrictions on the construction of sentences are introduced. Properties can be assigned to objects, properties of properties to properties, etc. But it is impossible to argue meaningfully that the properties of properties are present in objects.
Take a series of sentences: This house is red. Red is the color. Color is an optical phenomenon.
In these sentences, the expression “this house” means a certain object, the word “red” indicates a property inherent in this object, “to be a color” means a property of this property (“to be red”) and “to be an optical phenomenon” indicates a property of “Be color” belonging to the property “be red”. Here we are dealing not only with the object and its properties, but also with the properties of properties (“the property of being red has the property of being color”), and even the properties of properties of properties.
All three sentences from the given series are, of course, meaningful. They are built in accordance with the requirements of the theory of types. And, say, the sentence "This house is a color" violates these requirements. It attributes to the subject that characteristic which can belong only to properties, but not to objects. A similar violation is contained in the sentence "This house is an optical phenomenon." Both of these sentences should be considered meaningless.
A simple type theory eliminates the Russell paradox. However, to eliminate the “Liar” and Berry's paradoxes, simply dividing the objects in question into types is no longer enough. It is necessary to additionally introduce some ordering within the types themselves.
The elimination of paradoxes can also be achieved in the way of avoiding the use of too large sets, like the set of all sets. This path was proposed by the German mathematician E. Zermelo, who linked the appearance of paradoxes with unlimited construction of sets. Admissible sets were defined by him by some list of axioms, formulated in such a way that well-known paradoxes were not derived from them. At the same time, these axioms were strong enough to deduce from them the usual arguments of classical mathematics, without paradoxes.
Neither these two nor the other proposed ways to eliminate paradoxes are not universally accepted. There is no single belief that any of the proposed theories solves logical paradoxes, and not just discards them without a deep explanation. The problem of explaining paradoxes is still open and still important.
G. Frege, the greatest logic of the nineteenth century, was, unfortunately, a very bad character. In addition, he was intractable and even cruel in his criticism of contemporaries. Perhaps that is why his contribution to the logic and justification of mathematics for a long time did not receive recognition. And when the fame began to come to him, the young English logician B. Russell wrote to him that a contradiction arises in the system published in the first volume of his book “The Basic Laws of Arithmetic”. The second volume of this book was already in print, and Frege was only able to add a special appendix to it, in which he stated this contradiction (later called the “Russell Paradox”) and admitted that he was unable to eliminate it.
However, the consequences of this recognition were tragic for Frege. He experienced the strongest shock. And although he was only 55 years old at the time, he did not publish any more significant work on logic, although he lived for more than twenty years. He did not even respond to the lively discussion caused by the Russell paradox, and did not react to the many proposed solutions to this paradox.
The impression made by mathematicians and logicians with just open paradoxes was well expressed by D. Hilbert: “The state in which we are now in relation to paradoxes is unbearable for a long time. Consider: in mathematics — this model of authenticity and truth — the formation of concepts and the course of reasoning, as everyone studies them, teaches and applies, leads to absurdity. Where to look for reliability and truth, even if mathematical thinking itself misfires? ”
Frege was a typical representative of the logic of the end of the 19th century, free from any paradoxes, logic, confident in its capabilities and asserting it. to be a criterion of rigor even for math. Paradoxes showed that absolute severity, supposedly achieved by logic, was nothing more than an illusion. They, undoubtedly, said that the logic - in that intuitive form as it had at the turn of the XIX — XX centuries - needs to be thoroughly revised.
About a century has passed since a lively discussion of paradoxes began. The attempted revision of logic did not lead, however, to their unambiguous resolution.
And at the same time, such a state hardly worries anyone today. Over time, attitudes towards paradoxes became calmer and even more tolerant than at the time of their discovery. The point is not only that paradoxes have become something familiar. And, of course, not that they were reconciled. They still remain in the center of attention of logikov, the search for their solutions is actively continuing. The situation has changed primarily because the paradoxes were localized, so to speak. They found their specific, albeit restless place in a wide range of logical studies. It became clear that the absolute rigor, as it was painted at the end of the XIX century. and even sometimes at the beginning of the 20th century, is in principle an unattainable ideal.
It was also realized that there is no single, paradoxical problem that stands alone. The problems associated with them are of different types and affect, in essence, all the main sections of logic. The discovery of paradox forces us to analyze our logical intuitions more deeply and to systematically rework the foundations of the science of logic. At the same time, the desire to avoid paradoxes is not the only or even, perhaps, the main task. They are, although important, only a reason to reflect on the central themes of logic. Continuing the comparison of paradoxes with particularly distinct symptoms of the disease, one can say that the desire to immediately eliminate paradoxes would be like the desire "to remove such symptoms without worrying about the disease itself.
It is not only the resolution of paradoxes that is required, but their explanation, which deepens our understanding of the logical laws of thinking, is necessary.
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Logics
Terms: Logics