The use of non-finite methods in the study of the relationship forms a logical calculus based on the evaluation

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A. V. Titov

Abstract

Our approach to studying the relationship of various types of logical calculus is based on a study of evaluation as a morphism that preserves the structure of the algebra of formulas in the structure of the estimated values.

The use of non-classical logic in mathematics is currently limited. However, ever-growing and changing requirements for the mathematical apparatus used in formal models of complex objects and processes may significantly change this situation and lead to the development of mathematical theories based on the use of various types of non-classical logic.

Investigation of the interrelation between different types of logical calculus on the basis of evaluation is associated with the attraction of non-finite methods of structure theory, to which one can associate the methods of generalized non-standard analysis as a section of category theory.

Development of the approach to the study of formal logic types based on the use of non-finite methods of generalized non-standard analysis allows us to consider the set of logic algebra formulas with the introduced equivalence relation as a factor - algebra with a certain structure.

The use of methods applying modern mathematical theories allows us to reveal the mathematical structure of formal logic and to trace the relationship of different types of logical calculus, in other words, to identify the mathematical content of the considered type of logical calculus.

The validity of the use of non-finite methods in logical research is due to the fact that metamathematics is a theory that studies formalized mathematical theories. Formalized theory is a set of finite sequences of characters (formulas and terms) and a set of operations on these sequences. Operations replace elementary steps of deduction in mathematical reasoning. In this statement, mathematical logic (metamathematics) itself becomes a branch of mathematics. That is, the logic itself in such a statement becomes the subject of mathematical research.

This approach allows us to consider formal logic as a dynamic system, the development of which consists in the disclosure of a system of particular types of logical calculus, for the description of which it is proposed to use non-finite methods of generalized non-standard analysis. DOI: 10.21146/2074-1472-2018-24-2-129-136

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References

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