In the last class we talked about the argument from the heat, otherwise known as the Sorites argument. Today I want to describe the linguistic phenomenon that gives rise to that argument, gives rise to the paradox. Explain what that linguistic phenomenon is and why it gives rise to the paradox. The phenomenon is what's known as vagueness. So what's vagueness? Well, here's the definition along with some examples. So an expression is vague when there's no precise boundary between the cases to which that expression correctly applies and the cases to which it does not correctly apply. So, here's some examples. Poor, as we just saw in the last lecture. Bald, tall, accomplished, famous. Here, consider, what is the precise boundary line between baldness and non-baldness? Is there some precise line where if you have fewer hairs on your head than that then you're bald, but if you have more hairs on your head than that then you are not bald? No, there is no precise line. Baldness is vague. You could be more bald by having fewer hairs on your head. You can be less bald by heaving more hairs on your head. But baldness is vague. There's no precise line between baldness and non-baldness. How about tall? Is there some precise line where if you're taller than that then you're tall, but if you're shorter than that then you're not tall? No. Again, there's no precise line. Some people are taller than others, but there's no precise line where if you exceed that height, then you're tall. How about being accomplished? Is there some precise amount of accomplishment that you have to achieve in order to account as an accomplished person? No. Accomplished again is vague expression. There's no precise amount of accomplishment that's required in order for you to be an accomplished person. Clearly, some people are accomplished, but it's not clear exactly what the outlines of that category are. Some people are clearly included in that category. Some people are clearly not included in that category. But where exactly does that boundary line fall? That's not clear. It's vague. And again famous. How many people have to know who you are in order for you to be famous? Is Walter famous? Well someone might say sure, Walter's famous. Tens of thousands of students are taking his course. Yeah, but someone might say no, Walter's not famous, Tom Cruise is famous. So, how many people have to know who you are in order for you to be famous? It's not clear, it's vague. So those are some examples of vagueness. Examples of expressions that fit the definition of vagueness. Now, why do these expressions give rise to paradox? Give rise to the Sorites Paradox, the argument from the heap? Here's why. If there's no precise boundaries between cases to which an expression correctly applies and cases to which it does not correctly apply, then it's going to seem obviously true that making a minuscule change, just having a minuscule difference from a case to which an expression correctly applies or correctly doesn't apply, isn't going to make the difference between whether or not the expression correctly applies. Right? Having someone be just ever so slightly taller than a short person isn't going to make that person tall. If they're ever so slightly taller than a short person, they're still going to be short. If they're ever so slightly less famous than a famous person, they're still going to be famous. Right? It's going to seem that way as long as the boundary line between cases of correct application and cases of incorrect application is not clear. But the problem is that a series of minuscule differences or minuscule changes can add up to a big change. So, there could be a big change between two people, but if you just look at the minuscule changes it'll look like well, if one person isn't famous then the other one must not be famous. If one person isn't rich then the other person must not be rich. Again, because the minuscule differences are so minuscule that they seem not to involve crossing any threshold from cases of correct application to cases of incorrect application. Right? The differences are too small to allow for you to cross that threshold. Well, the reason is that the threshold isn't the precise one. So that's why it seems like when you have minuscule differences, they're not going to allow you to cross that threshold. If the threshold is fuzzy, then a minuscule difference isn't going to be enough to cross it. But, if you add up a whole bunch of minuscule differences and you get a big difference. A difference that is big enough to cross the threshold. So that's why vagueness leads to paradox, and here's some examples. Suppose someone says, well a person with no hair is bald. I mean clearly someone with no hair at all is bald, but someone with one more hair than a bald person is still bald. Right? It seems like if the only difference between you and me is that you have one more hair than I do, then if I'm bald, you must also be bald. Right? Someone who has only one more hair than a bald person is bald. But from those two premises, it follows by the principle of mathematical induction that someone with a billion hairs on their head is bald. I mean, I don't even know if it's possible to have a billion hairs on a human head, but if it is, presumably someone who did have a billion hairs on their head would not be bald. And yet, that conclusion follows validly from our two premises. Consider this other example. Someone who's only a meter in height is short. Right? A meter, for those of you who live in the United States, is about 39 inches. So someone who's only a meter in height is a short person. Now, you might think well someone who is only one millimeter taller then a short person is also short. If two people differ in height by a thousandth of a meter, then if one of them is short, you think the other one's gotta be short too. But from those two premises, again, it follows by the principle of mathematical induction that someone who's five meters tall is short, and that I can assure you is not true. Someone who's five meters tall, again for the benefit of people living in the United States, someone who's five meters tall is about a 195 inches tall. So that would be very, very tall. That would be taller than Dwight Howard. Okay, so again, why does this paradox arise? It arises because vagueness makes the second premise appear to be true. Vagueness makes it seem as if a minuscule difference between two things can't cross the threshold from correct application of an expression to incorrect application of an expression. So that's why the paradox arises. Okay, next time we'll see how the phenomenon of vagueness, how vague expressions can generate not just paradoxes, but also fallacies. See you next time.