Ugly duckling theorem

The ugly duckling theorem is an argument showing that classification is not really possible without some sort of bias.

The theorem is named after Hans Christian Andersen's 1843 story "The Ugly Duckling", because it shows that a duckling is just as similar to a swan as two swans are to each other. It was derived by Satosi Watanabe, in 1969.^{[1]: 376–377}

Example

Trait	Tennis ball	Basketball	Football
Small	Yes	No	No
Brown	No	Yes	Yes
Round	Yes	Yes	No

Suppose we have three objects: a tennis ball, an American football, and a basketball. These three balls have three "obvious" properties: their size, colour, and shape.

• The tennis ball is small, not brown, and round.
• The American football is not small, brown, and not round.
• The basketball is not small, brown, and round.

According to these properties, the "ugly duckling" is the tennis ball, because the other two objects share the categories "brown" and "not small".

The ugly duckling theorem relies on the ability to find additional properties from those which are provided.

Trait	Tennis ball	Basketball	Football
Small and brown	No	No	No
Small	Yes	No	No
Brown and round	No	Yes	No
Brown and not round	No	No	Yes
Round	Yes	Yes	No
Small or not round	Yes	No	Yes
Brown	No	Yes	Yes
Small or brown	Yes	Yes	Yes

The table to the right shows a selection of derived properties. In fact, these derived properties include every possible property up to equivalence: each one corresponds to one possible grouping (subset) of the original three balls.

There is no logical reason that the "original" properties are more important than the "derived" ones. Because the derived properties are found using fundamental logic, they are equivalent to the original properties. This list of all logical properties of the items has every item differ in exactly four places - when all logical properties are considered, all of the objects are just as similar as they are different.

The idea that we prioritize the three "obvious" properties exists outside the logic, so it is called arbitrary or biased. By applying the rules of logic alone, we get the result that all objects are equally different; there is no "ugly duckling".

References

1. Satosi Watanabe (1969). Knowing and Guessing: A Quantitative Study of Inference and Information. New York: Wiley. ISBN 0-471-92130-0. LCCN 68-56165.