Wednesday, October 06, 2004

Examples and Counterexamples

One of the most famous counterexamples in the history of philosophy is Hume's proposed counterexample to his own general principle: the missing shade of blue. The general principle at stake is the thesis, essential to Hume's project, that ideas are derived from impressions of which they are copies. Hume suggests a 'contradictory phenomenon':

Suppose therefore a person to have enjoyed his sight for thirty years, and to have become perfectly well acquainted with colours of all kinds, excepting one particular shade of blue, for instance, which it never has been his fortune to meet with. Let all the different shades of that colour, except that single one, be plac'd before him, descending gradually from the deepest to the lightest; 'tis plain, that he will perceive a blank, where that shade is wanting, said will be sensible, that there is a greater distance in that place betwixt the contiguous colours, than in any other. Now I ask, whether 'tis possible for him, from his own imagination, to supply this deficiency, and raise up to himself the idea of that particular shade, tho' it had never been conveyed to him by his senses? (T 1.1.1.10)

Hume's conclusion: "I believe there are few but will be of opinion that he can; and this may serve as a proof, that the simple ideas are not always derived from the correspondent impressions; tho' the instance is so particular and singular, that 'tis scarce worth our observing, and does not merit that for it alone we should alter our general maxim."

This has surprised Hume commentators considerably, and they have gone to great lengths to try to figure out how in the world Hume thinks his general principle can stand given that he himself admits a counterexample to it. My own view, however, is that Hume has a better understanding of the way counterexamples work than many of his commentators, and he is, in particular, quite right to regard the missing shade of blue in the light he does.

To see this, it will be helpful to recognize the actual function of examples and counterexamples in philosophy. It was long ago recognized in medieval philosophy that examples of all sorts serve as arguments. In particular, the idea was that we can run a sort of analogy with enthymemes:

example : induction :: enthymeme : deduction

This analogy is, I think, exactly right; examples are a sort of inductive enthymeme, as the medievals understood 'induction'. Another analogy will be helpful:

induction : division :: deduction : definition

The basic core of the medieval theory of induction is that inductions are based on divisions of the possible field of solutions. If we have three different possible kinds, and we are able to establish that some conclusion C follows from the properties of each of these kinds, we have established C for the universe of discourse. If the three kinds are all possible kinds (relevant to the discussion), it follows that we have established C. In principle the division can be as sweeping or focused as it needs to be for the discussion. There are, however, limits to this; one of the problems with induction by simple enumeration is that it tries to divide down to individuals, which works if you already have a fixed, well-defined population, but not so well if (for instance) you are trying to establish something as true for all possible individuals of a kind simply by going through the individuals.

Now, in an example, used as argument (as examples often are in philosophy) what we find is a particular part of the possible field of discourse on which an analysis is performed; and this is accompanied by an implicit or explicit 'and so on'. That is, we are given one part of the analysis, on the assumption that we can take what we learn from analyzing this particular case and use it for any other case that might come our way.

Seen in this light, we can begin to see some of the limits of examples as arguments:

First, it is not always clear to what division they should be related. Or, to put it in other terms: how far over the possible field do our conclusions from this example actually reach?

Second, we need to be able to do the analysis. That is, we need to understand what is going on in the example sufficiently to be able to trace out the conclusion that's being put forward.

Third, we need to be able to allow the 'and so on'. That is, we need to be able to apply what we learn in this case to other cases.

Fourth, we often need to be able to link the example with other cases relevant to the discussion at hand, not just with any other cases.

The four are actually closely bound together. In practical terms, they mean that we should be very cautious about strange, bizarre, complicated, or highly artificial examples put forward as arguments. In Hume's terms, some proposed examples or counterexamples are "so particular and singular" that they don't really contribute anything to the discussion. This is certainly true of the missing shade of blue:

- Hume is analyzing how the mind naturally works, the example is highly artificial.

- Hume wants a principle that will cover all ordinary cases, whereas it isn't at all clear that we could ever be sure the counterexample had actually occurred in real life.

- Hume is looking for a general principle that will move our understanding of the mind forward, the example presupposes that general principle for most cases beside itself.

- Nothing in the counterexample gives us any indication that we would be able to find more counterexamples beyond a handful of almost precisely similar scenarios, with precisely similar limitations.

- Hume is looking for a good empirical generalization; the counterexample is only worrisome if you want to prove that it is "absolutely impossible" for Hume's thesis to be false.

The missing shade of blue doesn't hurt Hume's case in the slightest. And, indeed, if this is the best sort of counterexample that could be provided for the generalization, it would be a sign that the generalization was a very, very good one. If no better counterexample can be proposed, the counterexample itself can serve as an argument for the thesis!

These sorts of issues are worth keeping in mind when we consider some of the examples and counterexamples that are proposed in analytic philosophy. I once attended a lecture on philosophy of causation in which one of the counterexamples began with something like the words, "Suppose there is a law of magic that when such-and-such is done something else happens at midnight." Red flags should go up immediately:
1) what do we actually know about a world in which this would be the case?
2) do we even know the scenario is actually possible rather than only apparently so?
3) how does this really help move forward the discussion of actual causation?
4) do we have the means to generalize from this case to other cases with any certainty?
And so forth. I'm not saying that complicated or strange examples and counterexamples are never helpful; but if they are helpful, we should be able to handle questions like this, which are concerned with their relevance, the force of conclusions drawn from them, the scope of those conclusions, and other such things that are key to an example's being a good argument.

(UPDATE: It should be said that the second analogy above is not as close as the the first. For one thing, it's complicated by the relations between division and definition. But the analogy does capture something, I think.)

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