Here's an intuition I have. Suppose that I somehow knew that a dozen of boxes have appeared ex nihilo for no cause (not even a stochastic one) in my office. I open half of them and each one was purple inside. Do I have good reason to think that the others are also purple inside? As long as I hold on to my knowledge that there is no explanation of the boxes' presence and character, I think not. It is rather like when I get heads six times in a row when tossing a fair coin—as long as I get to hold on to my knowledge that the coin is fair, I have no reason to think subsequent tosses will be heads.
This suggests to me that induction requires that the cases we do induction over be non-brute, that they have explanations. But not just any explanations will do. The cases need to have a common type of explanation. If one box was materialized in my office by aliens, and another was delivered by my best friend, and another coalesced from the drippings in a leaky ceiling, and so on, then I don't get to do induction across the cases.
Thus:
- That all observed Fs are Gs gives me knowledge by induction that all Fs are Gs only if there is a common type of explanation as to why each F is a G.
15 comments:
How did you decide which boxes to open? It seems to me that if your sampling was done using a randomizing device, you have excellent reason to think that all of the boxes are purple inside.
Things would be different, I think, if you were receiving the boxes one at a time in the mail and opened them as they were received. But then, in that case, I don't think that knowing who they came from improves your epistemic situation.
Well, if there are no explanations of any of them coming into existence, it's reasonable to treat them sort of like independent events. Suppose 12 fair coins were tossed on the table, and each one is covered up. You uncover six, at random. They're all heads. What does that tell you about the other six? Nothing!
I suggest a minor addendum to the proposed principle:
1'. That all observed Fs are Gs gives me knowledge by induction that all Fs are Gs only if I know that there is a common type of explanation as to why each F is a G.
This is important because (1) I have to have knowledge that there is this common type of explanation in order to gain knowledge that evidentially depends upon that fact. But -- I think more interestingly -- this is also important because (2) one presumably need not know what the common explanation actually is to gain knowledge by induction. One merely needs to have knowledge that there is one ... whatever it may be.
That would strengthen the principle. I don't really want to stick my neck out so far at this point. Moreover, it might give rise to a regress problem, in that the knowledge that there is a common type of explanation may need to be gained by induction.
"Suppose 12 fair coins were tossed on the table, and each one is covered up. You uncover six, at random. They're all heads. What does that tell you about the other six? Nothing!"
Sure. But then you're assuming that they are fair. You have no reason to make that assumption. Given that you have no reason to assume that they are fair coins, seeing all heads *does* give you reason to think that the rest are heads -- probably and approximately.
"... one presumably need not know what the common explanation actually is to gain knowledge by induction. One merely needs to have knowledge that there is one ..."
I'm skeptical. Peirce thought he had proved that there will always be some common explanation. The proof is purely set theoretic, and at least in his early writing, he treated it as a demonstration of a law of causality. But if the proof works, then saying that there is some common explanation is not informative.
I think there always is some kind of a common explanation, because I am a theist. But the assumption of a common explanation, or even of an explanation, is highly controversial.
"You have no reason to make that assumption."
Well, if none of the events have causes, we shouldn't expect any sequence of 12 to be any more likely than any other sequence of 12.
"Well, if none of the events have causes, we shouldn't expect any sequence of 12 to be any more likely than any other sequence of 12."
Sure, insofar as we shouldn't expect anything at all. But not expecting anything at all is not the same as putting down a uniform prior over the possible states.
But if we expect the others to be the same as these, then we are in effect taking more uniform sequences to be more likely.
Do I have good reason to think that the others are also purple inside?
Reason, but not good reason.
But what you would have is an excellent opportunity to try to figure out if there was something about the boxes or their appearance that had a causal link to make the contents purple.
Some people might even call that the genesis of science...
This is an excellent set up for thinking about induction! But there is room for more reasoning and less intuition appeals.
First of all, one could argue that you do have excellent reason to suppose that at least one other box is also purple on the inside.
As long as you know that there are, in fact, 12 boxes in your office, then you know before opening any that each will either be purple on the inside or not. The probablity that you picked the only boxes which are purple on the inside by chance alone is extremely low, and easily calulateable.
Null hypothesis: the six boxes opened are the only ones purple on the inside.
1/2 (probability that the first box you opened is purple on the inside) * 5/11 * 2/5 * 1/3 * 1/4 * 1/7 = 1/924 = 0.108% probability of occuring by chance alone.
Within statistics, this is considered strong evidence for rejecting the null hypothesis. This would give strong reason for suspecting that another box is also purple on the inside.
However, you could respond by arguing that it is possible that no boxes are purple until you open them, and that only the first six boxes will turn out purple. Since these boxes were created from nothing, there would be no reason to suppose that their innards must obey the laws of ordinary objects which have causal histories. Thus, the probability of this hypothesis cannot be calculated!
This is fascinating.
