Monday, April 09, 2007
Luminosity, Coziness, and Borderline Cases
Let a case be a world, subject and time triple < w, s, t >. Conditions either obtain or fail to obtain in each case, and are introduced by that-clauses: e.g. the condition that one is happy or that one feels cold (however, for simplicity I don't always state conditions in this cumbersome form in what follows).
A condition C is luminous iff the following holds:
(LUM) For every case A, if C obtains in A, then one is in a position to know that C obtains in A.
Tim Williamson's anti-luminosity argument is meant to offer a proof that no non-trivial conditions are luminous in this sense (a condition is trivial iff it obtains in every case, or else fails to obtain in every case).
Williamson proceeds, he claims without loss of generality, by way of a reductio of the claim that the condition that one feels cold is luminous. Imagine a series of times (T0, T1, T2.......Tn) between dawn and noon, each a millisecond apart. Fixing the subject and the world, we obtain a series of cases (A0, A1, A2........An) individuated by those times. One warms up very slowly, virtually imperceptibly, throughout this period. So the following are stipulated features of the case:
(COLD) In A0 one feels cold.
(WARM) In An one does not feel cold.
Moreover, one is constantly attentive to whether or not one feels cold, so that if one were in a position to know one feels cold, then one would know that one feels cold. So we endorse the following strengthened instance of (LUM):
(LUM+) For every case Ax, if one feels cold in Ax, then one knows that one feels cold in Ax.
Essentially, the proof demonstrates the inconsistency, given a series of cases as described, of (LUM+) with the following principle:
(REL) If in a case Ax, one knows that one feels cold, then in case Ax+1 one does feel cold.
The reasoning is very familiar. (LUM+) and (REL) together entail the following tolerance principle for one feeling cold:
(TOL) For every case Ax, if one feels cold in Ax, then one feels cold in Ax+1.
Sorites reasoning will easy demonstrate the inconsistency of (TOL) with (COLD) and (WARM). So, since Williamson thinks he can independently motivate all the relevant analogues of (REL), we should reject (LUM) for any non-trivial condition.
Most of the literature that exists on the anti-luminosity argument has trying to undercut the motivation for (REL) and its relatives. However, several philosophers--most explicitly John Hawthorne--have taken a different line of response. They've suggested that nothing in Williamson's argument rules out that non-trivial conditions might be cozy rather than luminous. A condition C is cozy iff:
In every case A in which C determinately obtains, one is in a position to know that C obtains.
The suggestion, of course, is that by restricting our luminosity claims to the determinate (i.e. non-borderline) cases of C, we can side-step the anti-luminosity argument. Implicit in this suggestion is recognition of the following feature of the argument as Williamson presents it: Williamson's argument targets the luminosity of a condition by targeting the luminosity of its borderline cases. Brian Weatherson has recently stressed this point:
'The counterexamples to Luminosity we get from following this proof through are always borderline cases of C obtaining. In these cases Luminosity fails because any belief that C did obtain would be unsafe, and hence not knowledge.' ('Luminous Margins': 374)
And in a recent paper Williamson himself writes:
'The strategy is to construct a sorites series between a case in which the condition clearly obtains and one in which it clearly fails to obtain. Luminosity must fail close to the boundary between cases where the condition obtains and cases where it does not, just on the obtaining side.' ('Contextualism, Subject-Sensitive Invariantism and Knowledge of Knowledge': 230)
But notice that this result only holds on the assumption that borderline cases of a condition obtaining are to be thought in more or less the same manner as the Epistemicist would have us think of borderline cases of a vague predicate. On a familiar rival picture of borderline cases of vague expressions, they give rise to truth-value gaps (for example, the account of vagueness offered by Supervaluationists). Extending that picture to the current context, we obtain the following characterization of what it is for a case B to be a borderline case of a condition C obtaining:
C does not obtain in B and ~C does not obtain in B.
(We'll need to make sense of a distinction between C failing to obtain and ~C obtaining on this picture. I won't worry about how to do that here).
Now, Williamson claims that his anti-luminosity argument is neutral with respect to the correct account of vagueness and hence, one would hope, with respect to what the correct account of the nature of borderline cases is. But notice that on the envisaged account of what it is for a case B to be a borderline case of a condition C obtaining, no borderline case of C can be a counterexample to the luminosity of C. Such a counterexample would be a case in which C obtained, only unknowably. But clearly on this conception, no borderline case of C obtaining is a case of C obtaining unknowably, since no borderline case of C obtaining is a case of C obtaining. Hence, as advertised, no borderline case of C obtaining can be a counterexample to the luminosity of C.
However, Williamson's proof does, as he claims, go through as before. If one thinks of borderline cases in the 'gappy' way just sketched, Williamson's premises entail that the last case of C determinately obtaining is a counterexample to the luminosity of C. But now note the following; this case will also provide a counterexample to the coziness of C.
What's the upshot of all of this? Well, whether the coziness response to the anti-luminosity result has any chance of success depends on what our view of borderline cases is. Hawthorne himself has tended to favor an Epistemicist-like account of borderline cases over a gappy one, so presumably this result wouldn't bother him too much. However, a gappy view is still the dominant one amongst philosophers of vagueness. It seems, then, that not many philosophers are in good shape to offer the coziness response to Williamson.
