In this chapter, I first have to introduce somebody: George Boole was born in Lincoln, Lincolnshire, England in 1815 and passed away in Ballintemple, Cork, Ireland in 1864. George Boole was the son of shoemaker and hence received only a primary school education, but he was a prodigy. George Boole taught himself Latin, some other languages, philosophy, and mathematics. As soon as he was sixteen years of age, he began teaching. He ran a boarding school, when he was nineteen years of age. In 1849, George Boole was appointed the first professor of mathematics at the Queen’s College of Cork in Ireland.

This George Boole invented what nowadays is known as Boolean algebra. It isn’t different from algebra known from mathematics classes, but merely a branch of it. Boolean algebra means to do calculations with dual numbers, which means that they are based on the number two. The dual number system is an accidental invention of the German philosopher Gottfried Wilhelm Leibniz. George Boole added the logical operators to the dual number system, so that some special calculations can be made. During my time in the German army, a comrade and roommate began to study informatics by correspondence course. He got sent a lot of material on Boolean algebra. So I know that Boolean algebra can be very abstract and complicated because it consists of applying really all theorems and methods of algebra to the dual number system. But the only true peculiarity of Boolean algebra is to interpret 1 and 0, the only digits of the dual number system, as TRUE and FALSE. These values are the only possible values of a so-called Boolean variable. Logical operations are done with these values.

What is not 1 can only be 0 in the dual number system. Expressed in the Boolean system NOT TRUE is equal to FALSE. On the other hand what is not 0 has to be 1 and this again can be expressed as NOT FALSE is equal to TRUE. NOT is a logical operator of the Boolean algebraic system. NOT turns a Boolean value into its opposite.

The other logical operators are AND, XOR, and OR. These logical operators combine the values of two Boolean variables to one result. The logical operator AND results in 1 or TRUE if both of the input variables contain the value TRUE. With other words only TRUE AND TRUE equals TRUE. Boolean algebra doesn’t accept a half-truth. So TRUE AND FALSE or FALSE AND TRUE always result in FALSE.

The logical operator OR delivers TRUE as a result if at least one of the input variables contains the value TRUE. The logical operator XOR delivers TRUE as a result if exactly one of the input variables contains the value TRUE and the other the value FALSE. The logical operators can be combined to new operators. So does NOT XOR perform a check on whether two Boolean variables contain the same value. Logical operators are needed for programming computers. Different programming languages have different notations for the logical operators, but all programming languages contain the logical operators because a programming language otherwise would be dysfunctional.