Lecture
Sharp criticism of classical logic has been subjected to the fact that it does not provide a correct description of logical following.
The main task of logic is to systematize the rules that allow deriving new ones from the accepted statements. The possibility of obtaining some ideas as logical consequences of others lies in the foundation of any science. This makes the problem of correctly describing logical following extremely important. Failure to solve it adversely affects not only the logic itself, but also the methodology of science.
Logical following is a relation that exists between assertions and conclusions that are reasonably derived from them, a relation that is well known to us from the practice of ordinary reasoning. The task of logic is to clarify the intuitive, spontaneously formed notion of following and formulate on this basis a uniquely defined notion of following. The latter must, of course, be sufficiently consistent with the intuitive representation it replaces.
Logical following should lead from true positions only to true ones. If the conclusions attributed to the justified, would give the opportunity to move from truth to false, the establishment between the statements of the relationship of following would lose all meaning. The logical conclusion would be transformed from a way of unfolding and developing knowledge into a means, erasing the line between truth and error.
Classical logic satisfies the requirement to lead from truth only to truth. However, many of its provisions on the following do not agree well with our usual notions about it.
In particular, classical logic says that anything can logically follow from contradiction. For example, from the controversial statement “Tokyo is a big city, and Tokyo is not a big city” follow, along with any other statements: “The mathematical theory of sets is consistent,” “The moon is made of green cheese”, etc. But there is no meaningful connection between the original statement and these supposedly arising statements. Here is a clear departure from the usual notion of following.
The situation is the same with the classical proposition that logical laws follow from any assertions. Our logical experience refuses to admit that, say, the statement “Ice is cold or ice is not cold” can be deduced from statements like “Two is less than three” or “Aristotle was the teacher of Alexander of Macedon”. The consequence that is derived must be somehow related to what it is derived from. Classical logic neglects this obvious circumstance.
An important role in all our reasoning is played by conditional statements, formulated with the help of the union “if ... then ...”. They perform many different tasks, according to their typical function - the substantiation of some statements by reference to others. For example, the electrical conductivity of copper can be justified by referring to the fact that it is metal: "If copper is metal, then it conducts electrical current."
Conditional statement in logic is called implication.
Classical logic interprets the conditional statement “If A, then B” in this way : it is false only if A is true, and B is false, and true in all other cases. It is true, in particular, when A is false, and B is true. The substantive, semantic relationship between statements A and B is not taken into account. Even if they are not related to each other, the conditional statement made up of them can be true.
So interpreted conditional statement was called material implication. According to her definition, such statements, for example, should be considered true: “If the moon is inhabited, then two and two are four,” “If the Earth is a cube, then the Sun is a triangle,” etc. It is obvious that, even if the material implication is useful for many purposes, it nevertheless does not agree well with the usual understanding of the conditional connection.
First of all, this implication poorly performs the function of justification. It is unlikely that in any reasonable sense such statements as: "If Napoleon died in Corsica, then Archimedes' law is not open to them," "If copper is an Egyptian deity, it is electrically conductive." It cannot be said that by putting an arbitrary statement before a true statement, we thereby substantiated this statement. Classical logic says: a true statement can be substantiated with the help of any statement.
It is difficult to attribute to true justifications such material implications as: “If lions have no teeth, then giraffes have long necks”, “If two and two are five, then Jupiter is inhabited,” etc. However, classical logic says: using a false statement, you can justify anything.
These and similar provisions on justification, defended by classical logic, are called paradoxes of material implication. They do not agree with the usual ideas regarding the substantiation of some statements with the help of others.
Thus, classical logic cannot be recognized as a successful description of logical following. The first to point this out in 1912 was the American logician K. Lewis. Then the logic was on the rise, it seemed flawless, and the criticism of Lewis in her address was not taken seriously. He was even accused of not understanding the substance of the matter. But he continued to deal with this problem and proposed a new theory of logical following, in which the material implication was replaced by another conditional connection - a strict implication. It was a big step forward, although it turned out that strict implication is also not without paradoxes.
A more perfect description of the conditional connection and logical sequence was given in the 50s of the 20th century. German logician V. Ackermann and American logicians A. Andersen and N. Belnan. They managed to exclude not only the paradoxes of material implication, but also the paradoxes of strict implication. The implication they introduced was called relevant (that is, relevant), since it can only be linked by statements that have some kind of common content.
At present, the theory of logical pursuit is one of the most intensively developing sections of non-classical logic. An interesting new approach is recently outlined by German logician X. Wessel. He proposed to separate the two tasks that were previously solved at the same time: first, to describe the basic rules of logical following, and only then to introduce different types of conditional links or implications. Evaluation of this approach is a matter of the future.
