@Banno

Actions speak louder than words, yes?

So what do my actions say? What do they tell you? What I value, what matters to me, that sort of thing, but also what I believe. My preferences and my expectations are two different things, and people will infer both from my actions.

If someone observes my behavior and says, "Pat seems to think that ..." I don't see how they could mean something like, "Pat is behaving like he's behaving that way." Not much of an inference, that. So what do people mean when they say, "You seem to believe X"? What do you seem like when you seem to believe something?

Some sites allow users to sort themselves in ways this one doesn't. Sub-forums for homework help, for instance. Or something labeled "for beginners" in some friendly way. We do get a regular stream of people describing themselves as new to philosophy , here to dip their toes, try out their pet theory on others, etc.

I suppose it could be an option to move certain posts into such a category, but the point is some people might use it on their own. It would be nice to have a social norm of being a bit kinder and gentler when commenting in such threads-- don't know whether moderation should have a different standard though-- and only people interested in helping others learn should be there.

(It feels like there are already too many categories here and the category system isn't used all that well, so recommending yet another category feels wrong. Still.)

You missed the process part. Sometimes you cherry-pick the evidence, and you know you're cherry-picking, and you know you shouldn't, but you do it anyway. A sort of cognitive akrasia. With others, it's easier: you just say something you know to be false. With yourself, it usually takes a more sustained effort.

In any case the take-home from psychology is that introspection is not provelidged. — Banno

Well there's this:

O wad some Power the giftie gie us
To see oursels as ithers see us!
It wad frae mony a blunder free us,
An' foolish notion:

I think Daniel Kahneman says somewhere in his book that cognitive biases are just the sorts of things we don't notice about ourselves but others do, so Burns was dead on.

Of course, that's primarily a matter of habit. We can learn to be more self-aware, more careful. We can learn to think better.

Must introspection be either infallible or non-existent? No middle ground?

What must be the case in order to successfully lie to yourself? — Moliere

I think most people's favorite method is convincing themselves, persuading themselves that they know something which they do not. (Second place is probably convincing themselves that they do not know something which they damn well do.) I'd count that as lying.

As we say goodbye to the two envelopes, I'd like to call attention to a couple oddities of the alternative analyses:

(A1) No-switchers imagine the first step as a choice or random selection between two items, and then the second step is trading that item for the one item left-- no choice.

(A2) Switchers imagine the first step as getting some determinate value-- there need not be a "choosing" here at all-- and then the second step is a random selection from two alternatives.

(B1) No-switchers see the options as getting either the smaller or the bigger of two items.

(B2) Switchers redefine smaller as "bigger of a smaller pair" and bigger as "smaller of a bigger pair".

How these symmetries arise seemingly just from the way you assign variables still puzzles me a bit.

Since Pr(L=L1) divides out, we don't need to know it. You are confusing the fact you can take a shortcut to get this result, with that shortcut being logically correct. — JeffJo

Doesn't this amount to saying that the loading of the envelopes and the selection of an envelope are independent events, in which case conditioning is pointless?

And then given that the player has no choice but to treat which envelope they "choose" as a matter of indifference-- they might as well flip a coin-- what possible reason could the facilitator have for not also treating which value is assigned to which envelope as a matter of indifference? They too could just as well flip a coin. Which leaves me puzzled about the point of your V=V1 stuff, if I'm following what you did there.

It's possible that this amounts to an ontological question, and Morris suggests that TLP is deliberately neutral on ontology, in at least some respects.

The world is structured in a certain way: (a) it divides into facts; and (b) facts have a certain structure of their own.

When we come to talk about sense-- pictures, thoughts, propositions-- we note that these are also structured in a particular way: the picture and the propositional sign are facts that each have a structure that mirrors the structure of facts, or, rather, possible facts, ways things might be, and either are or aren't. This is their sense. It is what they say. (What they cannot say, what is not part of the sense of a picture or a proposition, is the logical form itself, which they show.)

In "practical" or operational terms, this being the case means that learning about the structure of models is learning about the structure of facts and the world. Else representation is impossible. I guess the question is whether saying that facts have logical form amounts to saying facts have propositional form, are the expressions of propositions, rather than saying propositions also have logical form.