(The preceding ignores the effects of analogues of higher-order vagueness. I leave it to those much smarter than me to figure out how taking HOV seriously here would change the conclusions that could be drawn.)
A condition C is luminous iff the following holds:
(LUM) For every case A, if C obtains in A, then one is in a position to know that C obtains in A.
Tim Williamson's anti-luminosity argument is meant to offer a proof that no non-trivial conditions are luminous in this sense (a condition is trivial iff it obtains in every case, or else fails to obtain in every case).
Williamson proceeds, he claims without loss of generality, by way of a reductio of the claim that the condition that one feels cold is luminous. Imagine a series of times (T0, T1, T2.......Tn) between dawn and noon, each a millisecond apart. Fixing the subject and the world, we obtain a series of cases (A0, A1, A2........An) individuated by those times. One warms up very slowly, virtually imperceptibly, throughout this period. So the following are stipulated features of the case:
(COLD) In A0 one feels cold.
(WARM) In An one does not feel cold.
Moreover, one is constantly attentive to whether or not one feels cold, so that if one were in a position to know one feels cold, then one would know that one feels cold. So we endorse the following strengthened instance of (LUM):
(LUM+) For every case Ax, if one feels cold in Ax, then one knows that one feels cold in Ax.
Essentially, the proof demonstrates the inconsistency, given a series of cases as described, of (LUM+) with the following principle:
(REL) If in a case Ax, one knows that one feels cold, then in case Ax+1 one does feel cold.
The reasoning is very familiar. (LUM+) and (REL) together entail the following tolerance principle for one feeling cold:
(TOL) For every case Ax, if one feels cold in Ax, then one feels cold in Ax+1.
Sorites reasoning will easy demonstrate the inconsistency of (TOL) with (COLD) and (WARM). So, since Williamson thinks he can independently motivate all the relevant analogues of (REL), we should reject (LUM) for any non-trivial condition.
Most of the literature that exists on the anti-luminosity argument has trying to undercut the motivation for (REL) and its relatives. However, several philosophers--most explicitly John Hawthorne--have taken a different line of response. They've suggested that nothing in Williamson's argument rules out that non-trivial conditions might be cozy rather than luminous. A condition C is cozy iff:
In every case A in which C determinately obtains, one is in a position to know that C obtains.
The suggestion, of course, is that by restricting our luminosity claims to the determinate (i.e. non-borderline) cases of C, we can side-step the anti-luminosity argument. Implicit in this suggestion is recognition of the following feature of the argument as Williamson presents it: Williamson's argument targets the luminosity of a condition by targeting the luminosity of its borderline cases. Brian Weatherson has recently stressed this point:
'The counterexamples to Luminosity we get from following this proof through are always borderline cases of C obtaining. In these cases Luminosity fails because any belief that C did obtain would be unsafe, and hence not knowledge.' ('Luminous Margins': 374)
And in a recent paper Williamson himself writes:
'The strategy is to construct a sorites series between a case in which the condition clearly obtains and one in which it clearly fails to obtain. Luminosity must fail close to the boundary between cases where the condition obtains and cases where it does not, just on the obtaining side.' ('Contextualism, Subject-Sensitive Invariantism and Knowledge of Knowledge': 230)
But notice that this result only holds on the assumption that borderline cases of a condition obtaining are to be thought in more or less the same manner as the Epistemicist would have us think of borderline cases of a vague predicate. On a familiar rival picture of borderline cases of vague expressions, they give rise to truth-value gaps (for example, the account of vagueness offered by Supervaluationists). Extending that picture to the current context, we obtain the following characterization of what it is for a case B to be a borderline case of a condition C obtaining:
C does not obtain in B and ~C does not obtain in B.
(We'll need to make sense of a distinction between C failing to obtain and ~C obtaining on this picture. I won't worry about how to do that here).
Now, Williamson claims that his anti-luminosity argument is neutral with respect to the correct account of vagueness and hence, one would hope, with respect to what the correct account of the nature of borderline cases is. But notice that on the envisaged account of what it is for a case B to be a borderline case of a condition C obtaining, no borderline case of C can be a counterexample to the luminosity of C. Such a counterexample would be a case in which C obtained, only unknowably. But clearly on this conception, no borderline case of C obtaining is a case of C obtaining unknowably, since no borderline case of C obtaining is a case of C obtaining. Hence, as advertised, no borderline case of C obtaining can be a counterexample to the luminosity of C.
However, Williamson's proof does, as he claims, go through as before. If one thinks of borderline cases in the 'gappy' way just sketched, Williamson's premises entail that the last case of C determinately obtaining is a counterexample to the luminosity of C. But now note the following; this case will also provide a counterexample to the coziness of C.
What's the upshot of all of this? Well, whether the coziness response to the anti-luminosity result has any chance of success depends on what our view of borderline cases is. Hawthorne himself has tended to favor an Epistemicist-like account of borderline cases over a gappy one, so presumably this result wouldn't bother him too much. However, a gappy view is still the dominant one amongst philosophers of vagueness. It seems, then, that not many philosophers are in good shape to offer the coziness response to Williamson.
(The preceding ignores the effects of analogues of higher-order vagueness. I leave it to those much smarter than me to figure out how taking HOV seriously here would change the conclusions that could be drawn.)
Labels: Epistemology, John Hawthorne, Tim Williamson, Vagueness
# posted by Aidan @ 7:15 PM 
