ЛОГИКА МАТЕМАТИЧЕСКАЯ

Logic, Mathematical

Mathematical Logic (or symbolic logic) appeared as a result of the application of mathematical methods in the realm of formal logic, of the use of a special language of symbols and formulas. Mathematical Logic investigates logical thinking (reasoning and proof) as reflected in the systems of formal logic or calculi. Thus Mathematical Logic has for its subject-matter logic and for its method mathematics; it contains far-reaching generalizations.

Typical of the present stage of formal logic is the development of the ideas and methods of traditional formal logic. Contemporary Mathematical Logic includes a whole series of logical calculi, and is the theory of such calculi, their premises, properties, and applications. Besides its study of the formal structure of logical calculi (see Logical Syntax) Mathematical Logic also examines the relations between calculi and those substantive fields which serve as interpretations and models. This task reflects the problems of logical semantics. Logical syntax and semantics belong to metalogic, the theory of the means of describing the premises and properties of logical calculi.

The discovery of the formal investigation of logic is attributed to Aristotle (see Syllogistic). The Megarian school of stoics (3rd century B.C.) already knew some of the initial concepts of Mathematical Logic, whereas the idea of logical calculi was first formulated by Leibniz. As an independent branch of science Mathematical Logic established itself only in the mid-19th century, thanks to the works of Boole, who founded the algebra of logic. Later Ernst Schroder summed up and systematized the results of such development in his Algebra der Logik (1890–95).

Another trend in Mathematical Logic appeared at the end of the 19th century, arising from the need of mathematics to provide a foundation for its concepts and methods of proof. The sources of this trend are to be found in the works of Frege. The main contribution to its development was made by Russell and Whitehead (Principia Mathematica, 1910–13), and Hilbert. Two fundamental logical systems—the classical propositional calculus and functional calculus—were elaborated at the time.

Today Mathematical Logic investigates the various types of logical calculi and takes interest in semantical problems and metalogic in general, as well as in the problems of special scientific and technical application of logic. Alongside the studies by classical logic, constructive logic was created in order to substantiate mathematics. An analysis of the foundations of logic promoted the research into combinatory logic. The theory of many-valued logic was also created. Attempts to solve the problem of formalizing logical thinking led to the formation of the calculations of strict and material implication. The foundations of modal logic were laid as well.

At the same time Mathematical Logic exerted great influence upon contemporary mathematics itself. The essential sections of contemporary mathematics sprang up from Mathematical Logic, e.g., the theories of algorithms and recursive functions. Mathematical Logic is applied in electrical engineering (the study of relay-contacts and electronic systems), in computers (programming), in cybernetics (theory of automatic devices), in neurophysiology (simulation of the neuronic nets), and linguistics (in structural linguistics and semiotic). Such close interlacing of logical problems with the solution of special scientific problems and use of logic in concrete scientific studies were unknown to formal logic.

Математическая логика

Логика, развиваемая математическим методом. Характерным для М. л. является использование формальных языков с точным синтаксисом и чёткой семантикой, однозначно определяющими понимание формул. Потребность в такой логике выявилась в начале 20 века в связи с интенсивной разработкой оснований математики, возникновением множеств теории, где были открыты антиномии (см. Парадокс), уточнением понятия алгоритма и другими глубокими и принципиальными вопросами математической науки. Однако значение М. л. для науки в целом не исчерпывается её математическими приложениями, поскольку хорошо рассуждать и доказывать приходится во всех науках. Вот почему М. л. с полным правом может быть охарактеризована как логика на современном этапе. См. статья Логика (раздел Предмет и метод современной логики) и литературу при этой статье.

А. А. Марков.