Thursday, January 09, 2003

The Logic of Love

Assume that the statement

A is B.

is true. Then the statement

"Not A" is "not B."

is called the obverse of the original statement. The obverse is not always true. Interchanging the subject and the predicate of the obverse, we get another statement

"Not B" is "not A."

This statement is called the contraposition of the original statement. The contraposition is always true.

I learned the above logic from a young lady teacher of mathematics in the first year of senior high school. So I remember it well. However, you can understand it easily by drawing a small circle enclosed in a large circle and supposing that the inside of the small circle is A and that the inside of the large circle is B.

In the previous story A Mathematician's Desire, I mentioned about a comical essay written by the mathematician Masahiko Fujiwara. I found another short essay [1] of his that treated the obverse to be quite funny. The essay is entitled: Is ' "Not A" is "not B" ' true?

Without using jargons, Professor Fujiwara teaches the reader that the obverse is false in many cases, but can be true in some cases. He does this by the use of interesting examples. Examples of the false obverse are given by the original statements of daily observation, “The tulip is beautiful," "Snow is white," and "What bothers others is what you must not do." An example of the true obverse is given by the mathematical original statement, "If the polygon is the triangle, then the sum of its inner angles is equal to 180 degrees."

Finally the mathematician gives an example of the obverse that can be decided neither true nor false. The original statement of this example is "If the woman is your wife, then you may love her." He writes:
As for the statement, "If the woman is not your wife, then you must not love her," my wife's opinion and mine are different.
I guess that Professor Fujiwara is actually a good husband as well as a good teacher.
  1. M. Fujiwara, Asahi-Shimbun, Evening Edition (17 Dec. 2002).

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