But here the reasoning is really this:
Rule.—Every quadrangle is other than a triangle.
Case.—Some figures are quadrangles.
Result.—Some figures are not triangles.
Inductive or synthetic reasoning, being something more than the mere application of a general rule to a particular case, can never be reduced to this form.
If, from a bag of beans of which we know that 23 are white, we take one at random, it is a deductive inference that this bean is probably white, the probability being 23. We have, in effect, the following syllogism:
Rule.—The beans in this bag are 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 23 of the time.
If instead of drawing one bean we draw a handful at random and conclude that about 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 23 of the beans in the handful white, conclude that about 23 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.
But this is not the only way of inverting a deductive syllogism so as to produce a synthetic inference. Suppose I enter a room and there find a number of bags, containing different kinds of beans. On the table there is a handful of white beans; and, after some searching, I find one of the bags contains white beans only. I at once infer as a
