The Concept Language is shown in the box in the upper left and consists of four dimensions or properties each of which can take on one of two values. The exemplar is shown again on the right together with a description of the exemplar in the concept language.

     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.

     We can actually set a bound on the number of possible concepts when we know the number of possible concept instances. Recall that in this case the number of possible instances is 2 to the 4 or 16. These 16 possible instances can be grouped in 2 to the 16 or 65, 536 ways. This is what is referred to as the power set. The power set is the set of all possible subsets of a set. (You may remember that we referred to the power set when considering Wertheimer's discussion of perceptual grouping.) The power set includes the empty set as well as the set that includes all items. These two bounding sets can be excluded as possible interesting concepts. Thus, we can reduce the space of possible hypoptheses from 65,536 to 65,534!!