When you have exactly one non-refuted theory, you go with that.
The other cases are more complicated and difficult to understand.
Suppose I gave you the answer to the other cases, and we talked about it enough for you to understand it. What would you change your mind about? What would you concede?
If i convinced you of this one single issue (that there is a method for making the decision), would you follow up with a thousand other objections to Popperian epistemology, or would we have gotten somewhere?
If you have lots of other objections you are interested in, I would suggest you just accept for now that we have a method and focus on the other issues first.
[option 1] since there is no such thing as confirmation, it turns out that no conjecture is better or worse than any other.
But some are criticized and some aren't.
[option 2] conjecture that best weathered the criticisms you were able to muster
But how is that to be judged?
No, we always go with uncriticized ideas (which may be close variants of ideas that were criticized). Even the terminology is very tricky here -- the English language is not well adapted to expressing these ideas. (In particular, the concept "uncriticized" is a very substantive one with a lot of meaning, and the word for it may be misleading, but other words are even worse. And the straightforward meaning is OK for present purposes, but may be problematic in future discussion.).
Or is it something radically different from these two altogether?
Yes, different. Both of these are justificationist ways of thinking. They consider how much justification each theory has. The first one rejects a standard source of justification, does not replace it, and ends up stuck. The second one replaces it, and ends up, as you say, reasonably similar to Bayesianism. It still uses the same basic method of tallying up how much of some good thing (which we call justification) each theory has, and then judging by what has the most.
Popperian epistemology does not justify. It uses criticism for a different purpose: a criticism is an explanation of a mistake. By finding mistakes, and explaining what the mistakes are, and conjecturing better ideas which we think won't have those mistakes, we learn and improve our knowledge.
If i convinced you of this one single issue (that there is a method for making the decision), would you follow up with a thousand other objections to Popperian epistemology, or would we have gotten somewhere?
Yes, we will have gotten somewhere. This issue is my primary criticism of Popperian epistemology. That is, given what I understand about the set of ideas, it is not clear to me how we would go about making practical scientific decisions. With that said, I can't reasonably guarantee that I will not have later objections as well before we've even had ...
I was directed to this book (http://www-biba.inrialpes.fr/Jaynes/prob.html) in conversation here:
http://lesswrong.com/lw/3ox/bayesianism_versus_critical_rationalism/3ug7?context=1#3ug7
I was told it had a proof of Bayesian epistemology in the first two chapters. One of the things we were discussing is Popper's epistemology.
Here are those chapters:
http://www-biba.inrialpes.fr/Jaynes/cc01p.pdf
http://www-biba.inrialpes.fr/Jaynes/cc02m.pdf
I have not found any proof here that Bayesian epistemology is correct. There is not even an attempt to prove it. Various things are assumed in the first chapter. In the second chapter, some things are proven given those assumptions.
Some first chapter assumptions are incorrect or unargued. It begins with an example with a policeman, and says his conclusion is not a logical deduction because the evidence is logically consistent with his conclusion being false. I agree so far. Next it says "we will grant that it had a certain degree of validity". But I will not grant that. Popper's epistemology explains that *this is a mistake* (and Jaynes makes no attempt at all to address Popper's arguments). In any case, simply assuming his readers will grant his substantive claims is no way to argue.
The next sentences blithely assert that we all reason in this way. Jaynes' is basically presenting the issues of this kind of reasoning as his topic. This simply ignores Popper and makes no attempt to prove Jaynes' approach is correct.
Jaynes goes on to give syllogisms, which he calls "weaker" than deduction, which he acknowledges are not deductively correct. And then he just says we use that kind of reasoning all the time. That sort of assertion only appeals to the already converted. Jaynes starts with arguments which appeal to the *intuition* of his readers, not on arguments which could persuade someone who disagreed with him (that is, good rational arguments). Later when he gets into more mathematical stuff which doesn't (directly) rest on appeals to intution, it does rest on the ideas he (supposedly) established early on with his appeals to intuition.
The outline of the approach here is to quickly gloss over substantive philosophical assumptions, never provide serious arguments for them, take them as common sense, do not detail them, and then later provide arguments which are rigorous *given the assumptions glossed over earlier*. This is a mistake.
So we get, e.g., a section on Boolean Algebra which says it will state previous ideas more formally. This briefly acknowledges that the rigorous parts depend on the non-rigorous parts. Also the very important problem of carefully detailing how the mathematical objects discussed correspond to the real world things they are supposed to help us understand does not receive adequate attention.
Chapter 2 begins by saying we've now formulated our problem and the rest is just math. What I take from that is that the early assumptions won't be revisted but simply used as premises. So the rest is pointless if those early assumptions are mistaken, and Bayesian Epistemology cannot be proven in this way to anyone who doesn't grant the assumptions (such as a Popperian).
Moving on to Popper, Jaynes is ignorant of the topic and unscholarly. He writes:
http://www-biba.inrialpes.fr/Jaynes/crefsv.pdf
> Karl Popper is famous mostly through making a career out of the doctrine that theories may not be proved true, only false
This is pure fiction. Popper is a fallibilist and said (repeatedly) that theories cannot be proved false (or anything else).
It's important to criticize unscholarly books promoting myths about rival philosophers rather than addressing their actual arguments. That's a major flaw not just in a particular paragraph but in the author's way of thinking. It's especially relevant in this case since the author of the books tries to tell us about how to think.
Note that Yudkowsky made a similar unscholarly mistake, about the same rival philosopher, here:
http://yudkowsky.net/rational/bayes
> Previously, the most popular philosophy of science was probably Karl Popper's falsificationism - this is the old philosophy that the Bayesian revolution is currently dethroning. Karl Popper's idea that theories can be definitely falsified, but never definitely confirmed
Popper's philosophy is not falsificationism, it was never the most popular, and it is fallibilist: it says ideas cannot be definitely falsified. It's bad to make this kind of mistake about what a rival's basic claims are when claiming to be dethroning him. The correct method of dethroning a rival philosophy involves understanding what it does say and criticizing that.
If Bayesians wish to challenge Popper they should learn his ideas and address his arguments. For example he questioned the concept of positive support for ideas. Part of this argument involves asking the questions: 'What is support?' (This is not asking for its essential nature or a perfect definition, just to explain clearly and precisely what the support idea actually says) and 'What is the difference between "X supports Y" and "X is consistent with Y"?' If anyone has the answer, please tell me.