Continuous logic - II. Main generalizations

Outlines the basic results in the generalization of continuous-valued logic. The survey is based on Russian publications. We consider an order logic, which is a generalization of continuous-valued logic where operations of maximum selection (disjunction) and minimum selection (conjunction) are substituted with operation of selection of rth order argument, following the values of the arguments. Shows that this new operation is expressed in a superposition of disjunctions and conjunctions of continuous-valued logic. Various classes of logical determinants are considered; they are thought of as numerical characteristics of matrices, expressible in operations of continuous-valued logic. Namely, investigates order determinants, which generalize order logical operation of several arguments in matrix form, and determinants with various constraints on subsets of matrix elements. Properties of all logical determinants are discussed, compared with properties of algebraic determinants; techniques of computation of logical determinants are supplied. Also investigates a predicate algebra of choice, which generalizes continuous-valued logic in case of simulation of discontinuous functions; a hybrid logic of continuous and discrete variables; a logic-arithmetic algebra, which includes, in addition to continuous-logic operations, four arithmetical operations; a complex algebra of logic, where supportive set C is a field of complex numbers. A description of each algebra includes basic laws, which are compared with the laws of conventional continuous-valued logic. Several generalizations of continuous-value logic operations to operations over matrices, random and interval variables are discussed. Some applications of continuous logics are indicated.