What should you do? — Jeremiah

It depends on my utility function U. Utility functions map one's wealth to levels of 'utility', ie satisfaction. They are generally assumed to be monotonically increasing and concave down.

The expected dollar gain from switching after observing the envelope contains $X is positive, as it is 0.5 (2X - X) + 0.5 (X/2 - X) = 0.25 X

However the expected gain in utility from switching is

(0.5 U(W + 2X) + 0.5 U(W + X/2)) - U(W + X)

where W is my wealth prior to receiving the envelope.

This may be positive or negative, depending on the slope and curvature of the utility function U in the interval between S+X/2 and W+2X. If the slope is high and curvature is strong, expected utility will be maximised by not switching, otherwise by switching.

I'd draw a diagram, only I'm no good at doing that on a web page. But if you draw a few diagrams yourself, hopefully you'll see what I mean.

OK, with that information, we can conclude that it is consistent.

One way to understand that is to consider the interpretation where above(x,y) is always true, for any x and y (eg 'above' means 'is a block'). Then the axiom defining 'above' is true for all x, y and z without having to use the 'on(x,z)' part of the disjunction. Hence there is no relationship at all between 'on' and 'above'.

We could enforce the relationship by adding another item to the disjunction on the RHS of the axiom that defines 'above', viz:

∀x.∀z.(above(x,z) ⇔ ~on(z,x) ∧ (on(x,z) ∨ ∃y.(above(x,y) ∧ above(y,z))))

With that revised axiom, above(c,a) makes the theory inconsistent because it enables us to deduce above(c,b), which then conflicts with the axiom on(b,c).

The principle in cases like these is to remove all prior associations you have in your mind with the words that are used, because they will inject implicit assumptions into your analysis. If you find that hard to do, replace the words 'on' and 'above' by meaningless labels like 'P' and 'Q'. Once you've done that, consider various different truth tables ('interpretations') for P and Q that satisfy all the given axioms. If any of them make the statement you are testing true, then the statement does not make the system inconsistent.

Are you saying that your axiom set is

{<above axiom>, on(a,b), on(b,c)}

where <above axiom> denotes the relation you wrote in the OP with the two forall quantifiers,
or that it is

{<above axiom>, on(a,b), on(b,c), ~on(b,a), ~on(c,b), ~on(a,c), ~on(c,a)}

Or is the Block World a Model for the theory given by the axioms, such that the axioms are

{on(a,b), on(b,c)}

or

{on(a,b), on(b,c), ~on(b,a), ~on(c,b), ~on(a,c), ~on(c,a)}

and the relation in the OP is not an axiom but rather a feature of the model that is not entailed by the theory?

The question of consistency depends on the answers to these questions. In particular, if ~on(c,a) is not provable then it is possible for on to mean 'to the left of' and the blocks to be arranged in a ring, so that every block is 'on' one other and every block is 'above' every other one.

If you don't know what a Model is, don't worry, instead ask the question whether what you have described as a 'relation' in the OP is supposed to be an axiom. If not then what is it?

We can in principle, and there are real-life examples of this. One of them is ancient: a human is not [just] a featherless biped. — Snakes Alive

We can take a feather that has fallen off a bird, and use it, in the absence of any legs, to establish the concept of a feather. Then we can develop the concept of a featherless being as one that has no feathers. Thus we have separated the notion of feather from that of number of legs. That's what I was referring to when I mentioned the diversity of examples in our world.

If it were not possible to do that, 'featherless biped' would be a perfectly correct definition of a human.

In the hypothetical world I was describing, we can't separate the notion of green from glossy if everything that is green is glossy and everything that is glossy is green.

A class isn't one property, it's all the properties shared by a group. — Marchesk

In mathematics properties are sometimes defined as the equivalence class of all objects possessing that property. For instance, once can define the class 'three' as the collection of all sets that have two elements.

This raises an interesting question: If the only green things in the universe were also glossy (as opposed to matt), and no non-green things were glossy, would we be able to develop separate concepts of green and of glossy? I suspect we would not.

The way we learn properties when we are learning language as children is by looking at examples with and without the property, and being told words for them, until we get the idea that the word for all the things with the property relates to that shared property. That learning technique (and I can't imagine any other) would not be available in a world where green <--> glossy.

It seems reasonable to me to say that in the real world as well as in maths, a universal is the set of all objects that have the relevant property.

