## Wednesday, August 03, 2005

Suppose there was an inductively strong argument to the conclusion that induction is unjustified. What should we think of it? It seems paradoxical. If induction is justified, then we can infer from the argument that it isn't justified. But if induction isn't justified, then the argument's inference fails, so we can't attain the anti-inductive conclusion after all. At least, not via these means. But perhaps induction is unjustified on independent grounds, not related to this particular argument. I think that would have to be the case, in order to escape the paradox.

But that's an odd conclusion, isn't it? If faced by an inductively strong anti-inductive argument, we can deductively conclude that induction must be unjustified for some entirely different reason! The reasoning here is so convoluted, it's rather comical :)

1. Hmmm. If this outcome is convoluted to the point of being comical, what do you think of perfectly ordinary reductio arguments?

2. If induction isn't justified, then it doesn't matter that the argument fails, since induction is unjustified by assumption.

Unless I'm missing something, such an argument would be a perfectly valid way to show that induction is not justified.

It's only a paradox if, under the assumption that induction is not justified, the argument actually leads to the conclusion that induction is justified.

3. Yes, that's right. It isn't strictly speaking a paradox. But my point is that the way out of the paradox seems very odd. It allows us to accept the argument's conclusion -- that induction is not justified -- despite the failure of the argument itself. It strikes me as odd that we could deduce from a bad argument that its conclusion is true!

4. I think the problem comes from not spelling out more clearly what an "inductively strong argument to the conclusion that induction is unjustified" would be like.

Let's say that the "First Order Inductive Principle" (FOIP) says that if certain events happened in the past, it is likely that similar events will happen in the future. (Simplifying A LOT obviously!) Then if the world became "chaotic", with seemingly random events happening all the time without any natural law, regularity or pattern in them, we would be justified in rejecting the FOIP and not expecting any more that our naive first order indiuctions to the future will be correct. We would be inductively justified in doing so, because we would be using a second order inductive principle: maybe one that says that observed "trends" or "patterns" of events are likely to continue (in this case, the trend that first order inductions are failing).

And of course if there would be in the universe some periods of "chaos" and some periods of "order", and no regularity as to which succeeds which, we would reject the second order principle, relying for this on a third order principle... And so on. Like in Russell's theory of types, there is no "Principle of Induction" that covers all levels simultaneously. We always reject the principle of order n based on sound use of the one of order n+1.

5. "But it's not really odd if you think of it as a reductio ad absurdum"

Oh, I see it now. Yeah, that is a better way of looking at it. Thanks!

(And I like the argument in your latest comment too. Very clever!)

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