Knowledge of vague truths

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.