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Both conjunctive
and disjunctive hypotheses are shown. These are described with
the English names of the values on each dimension to simplify
reading and understanding each of these hypotheses.
Note that 26
possible hypotheses are shown here. Recall that there are only
16 possible instances of the concept. Many other hypotheses are
possible. For example, Black and Large or Circle; Black
or White and Circle; ... Clearly, the number of hypotheses
about a concept can be larger than the number of instances of
the concept! And, if you think back to your own experience with
this problem, you will undoubtedly note that you did not entertain
many hypotheses at all.
The number
of possible hypotheses can become large when our concept language
allows for a great many different expressions to be constructed
in the concept language. If "And" or conjunction
were the only logical connective allowed then only 15 different
hypotheses would be possible. Clearly, if we possess a bias for
conjunctive concepts, then this vastly reduces the number of
hypotheses that we would initially consider. It also increases
the likelihood that we may develop an hypotheses that is inconsistent
with the training set that we have seen.
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