One: The 2-ways of knowing.
Friends permit me to start with Hume’s fork .
He pontificated that there are two ways of knowing (justified propositions), 1) ‘matters of facts’, and 2) the ‘relations of ideas’. In the latter, he means logic, mathematics, proof and everything in between – all bachelors are unmarried, that kind of thing. These statements are true by definition and unfortunately there is nothing we can do about that.
For the former, ‘matters of facts’, knowledge is derived from experience.
If – after watching Vikings all night – you wake up 11am in the morning (now feeling very guilty, having wasted so much time watching Ragnar loot the west); you carefully peeped out of your window and you saw that the sun is shining, then it is the case that the sun is shining.
You cannot wish the sun not to shine (or wish the time not to be 11am). It is a brute fact about the world delivered by your experience.
There is another neat way to distinguish between these two ways of knowing. The negation of relations of ideas is impossible. For example, even if you want to, you cannot make yourself to believe (or any one for that matter that) all bachelors are not unmarried. You can try it, I will wait.
On the other hand, the negation of a matter of fact is a fair game. That is, for example, it is not impossible for the sun to not shine. The sun can shine and the sun cannot shine, there is no contradiction here.
Please, please bear this distinction in mind as we proceed to induction. It is the key to the famous problem of induction
Two: Induction, Schminduction
Let’s clear up a few more terms: inductive and deductive reasoning.
Hannibal Barca was an African (premise 1a)
All Africans are black (premise 2a)
Therefore, Hannibal Barca is black. (conclusion)
Given premise 1a, if premise 2a is true, then it is the case that, the beast himself, Hannibal Barca was black. This is an example of a deductive reasoning. 
On the other hand, here is what an inductive reasoning looks like:
All swan I have seen are white (premise 1b)
Therefore, all swans are white (Conclusion 2)
Given premise 1b, then it is not necessarily the case that the conclusion is true. In other words, conclusion 2 might be wrong. And why is that?
More technically (using Hume’s fork) the conclusion all swans are white cannot be a relation of idea, because its negation is not contradictory.
That is, it is not impossible to have a different colored swan (the reader should contrast this with the contradiction of the proposition all bachelors are not unmarried.)
To make matter worse, the conclusion all swans are white cannot be matter of fact either. If it is snowing outside now, then it is the case that it is snowing outside. That is a matter of fact. But for the ‘all swans are white’ conclusion, since we have not (and cannot) observe all swans in the world, then it cannot possibly be a matter of fact.
How then can we justify this ‘all swans are white’ proposition?
By thinking inductively: because it works, because that had been the case in the past!, or something like that.
Finally, and this is the brutal part: what proof do we have that the future will be like the past?
You might say because the past past had always been like the past future. But what you will be doing here is to use a principle (induction) to justify itself. Using a principle to justify the same principle, circularity per excellence.
After a few trials, you might want to face the brute fact: that there is no rational justification for induction.
Three: New Orleans
About a week ago, I was in this conference in New Orleans. In the hotel conference hall, we got this disturbing sound from the ceiling anytime something like a heavy object was moved around on the next floor.
The thing about the sound is that it gives you a very good impression that the building will collapse in (except that it had happened before, and the building did not collapse).
The first time it happened, there was a weird sigh in the conference hall, however subsequently, we got lesser sighs.
And by the end of the conference, almost no sigh. I thought aloud, this is epistemology in action.
Four: My Grandma’s Goat
My maternal grandma was a very practical person, very strong, and extremely dutiful. I take lots of pleasure visiting her especially when I was very young particularly because of the rustic terrain, simple, unsophisticated, peaceful, that kind of thing.
Also, she did almost everything. She was a small trader who sold everything, and she reared lots of animals. One of my favorites is the goat.
Take one of the goats, who got fed everyday by my grandma (or one of my uncles), the goat roamed around the street every day and came back home to rest and get fed again.
I can’t remember how long this roaming-around-getting-fed cycle was, but it went on for a very long time.
Now let’s unpack the goat’s epistemology.
Say, you ask the goat his thoughts about what he knows and don’t know, and the nature of the world. She might say something like this:
I have this person who cares about me so much, who gives me a nice place to rest after roaming around, everyday; I even get fed substantially every single day for f*** sake. What else can I say.
To the goat this is perhaps like the law of nature, say gravity.
This is brutal but I will say anyways, imagine you are the goat.
The probability that you are in good hands increases drastically every day. (every day the p value drops)
But the story changed on one cold Christmas eve when loads of family cramped up my grandma’s house. How else could she entertain all these beautiful people for God sake. The goat meat will go around for sure, she had thought.
So, a couple guys with huge chest in the family brought out some of the sharpest knife I have ever seen in my entire life.
All they did was simple, they called out the goat for his usual morning food. And then a very loud bleat followed.
We ate the goat meat on Christmas day. 
 David Hume, An Enquiry Concerning Human Understanding; Wikipedia, Hume’s Fork
 I have no idea if the Hannibal Barca was an African or whatever, the point of this is to show you what a deductive argument looks like.
 Adapted from Bertrand Russell’s chicken story.