Does inductive reasoning lead to knowledge?
In inductive reasoning, we form a claim based on a set of observations. The claim (or conclusion) derives from the idea of probability. But can probability really be calculated? Hume argues not, since claiming that a result is "more probable" implies that the past predicts the future.
Let us, for example look at:
Premise: All swans we have seen are white.
Conclusion: All swans are white.
Premise: Every person who has touched fire has been burned
Conclusion: Any person who touches fire will be burned.
Sextus Empiricus described the circular logic of induction. "Those who claim for themselves to judge the truth are bound to possess a criterion of truth. This criterion, then, either is without a judge's approval or has been approved. But if it is without approval, whence comes it that it is truthworthy? For no matter of dispute is to be trusted without judging. And, if it has been approved, that which approves it, in turn, either has been approved or has not been approved, and so on ad infinitum." (Against the Logicians trans. R.G. Bury (Loeb edn) (London: W. Heinemann, 1935) p. 179)
Hume, admits, however, that inductive reasoning is a force of habit, and "without the influence of custom we would be entirely ignorant of every matter of fact beyond what is immediately present to the memory and senses." (Enquiry, §5.1) The imagined possibility of an alternative to the inductive conclusion cannot be as strong as the instinctual conclusion.
Can anything really be proven? Well, even if it can't, we have a strong basis for our certainty.