Suppose that we know in lottery cases—i.e., if a lottery has enough tickets and one winner, then we know ahead of time that we won’t win. I know it’s fashionable to deny such knowledge, but such denial leads either to scepticism or to having to say things like “I agree that I have better evidence for *p* than for *q*, but I know *q* and I don’t know *p*” (after all, if a lottery has enough tickets, I can have better evidence that I won’t win than that I have two hands).

Suppose also that classical logic holds even in vagueness cases. This is now a mainstream assumption in the vagueness literature, I understand.

Finally, suppose that once the number of tickets in a lottery reaches about a thousand, I know I won’t win. (The example can be modified if a larger number is needed.) Now for each positive natural number *n*, let *T*_{n} be the proposition that a person whose height is *n* microns is tall but a person whose height is *n*−1 is not tall. At most one of the *T*_{n} propositions is true, since anybody taller than a tall person is tall, and anybody shorter than a non-tall person is short. Moreover, since there is a non-tall person and there is a tall person, classical logic requires that at least one of the *T*_{n} is true.

Hence, exactly one of the *T*_{n} is true. Now, some of the *T*_{n} are definitely false. For instance, *T*_{1000000} is definitely false (since someone a meter tall is definitely not tall) and *T*_{2000000} is definitely false (since someone a micron short of two meters tall is definitely tall). But if anything is vague, it will be vague where exactly the cut-off between non-tall and tall lies. And if that is vague, then in the vague area between non-tall and tall, it will be vague whether *T*_{n} is true. That vague area is at least a millimeter long (in fact, it’s probably at least five centimeters long), and since there are a thousand microns to the millimeter, there will be at least a thousand values *n* such that *T*_{n} is vague.

Moreover, these thousand *T*_{n} are pretty much epistemically on par. Let *n* be any number within that vague range, and suppose that in fact *T*_{n} is false. Then this is a lottery case with at least a thousand tickets. So, if in the lottery case I know I didn’t win, in this case I know that *T*_{n} is false.

Hence, some vague truths can be known—assuming that we know in lottery cases and that classical logic holds.

Of course, as usual, some philosophers will want to reverse the argument, and take this to be another argument that we don’t know in lottery cases, or that classical logic doesn’t hold, or that there is no vagueness.

## 11 comments:

Hmmm...One could still hold that vagueness requires unknowable cutoffs, though.

Pruss: There seem to be several huge problems here:

1) Why would anyone suppose that they know they will not win the lottery?

1a)For one thing, you're presuming that future-tense statements can be true (knowledge must be true belief, on top of whatever other criteria you're using). But fine, let's presume the falsehood of presentism for the moment. Really, the issue is still there: You [I]might[/I] still win! No matter how unlikely it is, [I]someone[/I] will win. There's practically a reductio ad absurdum here, isn't there...?

P1) If a million people each buy a ticket, they can each know that they will lose. {The premise in question}.

P2) Someone will win, making the belief of that person that they would lose false {By definition, a lottery has a winner}.

P3) Knowledge is warranted, TRUE belief + whatever else you care to add. {Again, definition}.

C) Therefore someone's belief that they would lose was both true and false {P1, P2, P3}.

P1 is clearly the premise that led to the absurdity, since the other two premises are true by definition.

1b) I don't think skepticism is looming here. Even if there were a Googleplex of lottery tickets, and I only have one, I can never have more warrant for "knowing" I will lose than for knowing I have hands. Indeed, however it is that I know there even IS a lottery or that lotteries exist at all is going to be [I]at best[/I] EQUAL to my knowledge that I have hands, no?

2) Isn't "tallness" only ever truly ascribed in a comparative way? A 2 meter tall person is only considered "tall" because most humans at the moment are usually shorter than that. It isn't that it's vague, so much as that it's comparative. It's like being fast or strong. These are only true by the standard of what would be normal in the setting....

Anyway, I don't think it follows that, if a person fails to be "tall" that they are therefore "short". They could be "average" in height. And I don't think it's ever going to be judged by microns because we are unable to detect that sort of difference. A person is tall based on inches or a few centimeters. So, say that I regard 183 centimeters as the point at which I call someone a "tall man". If they are 3 centimeters shorter than that, it is indeed possible that I'll say they are average (or, perhaps, on the tall side of average). But a centimeter or a millimeter of difference isn't enough for me to be able to make a judgment.

1. The future-tensedness is a red herring. One can say that the lottery winner was already picked, but you don't know who it is.

Truth is a necessary condition for knowing. So of the thousand people in the lottery each of whom is convinced she didn't win, 999 know that they didn't win, and the remaining one is mistaken in thinking she didn't win.

Knowledge is fallible, and hence one can say stuff like "I know that p, but I might be wrong."

2. If you don't like tallness, just substitute other things, like fatness. And, certainly, I agree that non-tall isn't the same as short.

Arcane Spork:

Yeah, the argument doesn't show that you could know where the vague boundaries are, just that you could know where they aren't!

1) I don't understand. If I have knowledge, and truth is a necessary condition of having knowledge, then I can't be believing falsely in this case....

2) Well, the thing is that, if we set an exact number of microns that counts as "tall", then anything short of that is not tall, but by doing this we've set ourselves up to not be able to distinguish between tall and almost-tall people in many cases.

Fatness is a good case, since there are certain health issues that are associated with being overweight (having a BMI higher than 25). So, if someone has a BMI of 24.99, they will probably experience the very same issues, but they cannot properly be categorized as "overweight". Is that the kind of vagueness you mean?

On 1):

You're confusing (P and Possibly not P) with Possibly (P and not P).

It's not possible that we know that p and not p

But it IS possible that we know that p and possibly not p.

Arcane: If I believe that P will happen, and then P turns out not to happen, I could still have had knowledge that P?

Seriously, I know I should probably drop this, but I really don't get it: I understand that knowledge can be of a contingent truth (ergo, it's fallible and could have been wrong). However, if there is a truth about what happened, then anyone who believed it would happen otherwise is wrong and never had knowledge.

And there's a tension with B-theory here. If you think there are future-tense truths, then you ought to think that a rationally warranted belief about the future can be knowledge (given that it is true at t*). On the other hand, we seem inclined (or, at least, Pruss seems inclined) to grant that a person has knowledge that they will lose a sufficiently large lottery just because of how much warrant they have, without any consideration for the truth or falsehood at t*.

Michael:

Suppose the lottery has 1000 tickets, and a winner has just been picked, but I have no idea which ticket it is. Let's suppose, unbeknownst to me, the winning ticket is #3.

Then:

I believe on good evidence that ticket #1 is not the winner.

I believe on good evidence that ticket #2 is not the winner.

I believe on good evidence that ticket #3 is not the winner.

...

My belief that ticket #1 is not the winner is knowledge, I contend.

Likewise, my belief that ticket #2 is not the winner is also knowledge.

But my belief that ticket #3 is not the winner is NOT knowledge. It's not knowledge because it's false. Of course, from my point of view, the three tickets are on par. But knowledge isn't just a function of my point of view.

Looking over the post, I didn't explicitly specify that I am not in fact the winner of the lottery. I should have specified that, but I took it to be obvious that I don't know that I'm not the winner if in fact I am.

In the application to vagueness, I did take care to specify that Tn is in fact false.

My whole problem is with the fact that you could indeed be the winner. Saying "if there are enough tickets, then I can know I will lose" seems to ignore that possibility. In other words, it seemed to me like you were saying all 1,000 of them know they will lose, when clearly one of them (if there is a fact of the matter about his future winning) does NOT have knowledge.

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