Soft Computing Evaluation Logic

2.1Graded Logic as a Generalization of Classical Boolean Logic

This chapter introduces graded logic as a system of realistic models of observable properties of human aggregation reasoning. After an introduction to aggregation as the fundamental activity in evaluation logic, we discuss the relationships between graded evaluation logic and fuzzy logic. Then, we present a survey of classical bivalent Boolean logic and introduce evaluation logic as a weighted compensative generalization of the classical Boolean logic. At the end of this chapter, we present a brief history of graded logic.

Three of the most frequent words in this book are means, logic, and aggregators. If we have n real numbers , then according to common sense the mean value of these numbers is a value M(x₁, …, x_n), which is located somewhere between the smallest and the largest of the numbers:

(2.1.1)

This property of function M is called internality. In our case, x₁, …, x_n are degrees of truth, and they belong to the unit interval . So, and . We can also rewrite relation (2.1.1) as follows:

(2.1.2)

Therefore, means ...