David Hume, a Scottish thinker of the commercial era, is the philosopher most often associated with induction. His formulation of the problem of induction can be found in An Enquiry concerning Human Understanding, §4. Here, Hume introduces his famous distinction between "relations of ideas" and "matters of fact." Relations of ideas are propositions which can be derived from deductive logic, which can be found in fields such as geometry and algebra. Matters of fact, meanwhile, are not verified through the workings of deductive logic but by experience. Specifically, matters of fact are established by making an inference about causes and effects from repeatedly observed experience. While relations of ideas are supported by reason alone, matters of fact must rely on the connection of a cause and effect through experience. Causes of effects cannot be linked through a priori reasoning, but by positing a "necessary connection" that depends on the "uniformity of nature."

Hume situates his introduction the the problem of induction in A Treatise of Human Nature within his larger discussion on the nature of causes and effects (Book I, Part III, Section VI). He writes that reasoning alone cannot establish the grounds of causation. Instead, the human mind imputes causation to phenomena after repeatedly observing a connection between two objects. For Hume, establishing the link between causes and effects relies not on reasoning alone, but the observation of "constant conjunction" throughout one's sensory experience. From this discussion, Hume goes onto present his formulation of the problem of induction in A Treatise of Human Nature, writing "there can be no demonstrative arguments to prove, that those instances, of which we have had no experience, resemble those, of which we have had experience."

In other words, the problem of induction can be framed in the following way: we cannot apply a conclusion about a particular set of observations to a more general set of observations. While deductive logic allows one to arrive at a conclusion with certainty, inductive logic can only provide a conclusion that is probably true. It is mistaken to frame the difference between deductive and inductive logic as one between general to specific reasoning and specific to general reasoning. This is a common misperception about the difference between inductive and deductive thinking. According to the literal standards of logic, deductive reasoning arrives at certain conclusions while inductive reasoning arrives at probable conclusions. Hume's treatment of induction helps to establish the grounds for probability, as he writes in A Treatise of Human Nature that "probability is founded on the presumption of a resemblance betwixt those objects, of which we have had experience, and those, of which we have had none" (Book I, Part III, Section VI).

Therefore, Hume establishes induction as the very grounds for attributing causation. There might be many effects which stem from a single cause. Over repeated observation, one establishes that a certain set of effects are linked to a certain set of causes. However, the future resemblance of these connections to connections observed in the past depends on induction. Induction allows one to conclude that "Effect A2" was caused by "Cause A2" because a connection between "Effect A1" and "Cause A1" was observed repeatedly in the past. Given that reason alone can not be sufficient to establish the grounds of induction, Hume implies that induction must be accomplished through imagination. One does not make an inductive reference through a priori reasoning, but through an imaginative step automatically taken by the mind.

Hume does not challenge that induction is performed by the human mind automatically, but rather hopes to show more clearly how much human inference depends on inductive -- not a priori -- reasoning. He does not deny future uses of induction, but shows that it is distinct from deductive reasoning, helps to ground causation, and wants to inquire more deeply into its validity. Hume offers no solution to the problem of induction himself. He prompts other thinkers and logicians to argue for the validity of induction as an ongoing dilemma for philosophy. A key issue with establishing the validity of induction is that one is tempted to use an inductive inference as a form of justification itself. This is because people commonly justify the validity of induction by pointing to the many instances in the past when induction proved to be accurate. For example, one might argue that it is valid to use inductive inference in the future because this type of reasoning has yielded accurate results in the past. However, this argument relies on an inductive premise itself -- that past observations of induction being valid will mean that future observations of induction will also be valid. Thus, many solutions to the problem of induction tend to be circular.