Because our world is so richly diverse, we can always find examples where two commonly-associated properties are not shared by an object, to allow us to distinguish between the two.

I return to the distinction I started with - between rational meaning and emotional meaning. I think for many people the question of whether there are other, advanced, non-human consciousnesses out there is of great emotional significance, even though it can never affect our lives one way or the other. It is emotionally meaningful, but not rationally meaningful.

Does all of physics include only events or things which can in principle be experienced by us? What about outside or light cone or other universes? — Marchesk

I would say yes, or at least all of physics that can be considered as science. In my lexicon other universes or events outside the light cone are metaphysics.

It depends on what his argument is. If all it does is to point to the logical gaps in attempted constructive rational arguments then I think it would stand. But if it tries to erect some universal principle of the non-provability of any metaphysical argument, I think it would run into the difficulty you are pointing to.

It doesn't really matter to my life whether universals are real, but it's interesting to think about sometimes, just like it's interesting to wonder whether the laws of physics really 'break down' inside a black hole, which is just as meaningful, except for the difficult math. — Marchesk

The difference between these two is that it is hard to imagine any experience that would answer the question about whether universals are real, but one can easily imagine one that would answer the question about physics inside a black hole. All one need do is sail one's spaceship inside the event horizon of a black hole and look around. Contrary to popular belief there's a long way between the event horizon and the point at which you get pulled apart by gravity ('spaghettified'). So there's plenty of time to observe.

Of course, one could never report back, but that doesn't matter. The doomed person's experience (and remember, we're all doomed - it's just a question of when) would differ according to whether the answer was yes or no.

I don't see how this is possible since many people have made rational arguments for various metaphysical positions. — Marchesk

Some have made arguments that they thought were rational. I don't agree with them on that. If they were rational they would be conclusively persuasive to anybody that understands logic, regardless of that person's prior opinion on the conclusion. Yet they are not.

I think it helps to make a distinction between rational meaning and emotional meaning. I tend to agree with Carnap that questions of ontology have no rational meaning. They leave unanswerable questions like 'how would our experience of a world in which only matter exists differ from one in which only minds exist?' or 'how would my experience of a world in which only I exist differ from one in which other minds also exist?'

But those questions, especially the second one, are very emotionally meaningful to many people. The fact that something cannot be defined, proved or a test devised for it does not mean that one will not be emotionally attached to one belief or another about it.

That's one reason I like Hume. Some see him as arguing for something like logical positivism, because he demolishes popular arguments in favour of certain metaphysical positions. But I see him as upholding emotions by arguing that they are the only things that matter and that determine what we do. He frees us from the perceived need to rationally our justify dearly-held metaphysical commitments, by demonstrating that such rational justification is impossible. But I don't think he's telling us to discard the commitments.

I'm a bit bemused by this Ramsey business. I can't see what perceived imbalance they are trying to correct. It's a long time since I was at uni but, when I was there, nearly all of the subjects in the Arts faculty were focused on Western culture, whether literature (Shakespeare, Goethe, Moliere), Music (Beethoven, Monteverdi), Art or philosophy. Most history subjects were Western history and most languages offered were European. There was even a department devoted to Classics, as in study of Ancient Greek and Roman history, literature, philosophy and language. The older universities like Oxbridge even have Divinity departments.

Maybe it's changed since 1985, but it would have to be a complete about-face for Western culture to be under-represented in Uni subjects.

I do have more recent experience in High school curricula and my observation is that they are tremendously Western-focused. Despite gestures like Welcomes to Country and aboriginal murals on walls, the literature studied is almost exclusively Western, and a high percentage of it Shakespeare.

I'm not bemoaning that - other than the over-emphasis on Shakespeare at the expense of other great writers like George Eliot - but I really can't see where the deficiency of teaching Western culture is supposed to be.

From what I can tell, radfems believe gender is a patriarchal institution that oppresses women (and also men, to a degree) by forcing them into artificial categories that suit the needs of those in power. When a transgender person "switches" genders, then, they are implicitly affirming the gender institution — darthbarracuda

Perhaps it turns on what is meant by 'switch genders' - specifically, whether the person in question declares that they are now of the gender that is the opposite to their sex, and requests others to change how they refer to that person, via gendered pronouns etc.

