Problem of justifying the inductive inference from the observed to the unobserved.
It was given its classic formulation by David Hume , who noted that such inferences typically rely on the assumption that the future will resemble the past, or on the assumption that events of a certain type are necessarily connected, via a relation of causation, to events of another type. (1) If we were asked why we believe that the sun will rise tomorrow, we would say that in the past the Earth turned on its axis every 24 hours (more or less), and that there is a uniformity in nature that guarantees that such events always happen in the same way. But how do we know that nature is uniform in this sense? We might answer that, in the past, nature has always exhibited this kind of uniformity, and so it will continue to be uniform in the future. But this inference is justified only if we assume that the future must resemble the past. How do we justify this assumption? We might say that in the past, the future turned out to resemble the past, and so in the future, the future will again turn out to resemble the past. The inference is obviously circular: it succeeds only by tacitly assuming what it sets out to prove, namely that the future will resemble the past. (2) If we are asked why we believe we will feel heat when we approach a fire, we would say that fire causes heat
i.e., there is a "necessary connection" between fire and heat, such that whenever one occurs, the other must follow. But, Hume asks, what is this "necessary connection"? Do we observe it when we see the fire or feel the heat? If not, what evidence do we have that it exists? All we have is our observation, in the past, of a "constant conjunction" of instances of fire being followed by instances of heat. This observation does not show that, in the future, instances of fire will continue to be followed by instances of heat; to say that it does is to assume that the future must resemble the past. But if our observation is consistent with the possibility that fire may not be followed by heat in the future, then it cannot show that there is a necessary connection between the two that makes heat follow fire whenever fire occurs. Thus we are not justified in believing that (1) the sun will rise tomorrow or that (2) we will feel heat when we approach a fire. It is important to note that Hume did not deny that he or anyone else formed beliefs about the future on the basis of induction; he denied only that we could know with certainty that these beliefs are true. Philosophers have responded to the problem of induction in a variety of ways, though none has gained wide acceptance.