Interestingly the person who I quoted is an atheist.

Look to David Hume and Enquiry Concerning Human Understanding.  In Books 3 and 4, he explains causation and the fact that it is not extant.  He then tells about the problem with induction.  Hope it helps.

Sorry, I don't know how to delete my comments.

There are neither any logical grounds for supposing that induction does or doesn't work. While logic does not say we shouldn't believe in induction, it doesn't say that we should either. That choice is not one we make a rational decision about. 

Yes. The probability that he obtains is not a frequentist probability, although it is similar. He makes the assumption that the probability of an event happening is the same in the future as the past. From this he uses Bayes' Theorem to deduce the probability of, say, a special coin landing on heads even though it has landed on tails the previous 30 times.

For instance, look at this problem. There are two outcomes for an event, and the probability of either outcome is between 0 and 1. Now lets say we repeat the event 99 times and only one outcome (A) is observed. Do you say that the probability that the next time the event is repeated the probability of A being the outcome is 1? No, because we cannot know that. If the probability of outcome B was 0.001 for each time the event happens, it would not be that surprising at all that we've only seen outcome A. We can't say the probability is 0.01 either, as that would be a frequentist probability based on the assumption that the next outcome will be B, which is obviously very wrong. What Laplace does is use Bayes' Theorem to calculate the probability of the next event being A or B. Note that at no point is the problem of induction solved. The whole solution is based on the assumption that the probability that A will happen is the same every time the event happens.

What do you believe the justification for his rule to be? How do you believe this circumvents the problem of induction?

Argument from inductive uncertainty is not automatically a fallacy in all cases.  Inductive uncertainty is a fact of logic that we can usually ignore, but it never actually goes away. 

I'm fairly certain that the sun will rise tomorrow, but I havent come to that conclusion based on some a priori mathematical proof. 

Yeah, what LJoll said.

And like I said, the problem of induction is not a problem most of the time.  It's only a problem when one tries to claim absolute certainty from an inductive argument. 

We can't prove there is an outside world and that the sun will rise tomorrow. Hm. Here is what I would say to that person.

We can't prove either one absolutely. This is true. Just because the sun has risen (or the Earth has revolved, however you want to look at it) billions of times before does not necessarily mean it will still do so 72 hours from now.

I can't prove that the outside world is real either. Perhaps everything is just a holographic image until the exact moment that it is observed. But perhaps those observations are lies. How do we know?

To that I would say, you're right, but who really gives a shit? If you tell me that there is no way to know at any moment that the sun will continue to behave the way I expect, I could just as easily ask you to tell me what the sun would then do.

If you tell me that the outside world may not even exist, I would then ask you to explain to me what the "perceived outside world" must be.

Whatever you try to substitute in its place would surely fall victim to the same flawed logic you are accusing me of. The result is a stalemate.

We come to these kinds of inductive conclusions because we must. If I am an animal in the wild and I make the observation that all the red berries I have ever encountered are good to eat, I continue to eat the red berries. Similarly, just because I have always had a mouth and a digestion system to eat and digest berries, doesn't mean that I certainly will tomorrow, but it would be stupid of me to automatically assume, based on that knowledge, that I don't have to eat any berries. Perhaps tomorrow I will wake up as an immortal that has no desire for food, but it would be stupid for me to count on it.

None of this means that my conclusions are right or that my reasoning is valid, but I am hungry, I want to live, and I need to make a decision. I know that I exist. I know that I don't want to cease to exist.

Just because the sun will not necessarily rise tomorrow does not automatically mean that I should stop believing that it will. Just because the outside world may not actually exist or may not be what I think it is (we could all be plugged into the Matrix, after all!), does not automatically mean that I should stop believing that the world is what it is.

We accept the reality of the world we are given (or maybe tomorrow we won't! Where does it end?!?!)

But what else are we going to do?

As for the suggestion that the outside world is no more justifiable than God's existence:

If you're arguing for an unrevealed god, I may let that slide. But if you're arguing for the god of a revealed religion that supposedly created then world, then are you saying that God created a world that we can't prove actually exists, and that's how we know God exists?

Hmm... something about that doesn't work. I'll let you figure that one out.

But if you're saying "some god that we just don't know about", then okay whatever. As long as we both acknowledge that you don't know anything about him and so it's probably pointless for us to even be talking about him, if he exists, which we can't really be sure of.

*sigh* 

Logically we can't prove that the Sun will rise tomorrow, but that is only an academic point.

We can attempt to set bounds on the probability based on enumerating all the scenarios we can imagine which might prevent it, and estimating their probability from what we currently know of physics and the movement of astronomic objects in our vicinity.

We know it would require a massive amount of energy to stop the Earth from rotating, which is the most intelligible physical event that would lead to such an observation. Extremely unlikely we would survive the impact big enough to create such an effect.

Inductive reasoning is all about estimating probabilities of various alternative explanations.

Speculation about fundamental nature of the Universe changing dramatically at any future point is pointless, since all the evidence we have to date is that it hasn't happened since the earliest instants after the Big Bang. 

It is not a logical fallacy, the mistake is to think it tells us anything at all useful about anything, except that 99% of philosophy is worse than a waste of time...

All you've done is accept the point and then apply the same point to arguments that no one has even made. Well done.

I think you've misunderstood the fundamental point. You've said that we can be fairly certain that the fundamental laws of nature won't change because we've observed the fundamental laws remaining unchanged through history. This relies on the assumption the future will look like the past. How can you know that is true? Because it always has been before? That makes the same assumption, so is a circular argument.