Dialogue on anti-induction.

post by momom2 (amaury-lorin) · 2021-08-26T23:21:19.326Z · LW · GW · 1 comments

My uncle with whom I shared thoughts on anti-induction remarked that humans are systematically anti-inductive in some situations : he gave the example of gambling, where people can think that losing a lot in a row means that they are poised to win soon.
But this is not a fair example in my opinion, because gamblers are not consciously anti-inductive : when their behavior is exposed as such, they do not defend their decision.
Among my relatives, the gamblers are notoriously irrational. A bayesian might say that a long streak of wins is very weak evidence that they will keep on winning, because they have a strong prior confidence in mathematical analysis of the game, but that hardly tells us anything about how anti-induction arose in the first place.

The following dialogue is intended to showcase a (moderately) intelligent anti-inductor in action, to try to understand the anti-inductors by putting myself in their shoes.

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Alice is an anti-inductor. She intuitively believes that things that have happened often typically don't happen again.
Aware of that fact, and of the existence of inductors, she has tried to look into anti-induction to know what it really means to her, and if it is an intuition she should abandon.

She has a friend, Iris, who is an inductor. They are both tentatively rational, as intelligent as each other (and coincidentally about as dumb as me).

Iris : See, when using induction, I have often been right, and you have often been wrong using anti-induction.
Alice : Then, your induction probably tells you that it means induction is right and anti-induction is wrong... How interesting.
I : I am aware of that, and I can accurately modelize your own view, including your modelization of me, and it's reciprocal, and I know you know I know... Let's accept that and move on.
A : We are very similar, in that our goals and cognitive patterns are the same, except when it comes to induction. We also have shared knowledge. I accept on these grounds that I might be wrong about anti-induction, with a strong negative emotional bias of course.
I : So do I. I care about you and want to make you recognize the good of induction for your own well-being. By symmetry, we will search for a non-(anti-)inductive argument to ground (anti-)induction, and we will try our best not to implicitly found our arguments on (anti-)induction.
A : How about... We accept there are inductors and anti-inductors. Let's approximate being right with fitness. This approximation is reasonable because anti-induction is used for very basic things, such as the sun not shining tomorrow. I expect inductors to die from starvation when they falsely believe that eating will feed them after they have done it a thousand times. Likewise, you expect anti-inductors to die from starvation from refusing to eat when they are hungry.
I : Exactly so ! Now, look : we have diverging conclusions about the state of the world. Let us observe it and crown one of the two competing theories !

Were I a non-inductive spirit, I would have perhaps no reason to fill my fictional world with inductors rather than anti-inductors (thereby supporting one side a priori), but I am not.
This does mean however that the following reasoning is only valuable insofar as it modifies my behavior relative to the real world and not a mere possible one (in some possible worlds, I would not update on counterfactually fictional evidence).

