Continuous logic – I. Basic concepts

A general description of a continuous (‐valued) logic is given, basic operations of the logic are defined, and some problems and particulars of their solutions are discussed. First, we define algebra of continuous logic and enumerate its basic unary, binary and ternary functions. All laws of continuous logic are compared with laws of discrete binary logic. We discuss how to enumerate all functions of continuous logic with specified number of variables and how to represent the functions in a standard form. Procedures of minimization of continuous logical functions and their decomposition into the functions with less clarity are exploited. The procedures are compared with their counterparts from binary logic. We also tackle problems of analysis and synthesis of continuous logical functions, and show that the problem of synthesis may not have a solution. Basics of differential and integral calculus are applied to continuous valued logic. We demonstrate that any continuous logical function has the points where a derivative does not exist. At the end of the paper we briefly discuss an incompleteness problem of continuous logic, application of continuous logic in mathematics, engineering and economy, give examples, draw a perspective of further development and supply extensive bibliography of Russian works in the field.