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July 2, 2004
2nd-Order Vagueness and Definiteness
Andre Gallois has twice presented the following view to me, and I'm now beginning to think the conclusion is right, as strange as it sounds, but when I was talking to him, we were unsure if his argument really shows it. I think I've figured out a way to make the reasoning explicit. The radical conclusion is that higher-order vagueness doesn't raise any problems, because there's only a second order. Once you reach that level, there aren't any other orders. The reason is that second-order vagueness is consistent with definiteness, and therefore there's no penumbral area between second-order vagueness and definiteness.
Let's move along a sorites series to see how it goes. Find something that's definitely red. Then move along for a bit. At some point you've reached definitely not red, but in is there a sharp line between the things that are red and the things that are not red? No. Some things are such that you'll want to say that they're neither red nor not red. This is first-order vagueness. So we adopt a third truth value, indeterminate, for the cases in this penumbral area. So far so good. But wait! What about the boundary between the penumbral area and the definitely red cases (or the boundary between the penumbral area and those that are definitely not red). That seems to be a matter of vagueness also. It doesn't seem as if you go from definitely red to being indeterminate whether it's red over some determinate line. So we need second-order vagueness. Then there's another vague region, in which it's indeterminate whether it's indetermate. Andre wants to argue that we need go no further. We don't need third-order vagueness to talk about the region between red and indeterminate whether it's indeterminate whether it's red. Why not? Well, if something can be both red and indeterminate whether it's indeterminate whether it's red, then there is no third-order vagueness. So if having second-order indeterminacy is consistent with being determinate, then there's no need for third-order determinacy, and infinite regress arguments fail. The only reason I can think of for resisting this view goes as follows. Suppose something is determinately red. That means it's not indeterminate whether it's red. It's definitely not indeterminate whether it's red, because it's definitely red. Well, if it's definitely not indeterminate whether it's red, then it can't be indeterminate whether it's indeterminate whether it's red. Therefore, something can't be both red and indeterminate whether it's indeterminate whether it's red. It has to be definitely not indeterminate whether it's red. This is a bad argument, because it assumes classical two-valued logic. We need a third truth value, indeterminate, if we're going to talk about cases in the penumbral area. That means denying the law of excluded middle. If excluded middle is false, we lose the double negation elimination rule. Not-not-p doesn't mean p, because not-not-p is still consistent with its being indeterminate whether it's p. Even stronger, consider not-indeterminate whether p, which is equivalent to not-not-determinate whether p. By the argument of the last paragraph, we can't conclude p, but it's consistent with p. What about indeterminate whether it's indeterminate whether it's p? It seems that also means we can't conclude p, but it should be consistent with p, right? Why not?
Posted by Jeremy at July 2, 2004 11:17 AM
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Comments
Andre has run this argument past me a few times, but I never got a grasp on it, largely because determinate/indeterminate were not properly defined. I assume you mean definite as 'determinate'? How are you understanding this? Are you defining them as duals? If so, what are the equivalence rules?
For it's not questioned (almost), as I argued with Andre, that DN, CP, RAA, MT, LEM, certain instances of contraposition fail in non-classical (NC) contexts. So we're obviously in an NC environment. I don't have time to write more, but I will. Put bluntly: the inferences made are dubious because NC and it's relation to FOL are not clearly spelled out. It's the logic that needs to be clear, particularly in NC environments, when one is evaluating higher-order vagueness. The interaction of an NC system is what generates problems. For example, on Heck's view, we can conclude: If P, then determinately P. But this holds only if one believes CP can be constructed from FOL. Can it? I argued no at my conference. So my confusion, prima facie, is that the interaction between FOL and NC is ill-defined, generating my confusion. Or maybe I'm simply confused.
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Posted by: KjK at July 6, 2004 8:21 PM
I cannot tell is this argument is contradicted by the following example, which avoids the mind-wrapping "indeterminate that it is indeterminate that it is indeterminate . . ."
Today is in raining. There is no vagueness or indeterminacy.
As it slows down to a drizzle, you say it has stopped raining. I say, "No, it is drizzling, and that is still raining." There is indeterminacy about whether it is raining.
The rain slows still more, so that it just feels like we are walking through a fine mist. "Certainly," you say, "even you would think it is not raining now, since it is not even drizzling anymore." I respond, "No, at this point, I don't really know if this sort of fine mist constitutes 'drizzle' or not. I therefore do not know if it is raining or not." There is now indeterminacy about whether it is drizzling, and indeterminacy about whether drizzle constitutes rain. This is second-order vagueness.
"Aha!" you now say, "I think that the fine mist has lifted. Now, there is no reason to deny that it has stopped raining." I look around puzzled. "Has it lifted? I can't tell. Maybe what I feel is just self-generated sweat from the humidity, but I thought we were still walking through the fine mist. I agree that if the mist has lifted that it has definitely stopped raining, but I am unsure whether the mist has lifted and I am sweating or if I am still walking through the mist, in which case I am merely unsure whether we are walking through a drizzle." Now, it is indeterminate whether there is a fine mist. It is indeterminate whether the fine mist qualifies as "drizzle", and it is indeterminate whether the drizzle qualifies as "rain." This is third-order vagueness.
You have long since concluded that it has stopped raining, but I hold the position that the rain-status cannot be determined until I stop sweating and work out whether a fine mist constitutes a "drizzle".
Posted by: Richard Bellamy at July 8, 2004 10:27 AM