The emergence of quantum mechanics, which replaced the classical mechanics of Newton, produced a genuine revolution in physical thinking. The revision of traditional ideas led to the emergence of the idea of a special logic of quantum mechanics. It was assumed that the theories of classical physics, describing the facts, are based on the laws of ordinary logic - the logic of the macrocosm; But quantum physics deals not only with facts, but with their probabilistic connections, and they argue in it, relying on completely different thinking patterns. The identification and systematic description of the latter is the task of the special logic of the microworld.
This idea was first expressed by the American mathematician D. von Neumann. In the mid 30s. XX century. he, along with another American mathematician D. Birktof, built a special quantum logic, which marked the beginning of another direction of non-classical logic. Later, the German philosopher G. Reichenbach built a special logic in order to eliminate the "causal anomalies" arising from attempts to apply the classical causal explanation to quantum phenomena. To date, dozens of different logical systems have been proposed, seeking to reveal the peculiarity of reasoning about quantum objects.
These “quantum logics” are seriously distinguished both by the sets of laws adopted in them and by the methods of their justification.
In the initial period of its development, quantum logic met both critics (physicists N. Bor, V. Pauli) and approval (physicists K. Weizsäcker, V. Heisenberg, M. Born). A lengthy controversy did not, however, clarify the question: is quantum mechanics really guided by particular logic? Even if this is the case, it must be admitted that research in this area did not have any noticeable impact on the development of quantum mechanics itself. Gradually, quantum logic even began to move away from it and look for applications in other areas. One of these emerging applications is the dialogue of two researchers who hold opposite points of view on the subject under discussion, but use the common language of the dialogue.
Science is irreconcilable to contradictions and successfully fights them. But in the life of many scientific theories, especially at the beginning of their development, there are periods when they are not free from internal contradictions.
Logic that requires the exclusion of contradictions must be considered with this circumstance. In addition, she herself is inherent in internal contradictions (logical paradoxes), periodically causing a lot of concern.
Classical logic approaches contradictions somewhat straightforwardly. According to one of its laws, anything that follows from a contradiction. This means that a contradiction is prohibited, moreover, it is prohibited under the threat that if it appears in the theory, any statement will be provable. Obviously, the theory will be destroyed.
However, no one really uses this permission to deduce from the contradictions all that is horrible. The practice of scientific reasoning sharply diverges in this paragraph with logical theory.
In response to this discrepancy, in the last decades, various versions of paraconsistent logic have been developed . Its somewhat unusual name is intended to emphasize that it interprets a contradiction differently than classical logic. In particular, it excludes the possibility of deriving any statements from contradictions. Proofability in the theory of contradiction ceases to be a deadly threat hanging over it. This does not eliminate, of course, the fundamental need to get rid of contradictions in the process of further development of the theory.
It is interesting to note that one of the first (as far back as 1910) doubts about the unlimited applicability of the law of contradiction was expressed by the Russian logician I. A. Vasiliev. “Suppose,” he said, “the world of a realized contradiction, where contradictions would be derived, would such knowledge not be logical?”
Vasiliev wrote not only scientific articles, but also poems. In them, his logical ideas were sometimes peculiarly refracted, in particular, the idea of imaginary (possible) worlds:
I dream of an unknown planet, Where everything goes differently than ours.
As the logic of the imaginary world, he proposed his theory without the law of contradiction, which for a long time was considered the central principle of logic. Vasiliev considered it necessary to limit also the effect of the law of the excluded middle and in this sense was one of the ideological predecessors of intuitionistic logic.
Vasiliev's innovative ideas were not understood by his contemporaries. They were interpreted incorrectly, declared illiterate. Vasiliev was seriously worried about such criticism and soon left his lessons with logic. It took half a century before its “imaginary logic” without the laws of contradiction and the excluded middle was appreciated.
The concept of causality is one of the central both in science and in the philosophy of science. A causal relationship is not a logical relationship. But the fact that causality is not reducible to logic does not mean that the problem of causality has no logical content and cannot be analyzed with the help of logic. The task of the logical study of causality is to systematize the correct patterns of reasoning, the premises or conclusions of which are causal statements. In this regard, the logic of causality is no different, say, from the logic of time or the logic of knowledge, the purpose of which is to build artificial languages that allow for a greater clarity and efficiency to talk about time or knowledge.
In the logic of causality, the relationship of cause and effect seems to be a special conditional statement - causal implication. The latter is sometimes taken as a source, not explicitly defined concept. Its meaning is given by many axioms. More often, however, such an implication is defined through other, clearer or more fundamental concepts. These include the concept of ontological (causal, or factual) necessity, the concept of probability, etc.
Logical necessity is inherent in the laws of logic, ontological necessity characterizes the laws of nature and, in particular, causal relationships. The expression “L is reason B” (“A implies causally B”) can be defined as “Ontologically necessary that if A, then L”, thus distinguishing the simple conditional link from the causal implication.
