## 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. But it's not really odd if you think of it as a reductio ad absurdum, as Protagoras suggested.

The anti-inductive induction does not warrant the conclusion that induction is unjustified on the basis of its inductive strength. Rather, the anti-inductive induction makes it a logical consequence of 'Induction is justified' that 'It is not the case that induction is justified.'

Our reason for rejecting
'Induction is justified' is that accepting under such circumstances would lead to a contradiction.

We could also cast it as a disjunctive argument (let P=Induction is justified.):
Either P or not-P.
Assume not-P -> hence, not-P.
Assume P, anti-inductive induction -> hence not-P
Hence in either case, not-P.

5. 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.

6. If alejandro is right, then the following (hypothetical) induction would be impossible:

I observe certain patterns with first-order occurances and make predictions for the future based on patterns in the past.

Based on the relative success of these predictions I reach a number of second-order conclusion, including that first order inductions provide a certain level x of support for their conclusions based on how often those predictions have turned out to be true.

I then go on to identify a number of patterns in my second-order judgements and make certain inductive predictions based on them. I then make a third-order judgement about the reliability of second-order inductions and give them a rating of y > x.

Suppose I continue on for several more levels of analysis and find that each time I go up a level of analysis the higher order inductive principle provides greater support to its conclusion than the one below it.

Eventually it seems this would provide a good inductive case for the claim:
For all natural numbers n, the (n+1)th-order inductive principle provides greater support to its conclusion than the nth-order inductive principle.

But if there is no general inductive principle, then there is no level of analysis on which this induction could be made (since its conclusion ranges over all levels of analysis).

This seems to me a good prima facie reason to think that principles of induction need not be limited to a specified level of analysis.

7. "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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