## Thursday, July 30, 2009

### Truth, logic and explanation

Consider the following two very plausible explanatory intuitions:

1. Roses are flowers or violets are yellow because roses are flowers.
2. "Roses are flowers or violets are yellow" is true because "Roses are flowers" is true or "Violets are yellow" is true.
Now, the intuition in (1), when generalized to the general principle that if p and not q, then p or q because p, yields:
1. "Roses are flowers" is true or "Violets are yellow" is true because "Roses are flowers" is true.
Explanation may or may not be transitive in general, but it seems correct in the case at hand to move from (2) and (3) to:
1. "Roses are flowers or violets are yellow" is true because "Roses are flowers" is true.

Observe that (1) and (4) are parallel. Now suppose we agree with the deflationist about truth that:

1. "Roses are flowers or violets are yellow" is true because roses are flowers or violets are yellow.
We now have two paths to explaining why "Roses are flowers or violets are yellow" is true. One explanation is (4) and the other is (5). Unless one of these two explanatory paths subsumes the other, it seems that we have a case of explanatory overdetermination. But neither path subsumes the other. First, the explanans in (4) does not explain the explanans in (5), since that "Roses are flowers" is true does not explain why it is that roses are flowers or violets are yellow, as the former is a fact about a sentence (we can also make the argument go with utterances, statements or propositions) while the latter is a fact about flowers. Second, the explanans in (5) does not explain the explanans in (4)—for that "Roses are flowers" is true may be explained by roses being flowers, but is surely not explained by roses being flowers or violets being yellow.

Thus, the deflationist who accepts (1) and (2) is pressed to accept that (4) and (5) are an overdetermining pair of explanations. But that is unappealing. Probably the deflationist will have to deny the Tarskian intuition in (2). I don't know how great the cost of that is.

So what should we say if we accept (1)-(4), and we are inflationists? We still have a bit of a puzzle, even if we deny (5). The problem is that the explanations in (1) and (4) are exactly parallel. But, we ask, what explains this parallelism? It seems too much to separately explain the truth of the disjunction by the the truth of the true disjunct, and to explain the disjunction by the true disjunct. There should be a way of unifying this. One way would be:

1. (a) Roses are flowers or violets are yellow because "Roses are flowers or violets are yellow" is true; (b) "Roses are flowers or violets are yellow" is true because "Roses are flowers" is true; and, finally, (c) "Roses are flowers" is true because roses are flowers.
(Or we can give a propositional variant. That would probably be better, but I'll stick to the linguistic here so I don't have to keep on saying "the proposition that...". And maybe to explain "Roses are flowers" being true we need a few more steps on the linguistic side—but maybe not, since it may be a logically simple claim about the natural kinds rose and flower rather than a quantified claim.) On this perhaps weird approach, the explanation in Tarski's Schema (T) sometimes goes in one direction and sometimes in the other. I don't fully like this weird approach, though, because step (b) is troubling. The obvious way to justify the explanation in step (b) seems to be: (bi) "Roses are flowers or violets are yellow" is true because "Roses are flowers" is true or "Violets are yellow" is true; and (bii) "Roses are flowers" is true or "Violets are yellow" is true because "Roses are flowers" is true. However, if (bii) has no further intermediate steps, then, by the same token, (1) shouldn't have any further intermediate steps, and (6) is wrongheaded. And if (bii) has further intermediate steps, then these steps will need to be expanded in the fashion of (6), which will result in circularity.

Maybe, though, we can get away with just making (6b) be immediate in the case where "Roses are flowers" is true. In that case, what makes certain complex apparently worldly facts true is stuff on the linguistic side, finally combined with something more basic on the worldly side. I suppose this is basically what Tarski was up to. A lesson of this approach would be that logically complex facts, like the fact that roses are flowers or violets are yellow, are very different from simpler ones.

Of course, if it can be shown that "explains" is used equivocally in (1)-(4), or that the instances of transitivity that I employed are unjustified, all of this goes out the window. But I do think that this may give some reason to be an inflationist about truth.

Heath White said...

Alex,

I am pretty thoroughly lost. First, AFAICT, (1) and (4) are not parallel. The left half of (1) is a disjunction, while the left half of (4) is not. (1) and (3) are parallel but they are instances of the same general principle.

Second, there seems like a pretty easy way to reconcile the explanatory overdetermination of (4) and (5), so long as we allow some fairly innocuous transitivity. (4) plus (6c) yields ‘“Roses are flowers or violets are yellow” is true because roses are flowers.’ (5) plus (1) yields the same thing. So I would say that ‘ “roses are flowers or violets are yellow” is true [ultimately] because roses are flowers’ and this determination goes through two different routes. That doesn’t really seem problematic to me.

Also, your allegedly problematic (6b) is just (4), which I thought had a pretty good justification.

All of this leaves me thinking that I’m missing something important in your post.

Alexander R Pruss said...

How odd that I didn't notice that (6b) is just (4)! But doesn't that just show that (6) reprises the original problem? (I'm kind of lost, too.)

What I meant by (1) and (4) being parallel is not that they share the same logical form--they don't--but that it seems that we have a parallelism between explanatory relations between claims about sentences and explanatory relations between claims about the world.

As for overdermination, I don't think an overdetermination problem disappears just because both explanations come back to a single ultimate explanation. When A causes both B and C, and B is sufficient for D and C is sufficient for D, and the path from B to D doesn't go through C, and the path from C to D doesn't go through B, then this seems to be a case of overdetermination even though there is a single ultimate cause. (If determinism were true, one could still have overdetermination, even though everything would be ultimately causally explained by the conditions at the time of the Big Bang.)