## Wednesday, April 26, 2023

### Cable Guy and van Fraassen's Reflection Principle

Van Fraassen’s Reflection Principle (RP) says that if you are sure you will have a specific credence at a specific future time, you should have that credence now. To avoid easy counterexamples, the RP needs some qualifications such that there is no loss of memory, no irrationality, no suspicion of either, full knowledge of one’s own credences at any time, etc.

Suppose:

1. Time can be continuous and causal finitism is false.

2. There are non-zero infinitesimal probabilities.

Then we have an interesting argument against van Fraassen’s Reflection Principle. Start by letting RP+ be the strengthened version of RP which says that, with the same qualifications as needed for RP, if you are sure you will have at least credence r at a specific future time, then you should have at least credence r now. I claim:

1. If RP is true, so is RP+.

This is pretty intuitive. I think one can actually give a decent argument for (3) beyond its intuitiveness, and I’ll do that in the appendix to the post.

Now, let’s use Cable Guy to give a counterexample to RP+ assuming (1) and (2). Recall that in the Cable Guy (CG) paradox, you know that CG will show at one exact time uniformly randomly distributed between 8:00 and 16:00, with 8:00 excluded and 16:00 included. You want to know if CG is coming in the afternoon, which is stipulated to be between 12:00 (exclusive) and 16:00 (inclusive). You know there will come a time, say one shortly after 8:00, when CG hasn’t yet shown up. At that time, you will have evidence that CG is coming in the afternoon—the fact that they haven’t shown up between 8:00 and, say, 8:00+δ for some δ > 0 increases the probability that CG is coming in the afternoon. So even before 8:00, you know that there will come a time when your credence in the afternoon hypothesis will be higher than it is now, assuming you’re going to be rational and observing continuously (this uses (1)). But clearly before 8:00 your credence should be 1/2.

This is not yet a counterexample to RP+ for two reasons. First, there isn’t a specific time such that you know ahead of time for sure your credence will be higher than 1/2, and, second, there isn’t a specific credence bigger than 1/2 that you know for sure you will have. We now need to do some tricksy stuff to overcome these two barriers to a counterexample to RP+.

The specific time barrier is actually pretty easy. Suppose that a continuous (i.e., not based on frames, but truly continuously recording—this may require other laws of physics than we have) video tape is being made of your front door. You aren’t yourself observing your front door. You are out of the country, and will return around 17:00. At that point, you will have no new information on whether CG showed up in the afternoon or before the afternoon. An associate will then play the tape back to you. The associate will begin playing the tape back strictly between 17:59:59 and 18:00:00, with the start of the playback so chosen that that exactly at 18:00:00, CG won’t have shown up in the playback. However, you don’t get to see the clock after your return, so you can’t get any information from noticing the exact time at which playback starts. Thus, exactly at 18:00:00 you won’t know that it is exactly 18:00:00. However, exactly at 18:00:00, your credence that CG came in the afternoon will be bigger than 1/2, because you will know that the tape has already been playing for a certain period of time and CG hasn’t shown up yet on the tape. Thus, you know ahead of time that exactly at 18:00:00 your credence in the afternoon hypothesis will be higher than 1/2.

But you don’t know how much higher it will be. Overcoming that requires a second trick. Suppose that your associate is guaranteed to start the tape playback a non-infinitesimal amount of time before 18:00:00. Then at 18:00:00 your credence in the afternoon hypothesis will be more than 1/2 + α for any infinitesimal α. By RP+, before the tape playback, your credence in the afternoon hypothesis should be at least 1/2 + α for every infinitesimal α. But this is absurd: it should be exactly 1/2.

So, we now have a full counterexample to RP+, assuming infinitesimal probabilities and the coherence of the CG setup (i.e., something like (1)). At exactly 18:00:00, with no irrationality, memory loss or the like involved (ignorance of what time it is not irrational nor a type of memory loss), you will have a credence at least 1/2 + α for some positive infinitesimal α, but right now your credence should be exactly 1/2.

Appendix: Here’s an argument that if RP is true, so is RP+. For simplicity, I will work with real-valued probabilities. Suppose all the qualifications of RP hold, and you now are sure that at t1 your credence in p will be at least r. Let X be a real number uniformly randomly chosen between 0 and 1 independently of p and any evidence you will acquire by t1. Let Ct(q) be your credence in q at t. Let u be the following proposition: X < r/Ct(p) and p is true. Then at t1, your credence in u will be (r/Ct(p))Ct(p) = r (where we use the fact that r ≤ Ct(p)). Hence, by RP your credence now in u should be r. But since u is a conjunction of two propositions, one of them being p, your credence now in p should be at least r.

