# Mathematical Methods - Legal Informatics

## Mathematical Methods

Formalization and modeling of the processes of information collection, movement and transformation are associated with the use of mathematical methods that implement the necessary computational and logical operations, including automated information systems. Therefore, legal informatics is closely connected with mathematics and uses methods of various mathematical sciences.

In recent years, the study of information processes in the field of law uses probability theory, mathematical statistics, mathematical logic, operations research, and many other science and math disciplines. Mathematical methods, specifically refracted in legal theory, enrich and enhance the method of legal science, but, of course, do not replace it.

Today one can say that the efforts of specialists who apply the exact methods of mathematics in the legal field are concentrated in two directions: the first is the mathematical processing of the results of legal research; the second is the study of the structure of law by mathematical methods. These directions form the basis for the creation and application in the legal field of various automated systems for processing social and legal information.

The first direction was developed back in 1775 by Pierre Simon Laplace, who proposed using the methods of probability theory to evaluate testimonies, for analyzing elections and decisions of meetings and for determining the probabilities of mistakes in judicial sentences.

His followers Simeon Poisson in 1837 and Auguste Cournot in 1877 published a treatise "Probability study on the materials of criminal and civil judgments based on general rules of calculus of probability", respectively. and the monograph "Fundamentals of the theory of odds and probabilities", in which Ch. 15 was called the "Theory of Probability of Judicial Decisions." Applying it to civil affairs statistics ". In the USA, the baton of the right-wing studies was taken by the professor from Michigan J. Schubert, who in 1959 published the work "Quantitative Analysis of Judicial Conduct." In 1961, Stuart Nagel published a number of works, among which "Waiting for a verdict" contains a quantitative indicator of the ability to win or lose claims arising from harm, depending on the availability of a number of variables that are processed by the method of statistical generalizations.

Currently, in the framework of this direction, various mathematical methods are successfully used to solve the following problems: a quantitative description of legal phenomena; maintenance of the account and the reporting in legal activity by numerical processing of various statistical indicators.

The second direction is based on the idea of ​​reducing reasoning to calculations and has deep historical roots that go back to R. Descartes. He meant the possibility of creating an artificial language of science, gave it a detailed description of the enormous benefits that are associated with its application. Descartes suggested the presence of some natural order in our thoughts, which he compared with the order in the world of numbers. With all the infinite set of numbers, each of them has a single sign representation, therefore, each of them can be given a proper name, which will allow the actions with them to be recorded in a special compact language. Since for such numbers a universal language has been developed, then, according to Descartes, in time an even more universal language will be constructed that encompasses not only numbers but also any objects that may become the subject of research. Such a language will allow to denote any ideas by highlighting simple representations and fixing the elements of which each thought consists. This will exclude any possibility of error. Such language will oppose to words having an irrelevant meaning, clearly defined artificial elements. Instead of let's argue scientists will say "let's calculate".

The development of the idea of ​​a universal language of science is given much attention in the works of G. Leibniz, who laid the foundation of mathematical logic. According to Leibniz, the ideal of a general method, thanks to which it is possible to systematize eternal truths, to prove them, even to discover new ones, is as follows.

1. It is necessary to expand all concepts into simplest ones, just as in mathematics compound numbers are decomposed into a product of prime factors. The number of the simplest concepts in such a language can not be large.

2. Denoting each of the concepts with a special symbol, we get the "alphabet of human thought".

3. All sorts of combinations of simple concepts will give us a set of complex ones. And although the number of the first is small, however, as the formulas of combinatorics show, the number of their combinations can be almost inexhaustible.

4. It is necessary to enter special symbols for the basic relations between concepts and to establish the usage rules and combinations of these symbols.

Thus, the process of thinking was supposed to be reduced to a special kind of mechanical calculus, what, in essence, modern symbolic logic does.

Modern logic has created many systems that describe individual fragments of meaningful reasoning. To model the structure of legal norms, specially developed "normative logic", the subject of which is the logical structure and logical connections of normative statements.

Thus, assessing the principles of logical modeling of the structure of legal norms, legal relations and normative conclusions, V. Knapp and A. Gerloch point out that the underlying classification of legal norms is a simplified abstraction of valid legal norms that are complex in nature. For example, by examining the comparability and compatibility of legal concepts, these authors come to the conclusion that the incompatibility of the concepts "hereditary right" and suffrage It can not be proved by logical reasoning in the framework of any of the logical theories, since the presence of a common sign "right" makes these concepts formally comparable. To prove the incomparability of these concepts, in the opinion of the authors, one can not do without the apparatus of the theory of law.