About sense, I'm not quite there yet. — Posty McPostface

We already know that "how things might stand in logical space" — what atomic facts obtain and what don't — is the sense of a picture, and thus the sense of a proposition. This is what is expressed and what is asserted to be the case.

If I assert that the cat is on the mat, what I assert is that a certain state of affairs obtains, that such-and-such is how things stand in logical space. That's the sense I'm expressing. Is there an identification of facts (broadly) and senses in the offing here? "That the cat is on the mat" is a fact, and it's a sense I can represent, express, model. The question is whether this cat's being on this mat is itself an expression of the sense of the proposition that the cat is on the mat.

Propositions and reality are said to share the logical form. Some facts are isomorphic to other facts given the right mapping (the logical form of representation). But there's some trouble here about "sense": are we saying that we get these correspondences because there are underlying structures in common, or do we in essence just project linguistic structure onto reality? We're accustomed to saying that a proposition expresses something about the world. But there's an option here to say that the fact itself is an expression of the same thing, the sense, that the proposition expresses.

Is my puzzlement clear yet?

This could use some expanding on. — Posty McPostface

I mean what LW means when he talks about correspondence:

2.13 To the objects correspond in the picture the elements of the picture.

3.2 In propositions thoughts can be so expressed that to the objects of the thoughts correspond the elements of the propositional sign.

3.21 To the configuration of the simple signs in the propositional sign corresponds the configuration of the objects in the state of affairs.

Cool. I've been reluctant to rely on an interpreter because, well, what an interpreter gives you is an interpretation. But I'm coming around. We'll just treat Michael Morris as a virtual participant in our little group. He doesn't get the last word, but he gets a word, and that's bound to be helpful.

I want to talk about these two, which should be a way into logical form:

3.1431 The essential nature of the propositional sign becomes very clear when we imagine it made up of spatial objects (such as tables, chairs, books) instead of written signs.

The mutual spatial position of these things then expresses the sense of the proposition.

3.1432 We must not say, “The complex sign ‘aRb’ says ‘a stands in relation R to b’”; but we must say, “That ‘a’ stands in a certain relation to ‘b’ says that aRb”.

One natural way to approach representation is in terms of things: a picture that shows things that actually exist is on its way to being a true picture; if it shows them related as they are in reality, or better, were at a specific time and place, then it's true. There's some of this in TLP, because we get the correspondence between the elements of the picture and the components of the facts it presents.

But what LW says is that the picture represents its sense. And in 3.1431 the arrangement of some physical things can express the sense of a proposition — and I think here we're not talking about propositions about those tables, chairs and books. I think the example we want is something more like this: you're explaining, say, how a figure can be translated in plane geometry, but drawing it on a blackboard with multiple chalk colors is just a confusing mess, so instead you cut out a shape, lay it on a piece of graph paper and then slide it from one position to another. This would be a way of using things to say what you can also say in (x, y) notation. Both are models. Both express the sense of a proposition.

What's puzzling though is that we seem to still need the isomorphism between the elements of a model and some special designated objects — which gets to my confusion over @Banno's remarks. The model is a model of something: it agrees or disagrees with what it models, represents it rightly or falsely. Do we say that what is modeled also expresses the sense of a proposition, and that the model and what is modeled agree if they express the same sense?

Gee, thanks.

I would say that defining the space as [X, 2X] just makes it painfully obvious that knowing the value of one envelope is completely useless, and that you should not bother with some X/2 or 2X conundrum. You can calculate the value of swapping before even choosing, and you will be right.

"A proposition determines a place in logical space."

SO propositions are found in logical space. — Banno

Is that what 3.4 says?

Because you can't tell, you have two options: take Y into account anyway, or ignore it. These are two perspectives on decision making, and neither really causes unpleasant surprises, because all that changes is the reference system. — Dawnstorm

But doesn't it cause trouble?

Suppose the problem is presented to you this way: one of these envelopes is worth twice the other; you get to pick one, maybe look, are offered the trade. You might begin — as I tried once — by describing the sample space as [L=2R, R=2L]. This leads to trouble. Depending on which one is true you get a different value for |L - R|: if L=2R, then |L - R| = R = L/2, and if R=2L, |L - R| = L = R/2. It gets worse if you not only use two variables but make the variables dependent on your choice. That's our Y and U. What we are quite specifically unable to know is which is bigger, and we've chosen a way of describing the sample space that is only coherent if you know which is bigger.