(One may rightly worry about difficulties in dropping the restriction that we are working with real-valued probabilities.)

William said...

With some tweaking this could look a little like the Banoch-Tarski paradox, because I am pretty sure that with such tweaks, if you add up your infinitesimal probabilities over the interval using this reasoning, the total will be much more than 1 -- kind of like making two balls from one by manipulating infinitesimal amounts.

Rob said...

I'm sure this question is due to my ignorance, but in the CG paradox how does one know that there will be a time shortly after 8:00 AM when the Cable Guy hasn't shown up? If there was, let's say, a second after 8:00 when you knew the Cable Guy wasn't going to show up, then presumably the original time frame for his arrival should be from 8:00 and 1 second, instead of simply 8? Is the idea that, given the continuity of time, any time at which he arrived there would be some infitessimal amount of time between 8:00 and his arrival? If so, why could he not come at precisely 8:00 AM?

Daniel Lindquist said...

Unknown: He can't show up precisely at 8 because the timeframe he was stated to show up in is "8:00 excluded". But I think you're right that we can't know that there will be a time after 8:00 and before he shows up: if there was such a time, he could have shown up at that time instead, so we can't know of that time that it is a time when he will have not shown up yet. So I think this doesn't cut against Van Fraasen at all: the initial setup just gives us even odds for Cable Guy showing up at any moment between 8am and 4pm (exclusive of the first and inclusive of the second), meaning it's 50/50 whether he shows up in the afternoon or the morning, and the Reflection Principle doesn't do anything to suggest we should think otherwise than just that.

IanS said...

One way of stating the reflection principle is that, with the usual provisos, you should defer to your future self. In this example, that would be your self at 18:00. The catch is that you won’t know when it is 18:00. So it’s not so clear that reflection should apply. Or so I suggest…

You might imagine recording your credences on a (continuous) graph while an external observer who knows the time looks on. The observer would be sure to find that at 18:00 your credence was a real amount greater that 1/2. But the observer would be relevantly different from you, because she would know the time while you would not.

The only relevant time you can know is elapsed time (or equivalently, the time stamp) on the video. Arguably, that’s the only time that can define a future self to which the reflection principle should apply. And of course, it’s not true that at a given elapsed time, your credence would necessarily be greater than 1/2 – you might have seen the cable guy arrive in the morning.

You might arrange for someone to tell you when it was 18:00. Then you could apply reflection to ‘your future self at 18:00’. But that changes the setup, and then your credence need not be greater than 1/2. You would know that the start of the video was timed so that you would not see the cable guy arrive by 18:00, so not having seen him by then would give you no new information. The elapsed time on the video would give new information, but it would not be clear how to use it – if it were a tiny fraction of a second, you might reasonably suspect that the cable guy had arrived very soon after 8:00, so your credence might be less than 1/2.

Alexander R Pruss said...

Daniel:
I don't follow. If he can't show up at 8 and most show up some time later and time is continuous, then there will be a time after 8 and before he shows up: for instance if he shows up at t, then the moment half way between 8 and t is a moment he doesn't show up. Of course we don't know ahead of time when that moment will be. But that's the point of the videotape trick

Alexander R Pruss said...

Ian:

Are you proposing that a qualification to add to the RP is that you know what exact time it is at the time when you're being taken as an expert by your old self?

That seems too much to require.

IanS said...

Well, yes…, that’s what I’m suggesting, at least in cases like this where the exact time matters. In many cases, it doesn’t, which is no doubt why the qualification is rarely (never?) stated explicitly. (Or so I suggest…) For example, in setups with discrete steps, it’s the step that matters, not the clock time.

Not knowing the ‘time’ (or the step) to the relevant accuracy can be considered as self-locating uncertainty. Think about Sleeping Beauty. If she could glance at her day/date watch when she was woken, there would be no controversy, and no conflict with reflection. It’s precisely because she can’t that ‘thirding’, which is inconsistent with reflection, is plausible. I’m suggesting that your setup is similar.

To be clear, I don’t have great confidence in any of this. It may well be way off-beam. But it’s what I’m thinking for now.