Here's a short essay I wrote in the last 20 minutes for my philosophy class.
When Hume claims that we have no reason to believe an inference from the observed to the unobserved, he is correct, as long as we’re talking about a certain definition of the word reason. The way he uses the word reason is analogous to whether an inference logically follows from observed data (as in, is deductively valid based on observed data). If we see that A->B has been true any number of times in the past, we cannot logically conclude that A will imply B the next time. For example, if we see the sun rise one time per 24 hour period for our entire lives, we don’t technically have enough data to conclude that it will rise in the next 24 hour period, unless we assume that patterns that have held in the past will continue to hold in the future. Unfortunately, there is no way to prove that past patterns will continue into the future without assuming exactly that. The question “Why will past patterns continue into the future?” can only be answered by claiming something like “well, all past futures have resembled past pasts”, which doesn’t really get us anywhere, or leads to infinite regress, whichever way you want to look at it.
So does this conflict with my ordinary beliefs about the world? Not at all. In fact, I wonder why Hume’s argument was not stronger. The basic conclusion to be drawn from his argument is that inductive reasoning (reasoning with logical leaps) cannot be logically (deductively) defended. Well, of course that’s true! Otherwise, we wouldn’t have any kind of reasoning other than deduction (why would we make logical leaps if we didn’t have to?). His second claim, that the only way to defend inductive reasoning is to assume the future resembles the past, and that assumption can only be defended by again assuming the future will resemble the past, to infinity, actually seems much weaker than it could be. Any deductive argument is subject to the same kind of reasoning as well. For example, take the argument, “Bachelors are unmarried, Evan is a bachelor, therefore Evan is unmarried”. How do you defend that bachelors are unmarried? How do you defend that Evan is a bachelor? Well, all you can do is say “um, you just have to assume that’s the truth”. Or you can use inductive reasoning, but Hume has already provided good reason for why that would lead to the same problem.
The truth is that all reasoning has to be based on something. There’s always something you assume to be true in any argument. In deduction, you assume the premises are true, and in induction, you assume that the data you have is valid and the future resembles the past. To use a computer science analogy, if you write a function that adds two numbers together and outputs the answer, the function doesn’t do anything unless it gets two numbers to work with. Before data is passed to it, it is useless. Does the function itself know or care whether the numbers passed into it are somehow “true”? It’s a meaningless question. Just like when you think of the argument: “A->B, A, therefore B”. Does it matter to the actual process of deduction whether A->B is true or whether A is true? No. The process itself just takes what it’s given and gives the answer. The same can be said for induction (reasoning from the observed to the unobserved). If you were to write a computer function that used induction, the first line of it would be something to the effect of “the future resembles the past”. That’s just what induction does. The question of whether or not the future actually resembles the past is irrelevant to the process itself. So Hume is basically saying “Hey, look at that function! It says the future resembles the past! Why?”. Well, the answer is, “uh…that’s just what the function does.”