A Modern Introduction to Fuzzy Mathematics

7Fuzzy Logics

Logic is about the process of reaching an answer by thinking about known facts. And logic in general can be divided into formal logic, informal logic, symbolic logic, and mathematical logic. We know that in mathematical reasoning, a statement is either true (e.g. “3 is an odd number”) or false (e.g. “4 is greater than 5”), since nothing else is meaningful. However, outside mathematics, there are statements that are neither true or false. For example, the statement “Brazil will win the next world cup in football” is such a statement. Put simply, fuzzy logic is a mathematical logic to reason about statements that are true, false, or something in between. The discussion that follows is based mainly on [157, 217, 218].

7.1 Mathematical Logic

The word “logic” derives from the Greek word λόγος (logos), which means reason. In fact, logic is yet another ancient Greek invention. In particular, as was noted in Section 2.1, Aristotle is credited with the invention of logic. Logic was born when he begun to study the structure of arguments. Aristotle called syllogisms all these structures. An example of a syllogism is the following: John goes to the movies on Wednesday; today is Wednesday; therefore, John will go to the movies. This deduction is an instance of the general inference rule:

Assume that “” and “if then ” are true. Then, “” is also true.

Aristotle invented logic, ...