If you came at the problem from here, you'd realize at some point that the clever thing to do is introduce a single variable X that is orthogonal to your choice and orthogonal to which envelope has which value. |X - 2X| = X, no matter the rest. It gives you an invariant description of the sample space so that you can properly measure the consequences of your decisions.

What's disorienting is that the best way to describe the problem was given to us first, and then we are left to discover the wrong ways all on our own.

Yes, what? There were two options.

Are the elements of logical space obtaining and non-obtaining atomic facts, or are the elements of logical space the obtaining and the non-obtaining of atomic facts?

I think the latter is what's in the text, but I don't know how to understand that. Surely these obtainings and non-obtainings are not entities of some kind in addition to the atomic facts themselves. So I want to say it's the former, the atomic facts themselves, obtaining or not, that are the elements of logical space.

3. Treat X as known and Y as unknown. Then the switch gain has a distribution of X or -X with even odds, so the expected switch gain is zero. This is the approach defended by srap. The approach is coherent but it begs the question of why it is valid to model lack of knowledge about Y/X by randomness, but not lack of knowledge about X. — andrewk

This is more or less fair. As far as this part of the problem goes, I haven't gotten past my first comment on this thread, that there is a de dicto/de re problem.

(A) There is a 1/2 chance that I will pick the larger of the two envelopes.
(B) The envelope I pick has a 1/2 chance of being the larger of the two envelopes.

These may usually be functionally or instrumentally equivalent, and we might usually use the same tools to model our uncertainty, but they are still different, and this is the occasion when the difference matters. I have a 1/2 chance of picking the envelope valued at 2X, but that envelope does not have a 1/2 chance of having a larger value than the X envelope. 2X has no chance of being less than X when X > 0.

If you can show me how to respect this difference within a subjective framework, I'd be all for it.

Been away for a few days, so I'll try to get back into this.

We're going to work on logical space some more?

((urk. still don't have it.))

LW tends to talk about the logical constants this way too. I can't remember if that's in Frege, but it might be.

I don't think it is "he knows" that is implied by "the keys could be in the kitchen" but "he thinks". — Janus

We agree. I'm just making life difficult for Banno.

It's at least in "Concept and Object" (maybe that's "function"): he tries several ways of explaining the difference between a concept (or function) and an object, explains the trouble with talking about concepts (you're forced to talk about them as if they're objects), and then finally says, I can't tell you the difference but I can show you. And then that's pretty much the point of the predicate calculus: you can see the difference.

And he got it from Frege.

average total value — Srap Tasmaner

That we're driven to use such a phrase is apparently the whole problem.

Can this sort of thing be done rigorously? What would we have if we did?

You have an average total value of 22.5 and an average envelope value of 11.25. Both of those values always turn out to be wrong.

When I calculate the total value of our envelopes to be U + 10, and the average to be U/2 + 5, I'm right. Whatever U turns out to be, these calculations will turn out to be correct.

How do you calculate the total and average value of all the envelopes, including your 10? What are the numbers?

What’s the expected value of his box? — Michael

Unknown.

In DOND, after each case is opened I can tell you the total value and the average value of all the remaining cases. To the penny. With no guessing and no variables.

For DOND you accept any offer in the neighborhood of the expected payout, because the banker usually low-balls you. (There was extensive discussion among math types about whether it's a Monty Hall variant.)

Note, yet again, that all the values that could be in the cases are known from the start. There is no speculation about possible outcomes.

How did you choose an envelope in the first place?

Suppose you have chosen, perhaps by flipping a coin, if the facilitator then offers to tell you the value of either, how will you choose which value to learn? By flipping a coin?

And it doesn't bother you that if you know the value of A you want B, but if you know the value of B you want A?

We wouldn't, because we've opened an envelope in this example. I know that there's £10 in my envelope. If from this we can deduce an expected value of £12.50 in the other envelope then once we switch we have no reason to switch back. Instead we have a reason to stick — Michael

Here's a proof (which you won't accept) that opening the envelope is irrelevant, and that your reasoning should be symmetrical.

Suppose you choose an envelope and then the facilitator tells you the other envelope has $10 in it. Then you would choose not to switch because yours has an expected value of $12.50.