However, it should be noted that the use of the language of mathematics for the formalization of law is substantially limited. This is largely determined by the fact that, as A.G. Olshanetsky admits, "among the choristers there is not yet a single opinion on the logical nature, the logical specifics of legal concepts, their constructive role in the development of the science of jurisprudence, in the formation of the regulatory legal determinant, its logical movement in the regulative mechanism of social systems. The opinions of scientists in this regard are ambiguous, controversial, sometimes contradictory. In particular, the opinion is expressed that only certain concepts of criminal law possess certain logic specificity. In the concepts of other branches of law, the specific legal is either insignificant or there is none at all ... They have only peculiarities of a nonlegal nature. In the structure of ... their content, in the character of the characteristics forming it, there are no features that would make it possible to distinguish these concepts in a special class of scientific concepts. "

According to OA Gavrilov, there are five main reasons why mathematics can not become a universal tool for research in the field of law.

1. With the increasing complexity and integrity of the socio-legal object, the possibility of its dismemberment into formalizable elements is significantly reduced.

2. The main categories of social sciences are complex, multifaceted and multifaceted concepts connected by a multitude of informal ties, such as the basis, superstructure, productive forces, production relations, state, law, economics, politics, democracy.

3. The state and law as phenomena of a class society are integral social and political systems. They are characterized by a large number of qualitative features and connections that are neither quantitative, nor probabilistic, nor functional (in the mathematical sense of the word) and therefore can not be mathematically formalized.

4. Carrying out a comparative analysis of mathematical methods and traditional tools of legal science, one can not fail to see their complementary opposition.

5. A distinctive feature of the studies performed on the basis of traditional qualitative methods is their versatility and diversity, the flexibility of the coverage of phenomena.

A distinctive feature of mathematical research is their high accuracy. Applying the traditional techniques of legal science, the legal researcher gets a prize in completeness of the picture, but loses all the accuracy. And vice versa, applying quantitative methods of research, he wins in accuracy of the scientific description, but loses in its flexibility and comprehensiveness.

It should be noted that not all lawyers adhere to this point of view. Thus, VP Pavlov, investigating the possibility of mathematizing legal research, does not agree with the above point of view of OA Gavrilov.

In his opinion, the history of any science indicates that at the initial level of cognition, on which the accumulation of scientific facts about the observed properties of the phenomena under study and empirical regularities (in the form of trends in the phenomenon of interest to us in practical life) , experiments, measurements, descriptions, methods of generalizing the comparison of analysis and synthesis, classification and systematization. To implement these methods, traditional general scientific methods, such as the philosophical method of comparative jurisprudence, and the method of comprehensive research are widely used in jurisprudence. However, a truly theoretical level is achieved in the event that scientific hypotheses are put forward, laws are formulated and theories are created. This level corresponds to various methods for explaining specific phenomena, among which hypothetical, structural, functional, abstraction methods can be distinguished, including the idealization and generalization of certain concepts, and the method of substantiating hypotheses and constructing theories. This level is achievable only by attracting mathematics as the most universal instrument for analyzing the material world. The dialectical connection of these two levels lies in the fact that the establishment of empirical facts as the initial stage of cognition is always carried out on the basis of certain theoretical knowledge of the previous level, and the empirical facts themselves are the basis for raising the level of theoretical knowledge in the investigated field. Therefore, the complementary connection between traditional and mathematical methods is not in their opposite, but just in the fact that the universality of mathematical methods makes it possible to provide visibility, accuracy and completeness of the phenomenon under investigation. Thanks to this, the field for comprehending by means of traditional means those areas of the phenomenon being investigated that were hidden from the observer by the fragmentary nature of the empirical picture of the phenomenon is expanding.

Thus, the main obstacle to the mathematical description of legal norms is the ambiguity of the conceptual apparatus of legal science, which multiplies with uncritical use of mathematical tools for its analysis. The contradiction is that without the use of a mathematical apparatus it is impossible to ensure the completeness and accuracy of legal research, and the application of a mathematical apparatus is impossible in the conditions of the existing ambiguity of the conceptual apparatus of law.

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