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.
Hume's distinction between relations of ideas and matters of fact is one of the first formulations of a distinction that has been instrumental in philosophy ever since. Kant made the distinction famous, calling relations of ideas "analytic" and matters of fact "synthetic." Ever since, and particularly in the development of analytic philosophy in the twentieth century, the analytic/synthetic distinction has been a hot topic of debate.
It might not be clear what Hume means when he says it would be a contradiction to deny relations of ideas, but not matters of fact. Surely, there is something contradictory about saying "it's raining" when the sun is shining brightly. The point is that we need to refer to the world around us to verify matters of fact. The claim that two plus two equals five is a contradiction because nothing in our experience can possibly prove it true. The claim "it's raining" might have been true under other circumstances, and the claim must be compared with reality in order to be proven false.
We can know relations of ideas quite easily by means of what Hume calls demonstrative reasoning. There are well-established axioms and rules of inference according to which I can derive mathematical and other logical truths. Similarly, there are well-established means of knowing observable matters of fact. For instance, my claim that it's raining can be verified by stepping outside or looking out the window. However, Hume notes that unobserved matters of fact are more difficult to sort out. I know that the sun will rise tomorrow, but how? I won't be able to verify that claim directly until tomorrow, but I can still claim to know it with certainty today.
Hume suggests that we use the principle of cause and effect to reason through matters of fact. The principle of cause and effect, he suggests, we learn from experience. The question, then, is how we can ground general principles that we learn from experience. This question cuts right to the very heart of the inductive reasoning that is central both to the scientific method and to Hume's philosophy. All our general principles in philosophy and science are induced from particular examples. Induction essentially consists in observing and predicting the future based on what we have observed in the past. We are absolutely certain that the second billiard ball will move when it is struck, not through demonstrative reasoning, but because we have seen bodies collide in that way countless times during our lives and have never seen one instance to the contrary.
For induction to be a valid form of reasoning, we need to propose some sort of "uniformity principle" that establishes that the future will resemble the past. It may seem obvious that physical laws will not change in the future, but Hume's genius lies precisely in seeing that this is still a claim that needs to be proved and argued for. To his surprise, he finds there is no good reason to trust in any sort of uniformity principle. It cannot be established through reason alone, since its denial is hardly contradictory. It would seem that we learn of this principle through experience, but we cannot claim that it is confirmed in experience. A uniformity principle is needed to justify all inductive claims based on past experience, so we cannot prove the uniformity principle itself through induction. We need to prove the uniformity principle before we can say anything about induction or knowledge from experience, but it seems that we cannot prove the uniformity principle without an appeal to experience. This circularity could be schematized as follows:
Rather than try to hedge at this point, Hume bites the bullet and accepts the consequences of his reasoning: there is no way we can prove any kind of uniformity principle, and so induction is not a valid form of reasoning. Any reasoning about future events is mere conjecture and the claim that the sun will rise tomorrow is no more certain than the claim that aliens will invade the earth tomorrow. Hume is not necessarily claiming that there is no uniformity principle or that there is a good chance that the sun will not rise tomorrow. He is saying that if there is some hidden power that enforces a continued regularity in physical laws, it is beyond the power of our reason to detect it. Our belief in induction is not based in reason but simply in custom. Past experience has led us to believe certain things about future events (and indeed, this experience rarely leads us astray) but these beliefs are not rationally justified. Hume's argument is that we are committed to the belief that the future will resemble the past, but that we are not rationally justified in holding this belief. Reason is a far weaker tool than we might have supposed.