Although we're going to be focusing on fallacies in unit four today, I want to start off by talking about something that might or might not be a fallacy. But in any case, it exposes a linguistic phenomenon, that's called vagueness and that does create fallacies, okay. So what is the thing I'm going to talk about today? Well, it's a phenomenon I call the argument from the heap, or sometimes called the Seriates argument after the Greek word Soros, which means heap. What's the argument from the heap? Well let me give you an example. Consider the following argument, this is an example of the argument from the heap. Premise one, a person who owns only one penny, is poor. Premise two, a person who owns only one penny more than a poor person, is also poor. Conclusion, therefore, a person who owns a gazillion pennies, is poor. Now, why should we believe that those premises support that conclusion? Well, think of it this way. If premise one is true, then a person who owns only one penny, is poor. But if premise two is true, then it's going to follow that the person who owns two pennies is also poor, right? because they own only one penny more than a poor person and anyway that's obviously right. A person who owns only two pennies, is clearly poor. And then it's going to follow by premise two, that a person who owns three pennies is also poor, right. because they only own just one penny more than a poor person, so they're also going to be poor. And by premise two, it's also going to follow that a person who owns four pennies, is poor. Now, I hope you can see the pattern here, you can see where this is going. We can keep on applying premise two, to our results and we'll get that person who owns four pennies is poor, a person who owns five pennies is poor, a person who owns six pennies is poor, and we can go on and on and on, to a million pennies, a billion pennies, a trillion pennies, a quadrillion pennies, and even a gazillion pennies. Now, there's a name for that kind of reasoning, which was common in mathematics. It's called the principle of mathematical induction. The principal of mathematical induction says that, arguments of the following form are valid. where premise one says the number zero belongs to some category F. In our example the category of numbers, such that anyone who owns that number of pennies, is poor. If x belongs to F, then x plus one belongs to F, that's premise two. So in other words, that's saying that, if a particular number of pennies is such that anyone that owns just that number of pennies is poor, then adding one to that particular number of pennies, will also leave the person poor. And the conclusion is of the form all the natural numbers belong to 'F'. In other words, no matter how many pennies you own, you're still going to be poor. Right. All natural numbers belong to the category of numbers such that, a person owns that number of pennies is poor. That's the principle of mathematical induction, stated quite generally. Now, that principle, I'm quite confident is correct. Right, arguments of that form are valid. If arguments of that form weren't valid, then a whole lot of mathematics, the mathematicians accept today, would be false and I don't believe it is. So, the principle of mathematical induction, I'm prepared to say is correct. But then, what's going on in this argument? Right, if the principle of induction is correct, then it looks like, the argument is valid. So if the argument is valid, that means there's no possible way for the premises to be true, while the conclusion is false. So, is premise one true? Is it true that a person who owns only one penny is poor? It seems quite obviously, yes. Premise one is true. A person who owns only one penny, has got to be poor. There's no where in the world where one penny makes you rich. What about premise two? If you own just one penny more than a poor person, could you avoid being poor, by owning just one penny more than a poor person? I don't see how you can. One penny can't make the difference between being poor and not being poor. So it looks like premise two is true. But if the argument is valid and both of the premises are true, then that means the argument is sound. And remember, in a sound argument, that the conclusion has to be true. So let's consider the conclusion a person who owns a gazillion pennies, is poor. Is that true? Well, no. It's not true. I wish I owned a gazillion pennies. Owning a gazillion pennies, does not leave you poor. Someone who owns a gazillion pennies is very, very wealthy. So the conclusion seems to be obviously, false. So what's going on here? Well, in logic, this is what we call a paradox. A paradox is an argument, in which all of the premises of the argument seem to be obviously true, as in our argument from the heap. The argument seems to be obviously valid. Again, as in our argument from the heap, but the conclusion seems to be obviously false, again as in our arguments from the heap. So our argument from the heap is one example of a paradox, there are many other examples. Now some paradoxes are fallacies because despite appearances, even though they appear to be valid, they're really not. Really their arguments in which the premises don't support the conclusion, despite appearances. So some paradoxes are fallacies, but the question is, is this particular paradox a fallacy? Is the argument from the heap, a fallacy or is it just a paradox? And if it is a paradox, what's the solution? Should we say that a person could avoid being poor, even though they own only one penny? Should we say that you could avoid being poor by owning just one penny, more than a poor person? Or should we deny the principle of mathematical induction? Or should we say that, even if you owned a gazillion pennies, you're still poor? How should we solve this paradox? I'll leave that for you to think about.