7.1 Non-classical logic Classical and non-classical in modern logic. From the history of non-classical logic

Classical and non-classical in modern logic

The immediate result of a revolution that occurred in logic at the end of the 19th century and the beginning of the 20th century was the emergence of a logical theory, which eventually received the name of classical logic. Along with many other researchers, Irish logician D. Buhl, American philosopher and logician C. Pierce, German logician G. Frege stand at its origins. In their works, the idea of ​​transferring to the logic of those methods that are commonly used in mathematics was gradually realized.

Classical logic focused mainly on the analysis of mathematical reasoning. Many of its features are associated with this, which are now often regarded as its shortcomings. In the process of development, it turned out to be one of many logical theories. But this does not mean that it is now only of historical interest. Classical logic still remains the core of modern logic, retaining both theoretical and practical significance.

A variety of non-classical trends that have arisen later constitute in aggregate the rather indefinite and heterogeneous whole, which is usually combined under the name of non-classical logic. Some of these areas were formed in opposition to classical logic, others in the controversy with it. But for all she was a model of the approach to the logical analysis of thinking, the first theory, consistently and fully implementing the program of mathematical logic.

From the history of non-classical logic

Criticism of classical logic began at the beginning of the 20th century. And it was conducted from different directions. The result was the emergence of a number of new sections of modern logic. In a number of cases, it turned out that the ideas realized at the same time were actively discussed in ancient and medieval logic, but were thoroughly forgotten in Povey time.

In 1908, L. Brower, a Dutch mathematician and logician, questioned the unlimited applicability in mathematical reasoning of the classical laws of the excluded third, (removing) double negation, indirect proof. One of the results of the analysis of such reasoning was the emergence of intuitionistic logic, formulated in 1930 by A. Geyting and not containing these laws. Simultaneously with Brouwer, the idea of ​​the nonuniversality of the law of the excluded third was defended by N. L. Vasiliev.

As early as 1912, the American logician and philosopher KI Lewis drew attention to the so-called “paradoxes of implication”, which are characteristic of the formal analogue of the conditional statement in classical logic — material implication. Lewis developed the first non-classical theory of logical consequence based on the notion of strict implication, defined in terms of logical impossibility. To date, a number of theories have been proposed which claim to be more adequate than the description given by classical logic, a logical consequence and a conditional connection. The most famous of them was the relevant logic developed by the American logicians A. R. Anderson and ND Belnan.

At the turn of the 20s. XX century. K. I. Lewis and J. Lukasiewicz built the first in modern logic modal logics, which considered the concepts of necessity, possibility, chance, etc. Thus, the theme of modalities was revived, which Aristotle and medieval logicians were actively engaged in.

In the 20s. multivalued logic also began to emerge , suggesting that the statements are not only true or false, but other truth values ​​may also have; deontic logic, which studies the logical connections of normative statements; the logic of absolute estimates, exploring the logical structure and logical connection of evaluative statements; probabilistic logic, which uses probability theory to analyze problematic reasoning, etc. All these new sections of logic were not directly related to mathematics, the natural sciences and the humanities were already involved in logical research.

Later on, interesting logic of time was developed and found , which describes the logical connections of statements, for which the time parameter is included in a logical form; paraconsistent logic that does not allow to bring anything out of contradiction; epistemic logic, studying the concepts of "refutable", "insoluble", "provable", "convinced", "doubts", etc .; preference logic dealing with the terms “better”, “worse” and “equivalently”; the logic of change, talking about change and formation; the logic of causality, which studies allegations of determinism and causality, etc. The extensive growth of logic has not been completed even now.

In the following, some non-classical sections of logic will be considered. Comparison of the main ideas underlying the foundations of classical logic, on the one hand, and different branches of non-classical logic, on the other, is interesting from the point of view of understanding each of these sections of logic. Such a comparison also makes it possible to more clearly understand the general principles of the approach of modern logic to the description of thinking.