Friday, January 09, 2009

Counterexamples

I've always been puzzled at the crudeness with which analytic philosophers often handle counterexamples, and was thinking about it after having come across a seriously egregious bungling of counterexamples in my reading recently. Counterexamples are pretty common, and most people manage to muddle through them in a reasonably tolerable way, but there seems to me to be a lack of sophistication, as evidence by the sheer puzzlement with which people regard the relatively simple case of Hume's missing shade of blue, and sometimes straightforward incompetence, when handling them. Part of the problem, I think, is that everyone starts using them early, and no one really bothers to have in hand a good account of how they work; I never was taught one, nor have I met many people who show any indication of having been taught one.

This is related, I think, to a failure to appreciate the ramifications of the usual accounts of validity. There are tricky details in handling them, but roughly on these accounts validity is truth-preservation, where truth-preservation has the following characteristics:

(a) if all of the premises are assigned the value TRUE, the conclusion is assigned the value TRUE
(b) if the conclusion is assigned the value FALSE, at least one of the premises is assigned the value FALSE

The important point here is that these are characteristics that can only be had under inference rules; and since inference rules can vary from domain to domain, validity varies from domain to domain. This isn't really surprising; truth-preservation does not work exactly the same way in an intuitionistic system that it does in a classical, or an affine, or a linear system. And there are homelier examples; in some domains inference from part to whole will be truth-preserving, and in others it will be the fallacy of composition. But this has ramifications for counterexamples used to show invalidity; counterexamples can only show arguments invalid for the domains in which they can be identified. In another domain the counterexample might not exist and the very same argument might be valid. And so it is always crucial to know what domains are involved. This is one reason why the missing shade of blue is not a problem for Hume's thesis, for instance; it's a counterexample that only occurs in a domain other than that in which Hume is interested (the one he is interested in being that which contains only events that happen naturally and regularly). There are other reasons, of course, but it is one of them. For every counterexample, we have to ask: is this counterexample from the right domain? Often it is, but sometimes one finds counterexamples being treated as unproblematic when it actually isn't clear that they apply.

Often counterexamples get their bite not from being mere counterexamples, but from an implicit suggestion that goes with the counterexample that the counterexample comes from a domain that's relevant to the argument or its premises. Counterexamples work, in other words, on the condition that they are relevant (when accurately described, of course -- the fact that something can seem to be a counterexample only when misdescribed is what allows for what Lakatos calls 'monster adjustment', where one simply redescribes the example in a way that eliminates its appearance as a counterexample).

UPDATE: In the comments David Corfield reminds me of this discussion of mathematical difficulties and counterexamples at "n-Category Cafe".

Thursday, January 08, 2009

Truth and Beauty

I've been surprisingly busy of late. Will be back soon. Here's some Emily.

I Died for Beauty
by Emily Dickinson

I died for beauty, but was scarce
When one who died for truth was lain

He questioned softly why I failed?
"For beauty," I replied.
"And I for truth,—the two are one;
We brethren are," he said.

And so, as kinsmen met at night,
We talked between the rooms,
Until the moss had reached our lips,
And covered up our names.

Monday, January 05, 2009

Boswell, December 1764

I have put up at Houyhnhnm Land the first part of the series I had promised on Boswell's interesting meetings with Rousseau and Voltaire in 1764. Here is the teaser:

James Boswell, best known for his biography of Samuel Johnson, had a very interesting December in 1764. In his tour of the Continent, Boswell had come to Switzerland, and there he met Rousseau and Voltaire. Unfortunately, he never had a chance to write up the meetings in good form, as he had with Johnson; but even though the interactions are scattered through journal entries, memoranda, and correspondence in cryptic form, it still provides a valuable sighting of the two philosophers in their natural habitat.

Read "A Very Boswell Christmas, Part I"

Sunday, January 04, 2009

A Poem Re-Draft and Three New Poem Drafts

The first is the re-draft.

Blue

I had a dream of Midas once,
where I touched and all was gold;
but then I went and woke myself
and shivered in the cold.

Everywhere I've been, I've been in trouble,
everywhere I go, it's just the same;
everything I do just ends in heartache;
everywhere I am, I'm still to blame.

I have an angel on my shoulder,
but he's not the trusty kind;
he's apt to up and leave me
when I get into a bind.

Everywhere I've been, I've been in trouble,
everywhere I go, it's just the same;
everything I do just ends in heartache;
everywhere I am, I'm still to blame.

but it left me to my sin;
it went out to 'buy some cigarettes'
but never came back again.

Everywhere I've been, I've been in trouble,
everywhere I go, it's just the same;
everything I do just ends in heartache;
everywhere I am, I'm still to blame.

Thinking of You

In the miasma of my thought
I sometimes forget to think of you
And have to begin my thoughts anew
To think of you as I ought.

Under heavy burden of the care
That my imagination draws
I must calm my mind and pause
To remember that you are there.

Blessed God, Though I Should Die

Blessed God, though I should die,
I shall not weep nor tear my eye
If only you remember me;
And in the darkness of the night
When you are hidden from my sight
You whisper then to strengthen me.
When illness comes, this fool to slay,
In your arms I wait and pray,
From all these mortal chains set free;
When sudden evil like lightning falls
Or glory fades and fire palls,
Your love, and death, will make me free.

Unicorn

Make me, my Lady, a unicorn,
leaping out from primal morn,