Through probability, a causal relationship is defined as follows: event A is the cause of event B, only if the probability of event A is greater than zero, it occurs before B and the probability of occurrence of B when A is present is higher than the mere probability of B.
The concept of causality is defined using the concept of the law of nature: A causally attracts B only if A does not imply logical B, but A, taken along with many laws of nature, follows logically B. The meaning of this definition is simple: causality is not logical, the consequence follows from the reason not by virtue of the laws of logic, but on the basis of the laws of nature.
For causality, the following statements are true:
• nothing is the cause of itself;
• if one event is the cause of the second, then the second is not the cause of the first;
• the same event cannot be both the reason for the presence of an event, and the reason for its absence;
• there is no reason for the occurrence of a controversial event, etc.
The word "cause" is used in several senses, differing in their strength. The strongest meaning of causality suggests that having a cause cannot be, i.e. can neither be canceled nor altered by any events or actions. Along with this notion of a complete, or necessary, cause, there is also a weaker notion of a partial, or incomplete, cause. For a complete reason, the condition is fulfilled: “If event A causally implicates event B, then L along with any event C also causally implies B”. For an incomplete reason, it is true that in the event of any events A and B, if A is a partial reason for B, then there is an event C that A together with C is a complete cause B, and yet it is not true that A without C is a complete cause B. In other words, the full cause always, or in any conditions, causes its effect, while the partial cause only contributes to the onset of its effect, and this effect is realized only if the partial cause is combined with other conditions.
The logic of causality is constructed in such a way that it can provide a description of both complete and incomplete reasons. This logic finds applications when discussing the concepts of the law of nature, ontological necessity, determinism, etc.
The logic of change is a section of modern logic that studies the logical connections of statements about the change or formation of material and other objects. The task of the logic of change is the construction of artificial (formalized) languages, capable of making more clear and precise the reasoning about the change of objects — the transition from one state of an object to another, its state, the formation of an object, its formation. The logic of change says nothing about the specific characteristics of change and formation. It only represents a perfect language from the point of view of syntax and semantics, which allows to give rigorous statements of statements about changing objects, reveal the bases and consequences of these statements, reveal their possible and impossible combinations. The use of an artificial language when discussing the problems of changing objects does not mean replacing these ontological problems with logical ones, or reducing the empirical properties and dependencies to logical ones.
The development of change logic goes in two directions: the construction of special change logic and the interpretation of certain time logic systems as logical descriptions of change. In the first approach, the “instantaneous” characteristic of a changing object is usually given; in the second, the change is considered as the relationship between successive states of the object.
The first direction is, in particular, the logic of directivity. Its language is richer than the language of classical logic, and includes not only the terms “exists” and “does not exist”, but also the terms “arises”, “disappears”, “already is”, “still is”, “is no longer”, “ not yet ", etc. With the help of these terms, such laws of logic of directionality are formulated, as, for example:
• to exist is the same as starting to disappear, and the same as ceasing to arise;
• not to exist - the same as starting to emerge, and the same as stopping to disappear;
• becoming is the cessation of non-existence, and extinction is the occurrence of non-existence;
• already exists - it means it exists or arises;
• still exists - it means, it exists or disappears, etc.
The logic of direction allows for four types of existence of objects: being, non-being, appearance (formation) and disappearance. With respect to any object, it is true that it either exists, or does not exist, or arises, or disappears. At the same time, an object cannot simultaneously exist and not exist, exist and disappear, exist and arise, not exist and disappear, arise and disappear, etc. In other words, the four possible types of existence exhaust all modes of existence and are mutually incompatible. The logic of directionality allows us to express in a logically consistent form the idea of the inconsistency of any movement and change. The statement "The object is currently moving in a given place" is equivalent to the statement "At the moment in question, the object is and is not in this place."
Примером второго подхода к логике изменения является логика времени финского философа и логика Г. X. фон Вригта. Ее исходное выражение «А и в следующей ситуации В» может интерпретироваться как «Состояние А изменяется в состояние В» («А-мир переходит в Б-мир»), что дает логику изменения. В логике времени доказуемы такие, в частности, утверждения:
• всякое состояние либо сохраняется, либо возникает, либо исчезает;
• при изменении состояние не может одновременно сохраняться и исчезать, сохраняться и возникать, возникать и исчезать;
• изменение не может начаться с логически противоречивых состояний и не может вести к таким состояниям и т.п.
Примеры утверждений, доказуемых в различных системах логики изменения, показывают, что она не является самостоятельной теорией изменения и не может претендовать па то, чтобы быть таковой. Формально-логический анализ изменения объекта преследует узкую цель — отыскание средств, позволяющих отчетливо зафиксировать логические связи утверждений об изменении того или иного объекта.
Вместе с тем логика изменения имеет важное философское значение, поскольку тема изменения (становления) еще с Античности стоит в центре острых философских дискуссий.
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Logics
Terms: Logics