The Problem of Induction is the philosophical issue involved in deciding the place of induction in determining empirical truth. Thus, I know from direct sensations (vision, pain, ...) that you dropped a rock on my toe. Is it true, however, that a rock dropped on my toe will always cause pain? You may propose to test this by a series of dropped rocks, to which I will probably object. But even if you carried out your plan, would a series of painful swollen toes demonstrate that dropping rocks on toes always hurts? Such a conclusion is reached by what is called inductive reasoning, but the problem of induction is whether inductive reason works. That is, what is the justification for either:

  1. generalizing about the properties of an entire class of objects based on some number of observations of particular instances of that class of objects (for example, "All ravens we have seen are black, and therefore all ravens black"); or
  2. presupposing that a sequence of events in the future will occur as they always have in the past (for example, the attractive force described by Newton's Law of Gravitation, or Einstien's revision in General Relativity) is universal ("All the rocks I have released have landed on your toes, and therefore the next rock I release will also do so.")

However, any series of observations, however large, may be taken to logically imply any particular conclusion about some future event only if 'induction' itself works. And we may conclude that only inductively! So, for instance, from any series of observations that water freezes at 0C it is valid to infer that the next sample of water will do the same only if induction works. That such a prediction comes true when tried merely adds to the series; it does not extablish the reliability of induction, except inductively. The problem is, then, what justification can there be for making such an inference?

David Hume addressed this problem in the 18th century in a particularly influential way, and no analysis since has managed to evade Hume's critique. Hume looked at ways to justify inductive thinking. He pointed out that justifying induction on the grounds that it has worked in the past begs the question. That is, it is using inductive reasoning to justify induction. Circular arguments are valid, but do not provide a satisfactory justification for the supposition they claim to support. Prior to Hume, Sir Francis Bacon had made a strong claim that science was based on induction. Sir Karl Popper sought to 'bypass' the problem in the philosophy of science by arguing that science does not actually rely on induction, developing the notion of falsification instead. Popper replaced induction with deduction, in effect making modus tollens the centerpiece of his theory. On this account, when assessing a theory one should pay greater heed to data which is in disagreement with the theory than to data which is in agreement with it. Popper went further and stated that a hypothesis which does not allow of such experimental text is outside the bounds of science.

Nelson Goodman presented a different description of the problem of induction in the article "The New Problem of Induction" (1966). Goodman proposed a new colour, “grue”. Something is grue if it is green up until some given time, and blue thereafter. The “new” problem of induction is, how can we know that grass is indeed green, and not grue?