"Null hypothesis: the six boxes opened are the only ones purple on the inside.
"1/2 (probability that the first box you opened is purple on the inside) * 5/11 * 2/5 * 1/3 * 1/4 * 1/7 = 1/924 = 0.108% probability of occuring by chance alone."
Yes, but you have to look at the priors of the various hypotheses when doing confirmation. And in these cases the priors are either undefined or unfavorable to the inference.
Let's say you are going to open all 12 boxes in a random order, and now you've opened six, and they're all purple inside.
We have no reason to think any one of the 2^12 sequences of purple / non-purple is more likely than any other.
Our observations of the first six rule out a lot of possible sequences. They leave behind 2^6 different sequences, each of which starts with six purples, and then may include other stuff. So after opening the first six, we know we have one of these 2^6 sequences. No one of these 2^6 sequences is any more likely than the others, given that all of this happened with no explanation. In particular, the sequence PPPPPPPPPPPP is no more likely than PPPPPPNNNNNN.
It's clear that there is something wrong with the reasoning you offer as it would lead to the reverse gambler's fallacy (thinking that if you had heads a lot times, then you'll have heads again).
I tend to agree with March Hare on this one. Especially if you do not know how the boxes got there or why several that you opened in a row happened to be purple. Mathematically speaking, it would seem like a high probability that the others are purple too. Maybe. For several reasons. I need to reffer to things from my personal experience.
1. When I was a child, I watched this children's program on TV back in the late 60's on CBC which we could pick up in Detroit. There was this magician and he would have children play this game. Several cards would be laid out face down. One of the cards would be the "Bug". If a child turned over all the cards without turning over the card which was the "Bug" he/she won the game. In other words, out of ten or so cards one was the kicker, the "Bug" the other nine were benign. I cannot remember how many cards were used in this game, it was so long ago and I was 8 at the time. I know that it was possible to pick a number of cards in a row which were benign, and then pick the next one which was the "Bug". If there were ten cards and you picked 6 in a row that were not the "Bug", yes, it may seem that mathematically you won't pick the "Bug" and that the next card you would pick would be benign. However; you've gone from initially having a 1 in 10 chance of picking the "Bug" to a 1 in 4 chance of picking the "Bug".
2. It reminds me of these promotions where if you bought the coke bottle or the candy bar with a certain number or some such you won $10,000 or something like that. It seemed to me that each bottle of coke I purchased or each candy bar I purchased always had the message on the cap or on the wrapper that said "Sorry you're not a winner." I always found it safe to assume that my next purchase would have the same result, and I was better off using my money to procure healthier fare.
3. Variable Schedule reward. Because you got six purple boxes in a row, and you want six more, you will keep picking the remaining boxes even if they turn out not to be purple. Initially the math may seem on your side, and psychologically you are hooked. Slot machines are set up to operate this way, and that is why people continue to plunk quarter after quarter in them as they are loosing more and more, because over the 6 or so initial quarters they got a token award of a dollar or two and mentally they still expect a reward.
In another theoretical case I have several boxes some with carrots in them some not. Let's say I give on purpose the first 6 boxes with a carrot in them to my horse, Merlin. He picks the next box, it is empty. The one after that is empty, but the following one has a carrot (I see to it that it does), I know he would then go through all the remaining boxes empty or not because there is an off chance it will contain a carrot. I know in order for him to keep picking boxes, I have to at random put a carrot in some of them. This would be a great way for me to mess with Merlin's head. Now Merlin always found a way to retaliate and mess with my head.
4. This is how I mess with a deer's head during hunting season. 6 times in a row I place corn, apple slices and acorns on a pile and let him eat without any trouble. I may continue that a seventh or eighth or whatever time without being there to shoot him. Mathematically the deer should be able to eat a ninth or a tenth time without trouble, but that is not my intention, because at either the the eighth, ninth or tenth time, at one of those, I'll be there to harvest him.
I wrote another response, but this issue is complex, and you may be correct.
The one issue you might not be aware of is the debate between frequentists and Baysean statistics in the role of priors in rational thought. I say this because a prime argument against frequentist methods is the "fair coin toss" example, but it is one which frequentists have given extensive responses:
http://errorstatistics.com/2012/01/08/dont-birnbaumize-that-experiment-my-friend/
Also relevant to the frequentist vs. Bayesian debate is this article. The authors state the problems they discuss come up for Bayesians, but I don't think they do.
That said, I don't actually want to say in the ex nihilo case that events should be assigned probabilities that make them independent--I was only using independence as an analogy. Rather, they should be assigned no probabilities at all, and thus no predictions whatsoever can be made--neither the inductive prediction that the probably other boxes will be all purple nor the independence prediction that probably at least one of the other boxes will be non-purple.
If we want to work with interval-valued probabilities, all the contingent outcomes in this situation should get probability [0,1].
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