The Ugly Duckling theorem is an argument asserting that classification is impossible without some sort of bias. It is named for Hans Christian Andersen's story "The Ugly Duckling." It gets its name because it shows that, all things being equal, an ugly duckling is just as similar to a swan as two swans are to each other, although it is only a theorem in a very informal sense. It was proposed by Satosi Watanabe in 1969.
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Basic idea
Watanabe came to realize there is an unquantifiable number of shared properties between all objects, making any classification biased. Murphy and Medin (1985) give an example of two putative classified things, plums and lawnmowers:
"Suppose that one is to list the attributes that plums and lawnmowers have in common in order to judge their similarity. It is easy to see that the list could be infinite: Both weigh less than 10,000 kg (and less than 10,001 kg), both did not exist 10,000,000 years ago (and 10,000,001 years ago), both cannot hear well, both can be dropped, both take up space, and so on. Likewise, the list of differences could be infinite… any two entities can be arbitrarily similar or dissimilar by changing the criterion of what counts as a relevant attribute."
Unless some properties are considered more salient, or ‘weighted’ more important than others, everything will appear equally similar, hence Watanabe (1986) wrote: “any objects, in so far as they are distinguishable, are equally similar". However, since there is an unlimited number of properties to choose from, it remains an arbitrary choice what properties to select/deselect. This makes classification biased. Watanabe named this the "Ugly Duckling theorem" because a swan is as similar to a duckling as to another swan (there are no constraints or fixes on what constitutes similarity).
Mathematical formula
Suppose there are n things in the universe, and one wants to put them into classes or categories. One has no preconceived ideas or biases about what sorts of categories are "natural" or "normal" and what are not. So one has to consider all the possible classes that could be, all the possible ways of making sets out of the n objects. There are
As all possible choices of zeros and ones are there, any two bit-positions will agree exactly half the time. One may pick two elements and reorder the bits so they are the first two, and imagine the numbers sorted lexicographically. The first
Boolean functions
Let
However, the choice of boolean features to consider could have been somewhat arbitrary. Perhaps there were features derivable from the original features that were important for identifying the ugly duckling. The set of booleans in the vector can be extended with new features computed as boolean functions of the
Proof. Let x and y be two vectors. If they are the same, then their completed vectors must also be the same because any Boolean function of x will agree with the same Boolean function of y. If x and y are different, then there exists a coordinate
Solutions
A solution to the Ugly Ducking Theorem would be to introduce a constraint on how similarity is measured by limiting the properties involved in classification, say between A and B. However Medin et al. (1993) point out that this does not actually resolve the arbitrariness or bias problem since in what respects A is similar to B: “varies with the stimulus context and task, so that there is no unique answer, to the question of how similar is one object to another”. For example, "a barberpole and a zebra would be more similar than a horse and a zebra if the feature striped had sufficient weight. Of course, if these feature weights were fixed, then these similarity relations would be constrained". Yet the property "striped" as a weight 'fix' or constraint is arbitrary itself, meaning: "unless one can specify such criteria, then the claim that categorization is based on attribute matching is almost entirely vacuous".
Stamos (2003) has attempted to solve the Ugly Ducking Theorem by showing some judgments of overall similarity are non-arbitrary in the sense they are useful:
"Presumably, people's perceptual and conceptual processes have evolved that information that matters to human needs and goals can be roughly approximated by a similarity heuristic... If you are in the jungle and you see a tiger but you decide not to stereotype (perhaps because you believe that similarity is a false friend), then you will probably be eaten. In other words, in the biological world stereotyping based on veridical judgments of overall similarity statistically results in greater survival and reproductive success."