that this bean is probably white, the probability being 2/3. We have, in effect, the following syllogism:
Rule.—The beans in this bag are 2/3 white.
Case.—This bean has been drawn in such a way that in the long run the relative number of white beans so drawn would be equal to the relative number in the bag.
Result.—This bean has been drawn in such a way that in the long run it would turn out white 2/3 of the time.
If instead of drawing one bean we draw a handful at random and conclude that about 2/3 of the handful are probably white, the reasoning is of the same sort. If, however, not knowing what proportion of white beans there are in the bag, we draw a handful at random and, finding 2/3 of the beans in the handful white, conclude that about 2/3 of those in the bag are white, we are rowing up the current of deductive sequence, and are concluding a rule from the observation of a result in a certain case. This is particularly clear when all the handful turn out one color. The induction then is:
These beans were in this bag.-
These beans are white. |
All the beans in the bag were white. | |
| | |
Which is but an inversion of the deductive | | |
syllogism. | | |
| | |
Rule.—All the beans in the bag were white.—+ | |
Case.—These beans were in the bag.+-+
Result.—These beans are white.+
So that induction is the inference of the rule from the case and result.
