# Transformations on Universe of Discourse and Rules in Error Systems

Chapter
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 275)

## Abstract

This chapter chiefly introduces the transformations on universe of discourse and rules for judging error in error systems. The universe of discourse (domain) here states the permitted range of objects of interest. In our research regarding error logic, error set, and error system, there are several types of domains: (1) the universe of discourse for error set C = $$\{$$(u, x)$$\mid$$ x $$=$$ f(G, u), u $$\in$$ U, f $$\subseteq$$ U $$\times$$ $$\mathbb {R}$$ $$\}$$ is $$U_C$$; (2) the range defined by error system S $$=$$ S(U(t), JG(t), Z(t), YS(t), G(t)) is called system ’s domain; (3) in error system S, the domain for structure JG(t) is $$U_{jg}$$; (4) in error system S, the domain for subsystems Z(t) is $$U_z$$; (5) in error system S, the domain for system elements YS(t) is $$U_{ys}$$; and (6) in error system S, the domain for rules for judging error G(t) is $$U_g$$. Universe of discourse (domain) plays a critically important role not only in the research of error set and error system but also in the process of error elimination. Many errors can be eliminated by changing the domain to which they pertain. From the definition on transformations, changes on universe of discourse and rules can lead to the changes in conclusion. For instance, regarding the addition of two numbers e.g. $$1+1=1$$, under rules and corresponding domain in the field of binary operation $$1+1=1$$ is wrong and it is correct under the rules and corresponding domain in the field of boolean algebra. With changed conclusion, it can inversely lead to the changes in rules for judging errors. In this chapter, we first investigate transformations of individual factors since there exists circular relationship between cause and effect. Then we consider the methods and laws for simultaneous transformations of multiple factors.