Abstract: Hume argues that induction (or 'experimental reasoning') cannot be justified to someone who is not already willing to accept induction, and as a result, it cannot be justified at all. In Lewis Carroll's "What the Tortoise said to Achilles" (1895), the Tortoise argues that deduction cannot be justified to someone who is not already willing to accept deduction, and as a result, it cannot be justified at all. In each case, it seems that we require some further principle in order to license the proposed form of inference. In the case of induction, Hume tells us that we need the principle that nature is uniform before we are entitled to treat past observations as a reliable guide of the unobserved. In the case of deduction, the Tortoise tells us that we need the principle that the other premises jointly entail the specified conclusion before we are entitled to draw that conclusion. But in each case, the appeal to the required further principle seems to open up the same inferential gap and the same skeptical problem all over again.

Thus far, Hume and the Tortoise might seem like natural allies. However, Hume suggests that, while induction lacks any basis in reason, demonstration—the Humean analogue of deduction—does have such a basis. Granted, humans can go astray in any process of reasoning, and ought strictly to check and re-check even their simplest demonstrations for mistakes. But considered in themselves, the rules of demonstration are rationally justified and unfailingly dependable: "In all demonstrative sciences the rules are certain and infallible," even if "when we apply them, our fallible and uncertain faculties are very apt to depart from them, and fall into error" (T 1.4.1.1).

I examine Hume's account of the relevant difference between induction and demonstration, and consider how he might reply to Carroll's 'paradox of inference.' Hume's vindication of demonstrative reasoning turns, I suggest, on his understanding of the reciprocal relationship between our powers of conception and the facts about aprioristic 'absolute' necessity. In the case of a properly formulated demonstrative argument, we cannot so much as conceive or imagine that the conclusion is false while the premises are held as true. This psychological inability serves as the basis for our talk and thought about absolute impossibility, possibility, and necessity—and indeed, for Hume, to call some proposition absolutely necessary is just a way of displaying an attitude or state of mind that flows from one's conviction that the contrary combination of ideas cannot be imagined. So a demonstrative conclusion is absolutely necessary once its premises are granted. I make the case for this interpretation of Hume on the nature of demonstration and absolute necessity. I also address the main objections to this interpretation, including the concern that conceivability and impossibility can diverge, and worries that this mind-dependent theory of modality might seem to make the facts about absolute necessity, possibility, and impossibility subject to change.