My understanding of the de Beauvoirian (dB) position (I prefer to use that rather than 'radfem', which sounds pejorative) is that they are supportive of somebody changing their dress and other behaviour to that which society deems as being 'gendered' in a way that mismatches their sex, precisely because the dBs want to subvert and remove the notion of gender altogether, and a great way to do that is to undermine people's expectations (or worse, requirements) of the relationship between behaviour and sex. I imagine the dB would congratulate the person on having liberated themself from oppressive societal expectations of how their behaviour should be constrained by their sex. Further, I think dBs would be supportive of requests to use neutral pronouns, but not of using gendered pronouns that mismatch the person's sex.

But if the person makes claims to a specific gender (as distinct from a denial of having a gender, or just not mentioning it at all), and requests agreement with that from others, the dBs may see that as regrettable because it supports rather than deconstructs the notion of gender.

So my understanding is that a dB would be supportive of a 'gender switch' that involves change of behaviour, but would regard it as unfortunate if the person self-labels that change of behaviour as a gender switch.

This refers to those cases where there is no ambiguity as to the person's sex, whether from how they were born (eg unusual chromosomal variation) or from medical interventions. As far as I know, there is no representative dB position on such cases.

If A and B are our length, C is our common measure, and n and m are integers:

(1) A = mC, and B = nC, then we can cancel out C such that:
(2) A/B = m/n
(3) B = A(m/n) — StreetlightX

I've been thinking about this in terms of various universes that are limited as to what's in them. If we have only two objects, A and B, and they don't have regular markings like on a ruler, then I think the only measurement we can do is to say for each of them, how many complete copies of the other can be lined up alongside it without exceeding the length of the first. We do this in the same way we use an unmarked 30cm ruler to measure the length of a wall - by rolling the ruler end over end until we reach the end of the wall, and counting the number of rolls.

In this way, we'll end up with the measure of the shorter object in terms of the longer being 0, and the measure of the longer object in terms of the shorter being the ratio of the two 'true lengths' rounded down to the nearest integer.

Both of these measures satisfy the axioms required to be a true measure.

If we want to allow rational measures but have only two objects, we need at least one of them to have regular markings on it like a ruler. So if one object is a 30cm long straight stick with marks every 1cm, we can measure the length of the other object to a precision of 1cm, rather than 30cm.

I have a slight doubt about whether an object with 29 regular markings is one object or thirty, but I think I'll dismiss that doubt as too metaphysical for the time being.

If both objects have cm markings on them, each can be measured as a number of whole cm, so we can say the length of A is m cm and that of B is n cm, but we know the lengths are not exact.

I think I need to stop here and ask for input, because it's starting to look like this avenue of investigation doesn't involve irrational numbers or Pythagoras. Rather it is just saying that any physical object can only have regular markings on it with a finite level of granularity. So the problem here is the limitation of accuracy that gives us, and it arises for objects whose length ratio is purely rational, just as much as for pairs where the ratio is irrational. If the smallest measure gradation marked on either object is micrometres (10^-6 m) then, even if both objects have lengths that are whole numbers of nanometres (10^-9 m), the measure is in some sense 'inadequate' if one or both of those whole numbers is not a multiple of 1000.

A right-angled triangle of sticks, whose lengths are not those of a Pythagorean Triple, guarantees a measurement problem because one of the sticks will have a length that is an irrational multiple of the lengths of the others, but so does just having two sticks, one of whose lengths is not a whole number of the lengths of the marks on the other.

Something under-appreciated about the mathematics of limits, which shows itself in the enduring confusion that 0.99... isn't equal to 1, is that when you're evaluating the limit of something; all the steps to reach that limit have already happened. So 'every finite step' misses the mark, as the limit is characterised as after all steps have been done. — fdrake

I'd rather not put it like that, as it seems to imply that we need to 'go to infinity' in order to make sense of the limit. Then before we know it, people like the apologist William Craig are butting in making ignorant statements about the possibility of 'going to infinity', as if that actually meant something.

Yet limits can be, and usually are in formal maths texts, defined using purely finite concepts.

Consider the function f(x) that is L + (x-a) sin 1/(x-a) for x<>a and gives L at x=a. Unlike the function g(x) = (x-a) + L we can't look at an intuitive straight line that heads straight for the point (a,L), because the curve of f keeps wiggling, and the speed of wiggling increases without limit as x approaches a. But when we say that the limit of f(x) as x->a is L, all we are saying is that, for any small number e>0, we can find another number d>0 such that whenever x differs from a by less than d, f(x) will differ from L by less than e.

