One of the nice things about living in Ithaca is that my walk to work (on dry days) consists of strolling up “Cascadilla Gorge”:http://www.explore.cornell.edu/scene.cfm?scene=Natural%20Beauty&stop=CU%20-%20NB%20-%20Cascadilla%20Gorge. On the walk I pass by between six and twelve waterfalls. On “this video”:http://www.explore.cornell.edu/tidbit_template.cfm?scene=Natural%20Beauty&tidbit=The%20Hiking%20Path they say six, but I’m not convinced. It would be nice to be more precise, but it’s a little hard for a few reasons. These include.
# I’m not very good at keeping count while climbing up stairs;
# Some areas are marginal cases (at best) of being waterfalls; and
# Sometimes it’s hard to tell whether we have one waterfall or two.
It turns out the last case is a problem for some theories of vagueness. Sadly there are no clear-cut cases of it, though what’s called “Lower Falls”:http://www.explore.cornell.edu/tidbit_template.cfm?scene=Natural%20Beauty&tidbit=Down%20in%20the%20Gorge&tidbitnum=8 comes close. In any case, the theoretical possibility is enough to cause problems, even if I don’t get the nice benefit of walking past a counterexample each day.
Here’s what I want out of a counterexample. Hopefully you’ll agree it’s clearly possible. A part of a river where “Top” consists of water running rapidly over more-or-less vertically stacked rocks, “Middle” consists of a horizontal pool through which water from Top flows reasonably steadily, and “Bottom” consists of water running from Middle again over more-or-less vertically stacked rocks. The thought should be that it’s clear there’s a waterfall there, but it’s not clear whether Top-Middle-Bottom (hereafter TMB) is a waterfall, or whether “Top” and “Bottom” are separate waterfalls.
This kind of case is a really clear example of the need for penumbral connections in a theory of vagueness. Traditional many-valued theories give completely the wrong answer here. Since it’s not true to degree 1 that TMB is a waterfall, and it’s not true to degree 1 that Top is a waterfall, and it’s not true to degree 1 that Bottom is a waterfall, and it’s true to degree 0 that something else there is a waterfall, when we point towards TMB and say “There’s at least one waterfall there” we don’t say something true to degree 1. But this is just false. We do say something perfectly true when we point towards TMB and say “There’s at least one waterfall there.” Supervaluationists and contextualists and (most) epistemicists and (modern, i.e. Brianish) many-valued theorists get this right.
Only most epistemicists though. It turns out the case is also a problem for one prominent epistemicist – namely Timothy Williamson. (The reason is closely related to the problem I raised in “Epistemicism, Parasites and Vague Names”:http://brian.weatherson.org/epvn.PDF, though without all the controversial metaphysical assumptions of that paper.)
As is well worked over by now, semantic anti-realist intuitions pose a problem for the epistemicist. By anti-realist here I don’t mean the strongly anti-realist views associated with Kripkenstein or Quine, just the view well articulated by Fodor that “if meanings are real, they must really be something else.” The problem for the epistemicist, I think, is that there’s nothing else for the epistemicist’s meanings to be. Williamson’s many responses to this kind of worry are well chronicled by John A. Burgess in the paper I’m responding to in EPVN. What I want to focus on here is what he calls the ‘parasite’ strategy.
The semantic indeterminist says that something, we’re never very clear what, determines that some things are clear cases of a predicate, others are clear anti-cases, and others are borderline cases. Williamson’s thought is that this gives us enough material to say how something can create the real determinate border between the instances and the non-instances of a predicate. Just take it to be the upper boundary of what we indeterminists call the borderline cases. That is, what we call the non-cases and what we call the borderline cases are all non-cases, and what we call positive cases are positive cases. Now we have a semantically sharp boundary without using materials unacceptable to the semantic indeterminacy theorist.
The waterfall case shows why this won’t work. It handles well the small sharp drops in the river that look at first like indeterminate instances of ‘waterfall’. Williamson classifies them as not being waterfalls and fairly enough. But look what happens with the case we started with. We indeterminists say that Top, Bottom and TMB are all borderline cases of ‘waterfall’. So applying Williamson’s algorithm for converting an indeterminist theory of vagueness to a determinist one makes all three of them not waterfalls. But this is wrong – it is determinately true that at least _one_ of them is a waterfall.
As Burgess notes, Williamson has arguments against the presuppositions of the challenge he is trying to answer here. For the sake of his epistemicism these challenges better work because the answer to the challenge doesn’t work in waterfall cases.