Eventually you recognize that you would reason the same way whichever envelope you had chosen.

The mistake could also be found in the assumption that the envelope I hold has a determinate amount X of which the values of the other envelope is derived. — Benkei

If only the amount in the first envelope, the envelope you chose and perhaps are even allowed to open, is fixed, and the second envelope is then loaded with either half or twice the amount in yours, then switching is the correct strategy. This is the variant Barry Nalebuff calls the Ali Baba problem.

Agreed. But it would be nice, knowing that the argument leads to absurdity and is therefore false, to pinpoint the step we should disallow. Like figuring out where you divided by 0 in the *proof we learned as kids that 2 = 1.

Why say that he searched the kitchen because he believed the keys were there, rather than he searched the kitchen because the keys were there? — Banno

Because you're ignoring the de dicto/de re distinction that Sapientia isn't:

That the keys were in fact there is irrelevant to why he was looking for them in that place — Sapientia

Pat doesn't know where the keys are. He searches for them in the kitchen because that is one of the places they could be. — Janus

That has a nice ring to it, but only because it is idiomatically suppressing "he knows". Without that, you'd be claiming that a possible event is causing my actions. But that's not the kind of "because" we mean here. We want the "because" of reasons. Because "know" is factive, we're once again trading on de dicto/de re ambiguity, with a an extra layer of "possibility". Maybe I only believed they might be in the kitchen. If they are actually somewhere else, we have to decide what we would mean by saying, "They are in the living room, but they might be (might have been) in the kitchen."

And I still say it has to do with our notions of rationality. If I believe my keys are in the kitchen, it is not rational, ceteris peribus, for me not to look for them there. On the other hand, knowing myself fallible, it's rational to glance around on my way to the kitchen just in case my belief is mistaken.

There are two natural and apparently sound approaches, one of which, the one you mention, produces the correct result. The puzzle is figuring out what's wrong with the other one. (Our efforts have been hampered somewhat by some people thinking the other answer is actually right.)

Have a look at the SEP article on Formal Epistemology

Here's the first paragraph:

Formal epistemology explores knowledge and reasoning using “formal” tools, tools from math and logic. For example, a formal epistemologist might use probability theory to explain how scientific reasoning works. Or she might use modal logic to defend a particular theory of knowledge.

certainty is not actually a percentage, but a range — Kaiser Basileus

This might be trouble though. I think it turns out that to go this way, you need confidence to be quantifiable. Ramsey argues for this view in "Truth and Probability", which you should read as soon as possible. The principal arguments are based on wagering, but there's also this, which I cannot resist quoting:

I am at a cross-roads and do not know the way; but I rather think one of the two ways is right. I propose therefore to go that way but keep my eyes open for someone to ask; if now I see someone half a mile away over the fields, whether I turn aside to ask him will depend on the relative inconvenience of going out of my way to cross the fields or continuing on the wrong road if it is the wrong road. But it will also depend on how confident I am that I am in the right; and clearly the more confident I am of this the less distance I should be willing to go from the road to check my opinion. I propose therefore to use the distance I would be prepared to go to ask, as a measure of the confidence of my opinion

The rest of the point being that envelopes worth less than yours, yours being worth Y, have an average value of Y/2 0, as a matter of fact. The envelopes worth more than Y have no average value.

Edit: dumbness.

P(A1)Y+P(B1)2X > P(A2)Y+P(B2)X or more importantly P(A1)Y+P(B1)2X is not equal to P(A2)Y+P(B2)X — Jeremiah

Which is to say that mean[Y, 2Y] > mean[Y/2, Y].

I've been thinking some about how this works. If you tried, as the player, to broaden your view of the situation, it might go something like this:

1. Here's 10.
2. The other envelope is 5 or 20.
3. If it's 5, I'm just as likely to have picked 5.
4. Then I'd think the other envelope has 5/2 or 10.
5. If it's 20, I'm just as likely to have picked 20.
6. Then I'd think the other envelope has 10 or 40.

You could go on, which is why I got to thinking about how going smaller converges, but going bigger doesn't. (The space does have a lower but not an upper bound, so far as you know.) Point being there's no mean value for the space as a whole. Jumping in at any point Y shows you ever increasing gains to your right and ever diminishing losses to your left. You get just a little taste of that when you try to calculate your expectation for the other envelope.