You may have heard scientists sometimes refer to 'the laws of nature.' The problem of induction is that we never directly experience these 'laws': all we seem to experience is the constant conjunction of certain circumstances with certain events. To talk about a cause-effect relationship between two events, however, seems to require more than just a mere conjunction (which, for all we know, might be entirely coincidental and fortuitous). In short, part of the conceptual problem is that science is supposed to be an empirical enterprise, grounded in experience. But experience is always of particular events. So if our experience is only of particular events, and never of the laws of nature themselves, how can we rationally engage in any meaningful discourse about general laws and principles? This, incidentally, is one of the reasons why scientists sometimes don't like philosophers :)
But here's where John Stuart Mill comes in: he developed, classified and formalized a set of methods for testing causal hypotheses (thereby going beyond mere generalizations from few to many or from past to future). As in all cases of induction, though we may never be able to develop a proof based on logical necessity (the way mathematicians and logicians are used to), science can nevertheless make progress by testing hypotheses: any hypothesis that survives the day gets to make it to the next round. Those that are eliminated are discarded and no more neural energy is wasted on them (ideally).
Anyway, the following is a slideshow presentation I've been working on, summarizing these methods. It's a bit heavy on text to help anyone who may want to use it without having to sit in one of my classes and hear me yap about these ideas :)
And if you want a short and thought-provoking introduction to Utilitarianism, check out Harvard Professor Michael Sandel's fascinating lectures on Jeremy Bentham and John Stuart Mill.
Disclaimer: The slideshow is based on the eleventh edition of Copi and Cohen's "Introduction to Logic," so the credit should really go to them.You may also be interested in my slideshow presentation on logical fallacies.