(opinion essay)

Paradoxes have occupied scientists and amateurs since ancient times, inflaming the imagination and causing ongoing controversy. Some of them only seem paradoxical, since the answers to them are contrary to common sense; others have not yet been resolved or cannot be resolved in principle.

Boy or girl?

Suppose there are two children in a family, and one of them is a boy. Assuming a one-half chance of having a boy, what are the chances that the second child will also be male?

Intuitively, the answer suggests itself: 50%. However, in reality, the chances are one-third. There are three possibilities in total: older brother and younger sister, older sister and younger brother, and older brother and younger brother. All three possibilities are equally likely, so the chances of each of them are one-third.

However, this answer causes fierce debate among mathematicians. Critics believe that it is actually impossible to find an unambiguous solution to the problem if it is not known exactly how information about this family was obtained.

Hempel’s paradox

The paradox proposed by the German mathematician Karl Hempel in the 1940s is also known as the “Raven Paradox”.

It begins with the statement: “All ravens are black.” This sentence is logically equivalent to the theory: “All non-black objects are not ravens.”

Each time an observer sees a black crow, the first sentence receives empirical confirmation. When he sees a non-black object, such as a green apple, he receives confirmation of the second statement.

The paradox arises from the equivalence of the two theories. In fact, seeing a green apple gives us empirical evidence that all crows are black. However, this conclusion contradicts our feelings.

Observations of non-black objects can increase our confidence that such objects are not ravens, but we do not get additional evidence that all ravens are black.