That definition uses only finite concepts, and proving that it holds in this case also requires only finite methods.

This is a demonstration of the little-recognised fact that most limits, and indeed most of calculus, can be done without using any notions of infinity. Even notions such as lim(x-> infinity) f(x) or (using an extended definition of limits) lim (x->a) f(x) = infinity don't actually need to use a notion of infinity. The infinity symbol in those expressions can be interpreted using only finite methods like what was done above.

Infinity is great fun to explore, especially in set theory and topology. I love Cantor's work as much as anyone does. But we don't need it to do calculus. We need unconstrained divisibility, but not a notion of infinity.

It seems to me that philosophers of law would tend, as in many other 'philosophies of...' (science, mathematics, ethics), to divide into those that think there is some Platonic entity that is (in this case) the Law (capital L), to which the laws we make are better or worse approximations, and those that regard law as a purely human activity that has no existence other than that which humans give to it by their creation and use of it. No doubt there's still enormous amounts of discussion that can happen within either of the two camps, but it seems inevitable to me that it divide into these two camps on Platonic vs non-Platonic lines.

FWIW I'm a non-Platonist on most things I can think of, including law, ethics, science and mathematics. But that's just my disposition, not an assertion of correctness.

There's another 'philosophy of' that is interesting, and has nothing to do with one's degree of Platonicity. I think of 'The Tao of Pooh', or 'Zen and the Art of Motorcycle Maintenance'. These are about the philosophical insights one can gather from engaging in a certain concrete activity, whether reading children's stories or fixing a Suzuki. My partner does a great deal of craft, and seems to get great satisfaction from it, which makes me wonder if there is some 'Philosophy of Quilting' from which I would benefit if I could only catch hold of it. The philosophy can also be about how one engages in the activity in order to obtain certain benefits, as a form of praxis. This is particularly strong in Zen (eg 'Zen and the art of archery', or think of all that raking of pebble gardens) but is also used in other traditions.

To engage in a philosophical discussion of mathematics and measure, it seems a good idea to first engage with the enormous branch of mathematics that deals with exactly that, which is Measure Theory. Without measure theory, there would be no firm foundation for integration or for probability theory.

One of the great insights of measure theory is that measures are rarely unique. For any given collection of objects there will usually be many different ways to measure them. For instance in integration, which can be thought of as a measure of areas or volumes, we have Riemann integration (which people tend to learn in high school) and Lebesgue integration (which tends to be introduced in university pure maths courses). In many cases they give the same answer but there are important cases where they don't. There are other forms of integration as well that differ from these two.

The case in the OP seems to suggest that there is some unique, true, primordial measure of the items A and B, and that we just need to find it. Measure theory reveals that we can define many different measures for the objects. A simple one is that the measure of A is 1 and that of B is 0. In the simple universe that contains only A and B, that measure obeys all the axioms required of a measure, as does the measure that says A's measure is 0 and B's is 1, or that they are both 5.1.

When we introduce a third object C we might try to define a measure in terms of that, as 'the size of an object is the number of complete copies of C we can line up against it before we reach the end'. Then when Rosen says that A and B are incommensurable in terms of C, the Measure Theoretic way of saying that is that the C-based size function we just defined is not a measure, because it does not satisfy the axioms required for a measure, in a universe that contains A and B as well as C. There's a bit of mathematical work required to show that it fails the axioms, but it's not difficult. It involves taking the left-over bit in A or B after measuring via C and replicating it until it becomes bigger than C.

I don't know the detailed history of Measure Theory. It is possible that its invention was inspired by problems like that of Pythagoras. If so then, far from being a disaster, it has led to some of the most useful and productive parts of modern mathematics.

What about Einstein, though? He was rather the odd man out in this respect, don’t you think? — Wayfarer

I feel that Einstein eludes classification. He was undoubtedly deeply philosophical, and I think his ideas changed greatly during the course of his life, which I see as a sign of an open mind. His large collection of memorable sayings is so varied that people of all different conflicting philosophical positions like to claim him as one of their own, but I I doubt he belongs to any neatly labelled philosophical school, or maybe he belonged to many of them, one after another.

