Generalizations of the Continuous Logic

Abstract

An order logic is studied, i.e., a generalization of the continuous logic to the case in an arbitrary argument of order r is determined instead of the operations of determination of maximum (disjunction) and minimum (conjunction). The new operation is expressed as a superposition of disjunctions and conjunctions of the continuous logic. Different classes of logical determinants—numeric characteristics of matrices that can be expressed through the operations of the continuous logic—are studied. Order determinants that are the generalizations of order logic operations to arguments in matrix form are studied. Determinants with different types of constraints on matrix subsets defining the matrix characteristic are described. For logical determinants, the properties that partly resemble the properties of algebraic determinants and computation formulas based on the operations of continuous logic are described. A predicate decision algebra generalizing the continuous logic to modeling of discontinuous functions is elaborated. A hybrid logic algebra is generalized to hybrid (continuous and discrete) variables. A logical arithmetic algebra, which includes continuous logical operations along with the four arithmetical operations, is described. A complex logic algebra in which the carrier set C is a field of complex numbers is developed. For all these logical algebras, main laws are formulated and their similarity to and distinction from the laws of the continuous logic are described. Generalizations of continuous logic operations as operations over matrices and random and interval variables are investigated. Their fields of application are described.