Opening remarks / Symmetry structure & non-supervenient relations

0:00 This is the higher point. If it is the higher point, then he will have a very high probability for reaching the promised land. If it is the point at the bottom there only, he will have a low probability. So the question is now, what should we think now? What should be the expectation of arrival there now? And the proposition that we can prove from the Fox Slip is this, that supposing that at the future time you still have the same certainties, which means that if you give something probability one now, you still do at a later time. You don't forget your certainties. Then the reflection principle follows. Then if you have the Fox Slip, if you obey the Fox Slip, and you retain your certainties over time, then the reflection principle in the day, are you okay. Now, the question is, do I have time for a few? What time is it? How much time is left? Well, you have 36 minutes. 26 minutes. Why don't I ask to see if there's questions about it, or discussion about this, and I can give you the two papers, or otherwise I'm leaving. Different questions, sort of fun, that you can do, or is it true? It's a pretty serious question. Yeah? Thank you. So, here's the box with the pen. With its non-English construction, my current expectation of a quality whose value depends on what my opinion will be, that opinion had knowingly at a later time, cannot exceed the expectation that I could have from knowingly of that opinion. So I can put this out. Imagine to be in a position where you have this opinion and you know that you have this opinion, and you ask yourself, what's my expectation now? What's my expectation value? This is one of the things that we could argue about, whether this partiality is plausible, acceptable, and so on. And as I pointed out, the cannot exceed, can be actually thanked to, must be the same as.

2:30 So then I give you a lemma, which is that supposing myself to be knowingly of any opinion whatsoever, there's a certain quality Q, it's just depending on what my opinion is then, but that will certainly have the expectation value zero for me. So I define this quality. I call it the AXD indicator. It says if my probability for A is not X, then this quality If it is x, then it depends on A. And it will take the value of 1 minus x if A is true and minus x if A is not true. And then if you calculate the expectation value for 2, you will find that it's certainly going to be 0, because you get 0 plus x times 1 minus x. So this is the, this quantity is an indicator for the statement my probability tomorrow or probability at that time for A equals X. And if any condition whatsoever, that quantity will have expectation value zero for you. So now, according to the postulate, it must have expectation value zero for you now. All right, so here's the proposition that the fraction will now follow, and the argument is as follows. Well, we know that the expectation value that we must have for this formula is zero, but we can also directly calculate directly my expectation value for that formula now. And it will be, first of all, the probability that my probability for A is not going to be x times the value zero, which we would count then, plus One minus x, the probability that, yes, I do have an opinion and A is true, plus minus x, times the probability that I have an opinion and A is false. It's just a direct calculation of expectation value in the same way that Huygens was outlining.

5:00 And so, according to the postulate, that must be the same then as what we found in the lemma. And finally, the formula that a conjunction gets the probability of the first conjunction times the probability that the second conjunction has the first, we can, assuming that the probability of x is not zero, of course, we can translate the expectation value that I was calculating. Let me just go back for a moment. One here at the bottom, right? Here we have a conjunction, the probability of this conjunction, PTA equals X and A. And we're going to translate this into the probability of PTA equals X times the probability of A given that PTA equals X. And we get the very simple version, which is that we call this alpha and beta, that alpha beta minus X alpha. That's my current expectation value. According to the postulate, it must equal zero as well, but that just means that X is alpha, beta, divided by alpha, which is beta, which is the probability of A given that PPA equals X. I think perhaps I should go back to the postulate and then you can think about whether you're willing to accept this postulate. Thank you very much.

7:30 So what's the relationship with conditionalization? Reflection does not imply conditionalization. Somebody who is a Bayesian conditionalizer will certainly obey reflection. But no, this is meant to allow for other changes in your opinion. Little revolutions, so to speak, where you're trying to update normally, but you get evidence that... It makes your opinion very unattractive to you and you say something is going totally wrong. I have to revise. I throw away my prior. I try something else. That's allowed on the reflection. You can have little scientific evolutions in your head, so to speak, and still obey the reflection, provided only, of course, that you don't have a rule that predicts those things. But in order to satisfy reflection and not conditionalize, then your belief in the probability of conditionals would have to change. Of conditionals? Meaning conditional statements, event statements? Yeah. I'm treating event statements like any other statement. I'm not giving any... Nothing like Stonnecker's equation of conditional probability over the probability of a condition, you know. Does it generalize the frequency for something else than probability? Because you make this, you're making this link between symmetries and the theory about symmetries of what we can think of. Yeah, okay, so if an actual symmetry is as follows. First of all, the form of argument is a symmetry, right? It's got the style that... To say, if you have the function that goes from here to here, and the situation is essentially similar by a transformation to a function that goes from here to here, then this root must give you the same value as that root, right? It's a general form of argument, which is, you know, which substantiates symmetry thinking, so to speak.

