Hume opens this section by drawing a distinction between "relations of ideas" and "matters of fact." Relations of ideas are a priori and indestructible bonds created between ideas. All logically true statements such as "5 + 7 = 12" and "all bachelors are unmarried" are relations of ideas. Relations of ideas are intuitively or demonstrably certain, and a denial of such a proposition implies a contradiction.
Matters of fact deal with experience: that the sun is shining, that yesterday I went for a walk, or that it will rain tomorrow are all matters of fact. They are learned a posteriori, and can be denied without fear of contradiction. If it is sunny outside and I assert that it is raining, I can only be proven wrong by looking out the window and checking: my assertion cannot be disproved simply by an appeal to logic and reason.
While I may know many matters of fact from sensory experience or from memory, neither is the source of my knowledge that my friend is in France or that the sun will rise tomorrow. Hume suggests that we know matters of fact about unobserved things through a process of cause and effect. My knowledge that my friend is in France might have been caused by a letter to that effect, and my knowledge that the sun will rise tomorrow is inferred from past experience, which tells me that the sun has risen every day in the past.
Hume then asks how we know the principle of cause and effect: if I see one billiard ball rolling toward another, how do I know that the second ball will move when it is struck? He suggests that this knowledge cannot be a priori, since I can deny that the second billiard ball will move without contradiction. Cause and effect are themselves totally distinct: nothing in the movement of the first billiard ball can a priori suggest to me the movement of the second billiard ball. Hume thus concludes that our knowledge of cause and effect must be based on experience. From observed phenomena in the past we infer as yet unobserved phenomena in the future.
We base our knowledge of future events in past experience, but how do we know that the past is a good guide for future predictions? Hume distinguishes between "demonstrative reasoning," which is based on relations of ideas, and "moral reasoning," which is based on matters of fact. We cannot know that the future will resemble the past by means of demonstrative reasoning, since there is no contradiction in suggesting that the future will not resemble the past. Moral reasoning is also unhelpful, since it falls into a vicious circle. If all our predictions about the future are based on this principle--that the future will resemble the past--and that principle is derived from past experience, we cannot know that it will remain true in the future except by assuming that principle from the outset.
Hume suggests that we infer similarities between past and future but that there is no form of reasoning that can confirm these inferences. He confesses that he may simply have failed to identify an argument that could give a rational foundation for causal reasoning, but he challenges the reader to identify it. Even a child knows from past experience that a flame will burn. If this knowledge comes from some form of reasoning, it must be a form of reasoning so obvious that even a child can grasp it. Why, then, Hume asks, is it so difficult to identify? He suggests that the child learns, not through reasoning, but through the conditioning of custom.