Probability that there are two elephants given one on the left and one on the right.
In any case, if your language can't express Fermat's last theorem then of course you don't assign a probability of 1 to it, not because you assign it a different probability, but because you don't assign it a probability at all.
I agree. I am saying that we need not assign it a probability at all. Your solution assumes that there is a way to express "two" in the language. Also, the proposition you made is more like "one elephant and another elephant makes two elephants" not "1 + 1 = 2".
I think we'd be better off trying to find a way to express 1 + 1 = 2 as a boolean function on programs.
Here's the slides from my talk on logical counterfactuals at the Cambridge/MIRI workshop in May 2015. I'm planning to give a similar talk tomorrow at the Google Tel Aviv office (meetup link). None of the material is really new, but I hope it shows that basic LWish decision theory can be presented in a mathematically rigorous way.
This is super interesting. Is this based on UDT?
Basically the problem is that a Bayesian should not be able to change its probabilities without new evidence, and if you assign a probability other than 1 to a mathematical truth, you will run into problems when you deduce that it follows of necessity from other things that have a probability of 1.
How do you express, Fermat's last theorem for instance, as a boolean combination of the language I gave, or as a boolean combination of programs? Boolean algebra is not strong enough to derive, or even express all of math.
edit: Let's start simple. How do you express 1 + 1 = 2 in the language I gave, or as a boolean combination of programs?
Except that around 2% of blue egg-shaped objects contain palladium instead. So if you find a blue egg-shaped thing that contains palladium, should you call it a "rube" instead? You're going to put it in the rube bin—why not call it a "rube"?
But when you switch off the light, nearly all bleggs glow faintly in the dark. And blue egg-shaped objects that contain palladium are just as likely to glow in the dark as any other blue egg-shaped object.
So if you find a blue egg-shaped object that contains palladium, and you ask "Is it a blegg?", the answer depends on what you have to do with the answer: If you ask "Which bin does the object go in?", then you choose as if the object is a rube. But if you ask "If I turn off the light, will it glow?", you predict as if the object is a blegg. In one case, the question "Is it a blegg?" stands in for the disguised query, "Which bin does it go in?". In the other case, the question "Is it a blegg?" stands in for the disguised query, "Will it glow in the dark?"
This is amazing, but too fast. It's too important and counter intuitive to do that fast, and we absolutely devastatingly painfully need it in philosophy departments. Please help us. This is an S.O.S. our ship is sinking. Write this again longer, so that I can show it to people and change their minds. People who are not lesswrong litterate. It's too important to go over that fast, anyway. I also ask that you, or anyone for that matter, find a simple real world example which has roughly analogous parameters to the ones you specified, and use that as the example instead. Somebody do it [please, I'm too busy arguing with philosophy proffesors about it, and there are better writers on this site that could take up the endeavor. It would be useful and well liked anyway chances are, and I'll give what rewards I can.
Here's a question, if we had the ability to input a sensory event with a likelyhoodratio of 3^^^^3:1 this whole problem would be solved?
What if the universe permits hyper-computation?
Hmm, it depends on whether or not you can give finite complete descriptions of those algorithms, if so, I don't see the problem with just tagging them on. If you can give finite descriptions of the algorithm, then its komologorov complexity will be finite, and the prior: 2^-k(h) will still give nonzero probabilities to hyper environments.
If there are no such finite complete descriptions, then I gotta go back to the drawing board, cause the universe could totally allow hyper computations.
On a side note, where should I go to read more about hyper-computation?
At first thought. It seems that if it could be falsified, then it would fail the criteria of containing all and only those hypotheses which could in principle be falsified. Kind of like a meta-reference problem; if it does constrain experience, then there are hypotheses which are not interpretable as causal graphs that constrain experience (no matter how unlikely). This is so because the sentence says "all and only those hypothesis that can be interpreted as causal graphs are falsifiable", and for it to be falsified, means verifying that there is at least one hypothesis which cannot be interpreted as a causal graph which is falsifiable. Short answer, not if we got it right this time.
(term clarification) All and only hypotheses that constrain experience are falsifiable and verifiable, for there exists a portion of experience space which if observed falsifies them, and the rest verifies them (probabilistically).
I have to ask, how does this metaphysics (cause that's what it is) account for mathematical truths? What causal models do those represent?
My bad:
Someone already asked this more cleverly than I did.
I have a plausibly equivalent (or at least implies Ey's) candidate for the fabric of real things, i.e., the space of hypotheses which could in principle be true, i.e., the space of beliefs which have sense:
A Hypothesis has nonzero probability, iff it's computable or semi computable.
It's rather obviously inspired by Solomonoff abduction, and is a sound principle for any being attempting to approximate the universal prior.
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Terminology quibble:
I get where you get this notion of connotation from, but there's a more formal one that Quine used, which is at least related. It's the difference between an extension and a meaning. So the extensions of "vertebrate" and "things with tails" could have been identical, but that would not mean that the two predicates have the same meanings. To check if the extensions of two terms are identical, you check the world; it seems like to check whether two meanings are identical, you have to check your own mind.
Edit: Whoops, somebody already mentioned this.