More on Induction

Using the argument in the previous post, we could argue that the probability that “every human being is less than 10 feet tall” must increase every time we see another human being less than 10 feet tall, since the probability of this evidence (“the next human being I see will be less than 10 feet tall”), given the hypothesis, is 100%.

On the other hand, if tomorrow we come upon a human being 9 feet 11 inches tall, in reality our subjective probability that there is a 10 foot tall human being will increase, not decrease. So is there something wrong with the math here? Or with our intuitions?

In fact, the problem is neither with the math nor with the intuitions. Given that every human being is less than 10 feet tall, the probability that “the next human being I see will be less than 10 feet tall” is indeed 100%, but the probability that “there is a human being 9 feet 11 inches tall” is definitely not 100%, but much lower. So the math here updates on a single aspect of our evidence, while our intuition is taking more of the evidence into account.

But this math seems to work because we are trying to induce a universal which includes the evidence: if every human being is less than 10 feet tall, so is each individual. Suppose instead we try to go from one particular to another: I see a black crow today. Does it become more probable that a crow I see tomorrow will also be black? We know from the above reasoning that it becomes more probable that all crows are black, and one might suppose that it therefore follows that it becomes more probable that the next crow I see will be black. But this does not follow, since this would be attempting to apply transitivity to evidence. The probability of “I see a black crow today”, given that “I see a black crow tomorrow,” is certainly not 100%, and so the probability of seeing a black crow tomorrow, given that I see one today, may increase or decrease depending on our prior probability distribution – no necessary conclusion can be drawn.

On the other hand, we would not want in any case to draw such a necessary conclusion: even in practice we don’t always update our estimate in the same direction in such cases. If we know there is only one white marble in a bucket, and many black ones, then when we draw the white marble, we become very sure the next draw will not be white. Note however that this depends on knowing something about the contents of the bucket, namely that there is only one white marble. If we are completely ignorant about the contents of the bucket, then we form universal hypotheses about the contents based on the draws we have seen. And such hypotheses do indeed increase in probability when they are confirmed, as was shown in the previous post.

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