10:00 The problem stays the same under certain transformations and so Eugen's idea both for collisions and for games of chance was I know something to which I can equate any problem, a kind of every problem is equivalent, a transformation of the problem to one that we can solve. But there's another connection too which is that the... There is also a symmetry argument for normal updating. Now, as I was just explaining to James, the reflection principle is supposed to be something that still holds, even if you think that you might not be able to do a normal updating, namely if you get evidence that it's so recalcitrant for you that you think there's something wrong with it. Two questions. Are you certain that you don't fall prey to a version of the Dutch Poop argument? Yes, I am. Because the reflection principle was actually first developed precisely in order to show that certain kinds of Dutch Poop arguments, which extend the usual time, are still avoided. So, you know, there's a version, there's a strong version of that group of arguments that goes, that uses your picture opinion. So the bookie says to you today, let's talk about what your opinion might be tomorrow morning. And let's make a bet on conjunctions of your opinion tomorrow morning is this and what happens is that. It gets money out of you. So, yes, it's exactly designed to avoid these strong, detrimental items. This is more so open to me than the original question, but obviously this is to you.

12:30 Why is this more so open to you? Well, you see, of course, if I just imagine myself in the position where I have this opinion, Right? Then I can just use my present means of calculating expectation values. In other words, I can say, yeah, point in spot, that's how to do that. And so now I'm saying that the question about what I think about my future, I can reduce to a question just that is an exercise of my current. Right? Now, oh yes, now the question is, is this possible? Is this possible? And I think that it is plausible, well, I think that it has this implication that someone who holds this, someone who, I should say, satisfies this principle, is not necessarily someone who has a great deal of confidence in his future opinion, but he expresses a commitment to manage it as well as he can. What if his ability to manage it is different from his ability to manage it? Well, you see, predictably or unpredictably. When it comes to unpredictably, then of course, we have what I said before, that as long as he says, as a result of my illnesses or drunkenness and so on, my opinion can get, my probability can get too high or too low, there's no conflict. But if he says, I can predict. Something like this. I will be drunk tomorrow morning. Well, maybe you don't get drunk in the morning. I will be drunk this evening, right? And when I'm drunk, I am overconfident about my driving, and so I will be of the opinion that I can safely drive home. And that will be a mistake. I suppose you said that. Then I would say, well, what you are saying now shows that you don't have a commitment to manage your opinion very well.

15:00 Because you are actually saying that you're going to put yourself in conditions where you can't manage your own people. If I try to work out a theory of rational opinion, I'm going to of course allow that there are people who, you know, do not do that. Thank you very much for your time and I look forward to working out your expectations in the next few minutes. Yes. Well, again, you see, the question is predictably or not predictably. If it's predictable, then you should be calibrated. So, for instance, someone says, you know, I know when it's going to rain because my chill veins ache. I don't actually know what chill veins are, but people say that. And then after a while, he begins to realize that his chill veins ache more often. They become, you know, they come too often. In that case, the opinion he forms on the basis of his pain should be recalibrated, he should start lowering the probability of rain given the pain, you see? And again, if it's unpredictable, then of course it can go either up or down. Now, but there are paradoxes, you're right, there are paradoxes in this neighborhood. I think one form that I think illustrates the sort of general thing is this, I saw this actually as a bit of graffiti once. It said, materialism is due to a dietary deficiency. I think it's very much actually due to a dietary deficiency. I said this once in the... I gave this example once in an audience with mostly physicists, and one of them maybe said, that's a contradiction. You're contradicting yourself. Of course, it didn't really a contradiction, but you couldn't very well have a materialist who held that this materialism was just due to the idea of efficiency. But I think in the same way, you know, if I express opinions about my future opinion, And I also, in these opinions, include things about how uncontrollably, in ways that have to do with, say, physical conditions and so on, they will go in a certain direction, and that I will not be able to manage them.

17:30 There's something very strange about that, you know? You won't say to the person, well, come on, you just told us what's going to happen. So in your, you know, reasoning, you can take into account what you've just told us. Well, let's take a bow.