Bertrand Russell wrote an important essay called “On Induction” in his little book The Problems of Philosophy. His treatment of induction harkens back to the skeptic David Hume, who maintained that there was no “necessary connection” between the occurrence of two events A and B, but only a “habit” of expecting that the occurrence of A, say the movement of one billiard ball, will lead to B, the movement of another ball that A contacts. Hume was talking about causality in nature.

Using careful analysis and re-exploring the problem in new language, Russell showed that Hume was right: induction could never be justified. As Russell puts it:

The problem we have to discuss is whether there is any reason for believing in what is called ‘the uniformity of nature.’ The belief in the uniformity of nature is the belief that everything that has happened or will happen is an instance of some general law to which there are no exceptions.

But can you establish that there is a “uniformity of nature?” In answering, Russell the empiricist comes out looking very much like David Hume the empiricist. There are enough examples that show that the uniformity of nature is difficult to show. Russell notes that domestic animals expect food when they see the person who usually feeds them. But the man who feeds the chicken every day throughout its life at last “wrings its neck instead, showing that more refined views as to the uniformity of nature would have been useful to the chicken.”

Also, Russell says, “Food that has a certain appearance generally has a certain taste, and it is a severe shock to our expectations when the familiar appearance is found to be associated with an unusual taste.” So you expected the green apple to taste as it did the last five times you had one or that the penicillin that cured your strep throat a dozen times will do so again. But all you are doing here is expecting that the future will resemble the past. The problem is, Russell says, that you can have experience of “past futures,” but you have not yet sampled those “future futures.”

Russell allows that the principle we have been examining is called “the principle of induction,” which runs as follows:

When a thing of a certain sort A has been found to be associated with a thing of a certain other sort B, and has never been found dissociated from a thing of the sort B, the greater the number of cases in which A and B have been associated, the greater is the probability that they will be associated in a fresh case in which one of them is known to be present.

Even though you believe that probabilities are reliable, you really don't know those either. For even if the sun has always risen and so you expect to see it rise tomorrow, what can you use for your justification? That it has always risen in the past? But you cannot assume the past in predicting the future. To do so would be “begging the question,” that is, assuming the very thing to be proven.