A : We have observed without bias in samples or experiment, and it has been shown everyone on fictional-Earth except for me is an inductor. Wow, I did NOT expect that !
I, triumphantly : You know the bias of being surprised at your own failures. Now, be fair and accept to change your mind and join the inductive side of the force !
A, thoughtfully : You could have worded it differently... Is there a meaningful difference between what you said and "Now, because of the laws of rationality, you must change your mind etc." ? In that case, I will argue that the laws of rationality have had this implication everytime a similar reasoning followed a similar experiment indeed... Which seems to prove that we cannot think in the same way this time !
I : There are no "laws of rationality", there are only the actual laws of rationality ! How could correct methods of thinking change from one experiment to the other ?
A : Well, it has definitely never happened before.
I : The fact that anti-induction support a contradictory proposition does not mean that the original one relies on induction, in general and in this particular case.
A : Although I don't see why what you just said would be true it sounds very reasonable, so let's accept it and continue our research.
I : What is there to continue thinking about ? We have shown that induction improves fitness, and we have previously agreed that the fittest heuristic would be the truest.
A : Yes, but that is only a true fact about the past. Induction has managed to prevail until now (rather, until the moment we observed the people of the world), but how do we know it will remain the best heuristic tomorrow ? Or in five minutes ? Heuristics like this one are as fundamental as modus ponens, of the kind that does not change on small scales of time, and tend to be invariant general fitness (dis)advantages.
A : (As a side note, we both feel like it's more than a mere behavioral heuristic, but rather a logical truth, but we cannot find any supportive reasoning.)
I : So what ?
A : So I expect this particular one to be unlike all the other ones. This is anti-induction alright, but is it false ?
I, after a moment of reflection : You used very vague wording, and perhaps you think there's no flaw because you are confused. Can you detail your proposition ? Perhaps tabooing words like "other" and "particular".
A : So, I expect anti-induction to be true tomorrow, since induction was true until five minutes ago, because (anti-)induction belongs to the reference class of fundamental behavioral heuristics, like will to survive or will to reproduce. The kind of things that we expect every single living being to have because anything else is simply improbable to happen to live.
I : Geez... I'm still not sure it's not just confusion, but I see why I can't spot a mistake : (anti-)induction is messy, does not yield logical certainty, and uses ill-defined reference classes. Using arbitrary classes can make a committed (anti-)inductor believe anything, there are just too many potential reference classes with interesting properties out there.
A : And yet our reference classes are not arbitrary. We would reject a reference class used for induction comprising gems blue before today and green after. My point stands very strongly if "fundamental behavioral heuristics" is a very legitimate class.

None of my tentatives to formalize induction in formal logic led to interesting results. Please tell me if you know a paper on the subject !
So far, my best definition is that there is an interest function of propositions (personal ?) and interesting reference classes, defined as {x : P(x)} for an interesting property P. Induction states that if A is an interesting reference class, 

Vx€A, (O(x)=>Q(x)) -> Vx€A, Q(x) 

where Q is interesting and O(x) means "x is observed", assuming a meaningful "observed" property.

I : This is starting to become too subjective, so I'll try a new approach. You said earlier that inductors should die of starvation. Please tell me, Alice, how is it that you managed to survive until now ?
A : I had not thought about the absurdity of eating. From now on, I will not eat anymore when I am hungry.
I : Funny how the thought did not occur to you before... Perhaps your system 2 is anti-inductor, and your system 1 is inductor ?
A : Perhaps, but we have little evidence in favor of that, it's just a wild conjecture, perhaps motivated by your desire to prove anti-induction wrong. I am fictional anyway, so it's not evidence that there is no rational anti-inductor.
I : Tell me, Alice, how is it you believe the word induction is used in the same meaning it had since the beginning of our conversation ?
A : By the double-barrelled jumping jiminetty and flying spaghetti monster !

Alice refused to partake further in the debate, troubled. Thank the lord she did not think about the fact that she had used anti-induction until now ! She would then have legitimately expected to stop believing her truth any moment, without any explanation as to why ! Fated to be wrong like someone who receives overwhelming evidence they are a Botzmann brain is fated to stop thinking ! In that sorry case, she would weep for her soon to be lost love for truth, and digging further, her soon to be lost consistent utility function (for she followed one until now).

I : Now, that is a devastating argument from my viewpoint : that anti-induction leads to systematic uncertainty. Unfortunately, that is conditional on the fact that obervedly-unchanging truth is unchanging, which I suspect is an inductive system 1 reasoning.
I : I predict that anti-inductors will fail to think, and anti-inductors incapable of self-reflection would take me up on that bet while anti-inductors capable of self-reflection will fail to think, like Alice. Now, I just need to observe the world and see whether people actually think consistently, right ? Or maybe bet a lot of fitness on that ?
I : Another potential argument is the fact that Alice cannot explain how she managed to avoid starvation until now. Perhaps induction allows me to explain more facts of the world than anti-induction ? The big issue there is a posteriori explanation, since it may not be enough to deduce knowledge of the future without using (anti-)induction.

I : I sure hope my reasoning was not too motivated by the fact that I actually believe induction to be superior to anti-induction... As long as Alice was there, at least she could compensate by being equally motivated to defend anti-induction, but what would she answer to the above arguments ?

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