I will consider two instructive thought experiments. First, suppose there are people who have the following psychological deficiencies: (a) they have no concept of the natural powers or dispositions of objects, just concepts of manifest properties; and (b) they have no natural tendency, innate or acquired, to expect certain effects given the regularities in their past experience. All they have is an ability to register particular observable facts and remember what they have registered. For example, they have observed individual instances of the sun rising many times, but they cannot form the thought that the sun has the power or disposition to rise every day; nor do they have a natural psychological tendency to expect that the sun will rise when the hour comes. They do not think in terms of powers and they have no natural instinct to project into the future what they have experienced in the past; but they do observe individual instances of what are in fact broader laws of nature.
I suggest that in this state of psychological impoverishment they will not make inductive inferences from what they have observed. They will view what they have observed as simply a set of particular facts with no implications for what they have not observed. They will believe, for instance, that the sun sometimes rises or has risen such and such a number of times, but they will not believe that it always will rise or even that it is probable that it will. They will not see their limited experience as pointing to anything beyond that experience—to a generalization of law. This is because they lack the concepts or psychological habits that can ground such inferences. They won’t have the thought that the objects they observe have the power to bring about certain effects in virtue of their intrinsic nature, and they won’t find themselves instinctively expecting the repetition of what they have observed in the past. Nature will seem to them to consist of individual facts that point to nothing beyond themselves. They will not believe in general statements based on an incomplete set of particular statements. They will perform no inductive inferences.
If this is right, then the traditional model of our reasoning about the future and the unobserved must be wrong. That model has it that we form our general beliefs by observing particular cases and then extrapolating—from the particular to the general, from some to all. Critics of induction have doubted that such an inference can be regarded as valid, and it would seem that our imaginary people would agree: they have no tendency to generalize beyond the particular. There is nothing in the particular, considered by itself, that could warrant such an inference. Insofar as any generalization is possible, it must be grounded in something like the concept of natural power or a brute tendency to form expectations. Individual instances are impotent to produce knowledge that goes beyond them without non-trivial supplementation. Remove the supplementation and the inference grinds to a halt. So it cannot be that induction rests upon an inference from some to all.
Now consider beings with the following unusual psychology: they have no memory of the past but they do have a faculty of precognition.  They can tell what will happen but they have no knowledge of what hashappened. Suppose they wish to fill this gap in their knowledge by making inferences to the past from the future–for example, they want to know whether the sun rose in the past. They consult their precognitive faculty and ascertain that the sun always rises in the future; then they extrapolate backwards to infer that it rose in the past. Thus they reason from the premise, “The sun will rise” to the conclusion, “The sun rose”. Skeptics will point out that the latter doesn’t follow from the former—the sun might not have risen in the past despite rising in the future. Many things will be true in the future that were not true in the past, so how can we infer the past from the future? These imaginary beings are just like us from an evidential point of view; they merely reverse the temporal direction of our inductive inferences. They proceed from knowledge of particular cases to generalizations about the unobserved, moving from the future to the past. So the problem of induction has nothing essentially to do with reasoning from the past to the future: it applies equally to reasoning from the future to the past. It is temporally symmetric and faculty-relative.
We can even stipulate that our imaginary beings lack the power of perception, so they have a problem knowing what is true at the present time. They can know the future by precognition, but they can’t know the present by observing it with their senses (or the past by remembering it). Wishing to remedy this lack they turn to the future for guidance: they infer that now such and such facts obtain because in the future they will obtain—for instance, objects presently fall to earth and bread presently nourishes. They can’t know these things by perceiving them, but they can at least infer them by working backwards from the future. It seems that they are no worse off (and no better off) from an evidential point of view than we are: they perceive the future and use it to infer the past and present, while we perceive the present and remember the past in order to infer the future. In both cases there is a leap from the known to the unknown of essentially the same type—extrapolation from knowledge of particular facts to beliefs about facts extending beyond that knowledge. Thus there can be a problem of induction with regard to the present: for how can the imaginary beings know about the present on the basis of their knowledge of the future? What counts as inferential depends on the cognitive faculties the knower happens to possess. The imaginary beings may harbor a Hume who torments them with the question of whether they know anything about the present, given that they have direct access only to the future (and similarly for the past).
If we humans had precognition, we would have no problem of induction with regard to the future: we would not need to base our beliefs about the future on our experience of the past, and so would not need to rely on concepts like natural power or fall back on brute inductive instinct. There would be no problem of principle about predicting the future. The problem of induction, as it exists for us, arises from a temporal asymmetry in human knowledge, but that asymmetry could be reversed in conceivable cases.
 I won’t go into the question of whether precognition is logically possible, given the direction of causality. It seems to me hard to rule out on conceptual grounds (especially if we help ourselves to a bit of divine assistance), but it suffices that we can intelligibly raise the possibility and consider its philosophical consequences.