8.1 Modal logic Modal concepts

From the point of view of the humanities, modal logic is of particular interest from non-classical branches of logic. A separate chapter is dedicated to her. In the future, from different sections of this logic, assessments and norms that play a special role in the social and human sciences are considered in more detail.

Modal expressions are statements that include modal concepts, or modalities (from lat. Modus - measure, image, method).

Examples of modal concepts include: “possible”, “necessary”, “accidentally”, “convinced”, “knows”, “believes”, “necessarily”, “allowed”, “prohibited”, “good”, “indifferent” , “Bad”, “better”, “worse”, etc.

There is no exact and complete list of modal concepts. Their range is wide, in the language they can be expressed in different contexts in different words.

Modal logic is a section of modern (non-classical) logic that deals with reasoning, which includes modal expressions.

For logic that deals only with non-modal utterances, the thing exists or does not exist, and there are no other options. But both in ordinary life and in science, one constantly has to speak not only about what is in reality and what is not, but also about what should or should not be, can or cannot be, which is good to be, and what is bad to be, etc.

The actual course of events, in which there is only “yes” and “pet”, can be viewed as the realization of one of the many conceivable possibilities, and the real world in which we are located - as one of an infinite number of possible, gradually realizing worlds. N. A. Vasiliev expressed this idea in the following way:

... In a possible vast ocean, the Real is a small Gulf Stream.

The language of non-modal expressions is too poor to be able to convey reasoning not only about real events (taking place in the real world), but also about events possible (occurring in some possible worlds) or necessary (occurring in all such worlds). In this language, it is impossible to speak of the obligatory and permitted, proved or only supposed, good or indifferent, etc.

The desire to enrich the language of logic and expand its expressive possibilities led to the emergence of modal logic.

Even Aristotle began the study of such the most common modal concepts such as “necessary,” “possible,” “accidental.” In the Middle Ages, the range of modalities was significantly expanded, and it also included "knows", "believes", "was", "will", "necessarily", "is allowed", etc. In the last century, when modal logic began to develop especially rapidly, modal concepts were attributed to “provable”, “refutable”, “better”, “worse”, “indifferent”, “convinced”, “doubts”, “rejects” and etc.

In principle, the number of groups of modal concepts and the points of view they express is not limited. Modern logic identifies the most important of these groups and makes them the subject of a special study. It also studies the general principles of modal evaluation, which are valid for all groups of modal concepts.

Thus, modal concepts, or modalities, serve to evaluate statements given from one point of view or another.

All statements containing at least one of the modal concepts are considered modal expressions.

Modal concepts, as it is easy to see, are very different in their content. It would seem that there is a common between the concepts of "provable" and "good", "convinced" and "necessarily", etc.?

Common to all modal concepts is the role that they play in utterances. With the help of these concepts the connection fixed in the statement is concretized, its character is clarified.

Take, for example, the statement "The criminal always leaves traces." It allows a double specification: quantitative and qualitative. You can use some of the words "all", "some", "most", etc. and clarify whether all the criminals in question, or only some. This will be a quantitative specification of the statement. You can also try to clarify the qualitative nature of the connection established in it. For this, modal concepts are used. The result of their application will be the statements: “It is assumed that criminals always leave traces”, “It is proved that criminals always leave traces”, “It is possible that criminals always leave traces”, etc.

Modal concepts are concepts that specify the qualitative nature of the relationship established in the statement.

Each of the groups of modal concepts gives a characteristic of the connection established in the utterance from a certain unified point of view.

Thus, the concepts of “provable”, “refutable” and “insoluble” are used for a theoretical-informative concretization of statements; for a normative one — the concepts of “mandatory”, “allowed” and “prohibited”; “Indifferent” or the notions of “better”, “worse” and “equivalently”, etc.

Modal logic explores the most interesting of the groups of modal concepts and then extends the results to their other groups.

We emphasize once again that modal concepts of different groups perform the same function: they clarify the connection established in the utterance, concretize it. The rules for their use are determined primarily by this function and do not depend on the content of the statements that include them. Therefore, these rules are similar for all groups of concepts and have a formal character. In terms of their logical characteristics, the concept of “necessarily” is not much different from the concept of “convinced”, “normatively indifferent” - from “accidentally”, etc.