His 'God does not play dice' and 'the moon is still there when nobody is looking at it' are indications of a lifelong inclination towards an anti-idealist position, perhaps materialism. But I very much doubt he was a reductive materialist, and I suspect he would have recoiled in distaste at the sort of scientism that is popular in some circles these days. After all, he said it was God that doesn't play dice, not The Universe. He was such a chameleon that he probably meant different things by the word God from one day to another, but I feel convinced that he thought there were great, deep mysteries out there, that we could never approach just by empirical experiment.

What are your thoughts on Einstein?

The notion of a completely objective reality is the bedrock principle of science

This sounds like the sort of thing Stephen Hawking would say, and it is wishful thinking on the part of people that would like to co-opt science in their evangelical quest to spread reductive materialism.

But science will not be co-opted.

The claim is belied by the rest of the post, in which four scientists: Planck, Bohr, Fuchs and Schack - two of them in the veritable pantheon of physicists - disagree with it. We could add Heisenberg and Bohm to that for a start.

In no sense at all is that notion of objectivity a bedrock of science. Science is silent on that topic. That people like Hawking claim it is a bedrock does not make it so. It may be a bedrock of scientism, but it is certainly not a bedrock of science.

Why would they announce their apathy about death if apathy was the norm?

My understanding is that fear of death was present in Ancient Greece, but not because they thought it was non-existence. It was because they thought they'd be condemned to live eternally in the gloomy Underworld. Epicurean writing against fear of death was not saying 'stop being afraid of ceasing to exist' but rather 'you can stop being afraid of existing forever in gloom, BECAUSE the good news is that you will cease to exist'.

It was that idea of annihilation that was radical and new from the Epicureans.

Excellent opening post. It had never occurred to me that the widespread fear of non-existence is a relatively new phenomenon, but when you put it like that it sounds very right.

Of course nearly all humans have an instinctive, visceral fear of things that could make us die, like being unable to breathe, caught in a fire, or being attacked by a knife-wielding assassin. But that has nothing to do with the existential fear of non-existence.

I find persuasive your allusion to the modern Western obsession with individualism and individual rights, and that that may be at the root of the modern fear of no longer being alive.

It makes me wonder whether that fear is less common in more communalist cultures, such as the Chinese. I also get the sense that earlier cultures - whether nomadic tribal, rural feudal or municipal, were much more communalist. When survival is harder to maintain, one becomes much more conscious of one's dependence on the people around us. Oddly, we are all now more dependent on others for our survival than ever before, yet this illusion of rugged individualism is stronger than ever, in the West at least. I doubt that people in nomadic tribes, whether in ancient or modern times, spend much time quivering about the inevitability of their ultimate cessation.

The earliest writing I am aware of that is filled with dread of non-existence is The Death of Ivan Ilyich by Tolstoy. And I am not aware of anything from a non-Christian background that projects such dread. Although the existentialists were mostly non-Christian, they emerged from a Christian culture and I think were very much shaped by it.

Did you mean to reply to somebody else? I can't see the connection between an absurdist joke about a duck and the old canard about fine tuning.

This reminds me of the question: "What is the difference between a duck?" — angslan

That was a joke one of my schoolfriends told me back in about 1975. His answer was
'One of its legs is both the same'
I found it hilarious no matter how many times I recalled it.

Have you heard that answer - or a different answer? I've always wondered whether my friend made up the question (it seems not) or the answer.

Normal is a probability distribution with density function exp(-(x-u)^2/2s^2)/sqrt(2 pi).
Normal is a vector pointing away from a plane at right angles.
Norm is the length of a vector.
Normal is a subgroup which is invariant under conjugation by members of the group of which it is a part.

Any other uses of the word 'normal' are highly suspect and best avoided.

@Michael @Srap Tasmaner
It occurred to me that there may be a parallel between this puzzle and Nick Bostrom's simulation hypothesis. We are analogous to Beauty, and the coin coming up tails is analogous to beings in some universe developing the ability to perform simulations so intricate that consciousness arises in the simulands, call it p.

Say the average number of conscious simulands created in a universe in which those simulations are developed is N. Let the probability of conscious life arising in a universe be q and M be the number of conscious beings arising in such a universe.

We then wonder what is the probability that we are in a simulation. I think Bostrom argues that it is

pN / (pN + qM)

which he thinks would be close to 1 because N would likely be much bigger than M because in such a world, using computers or their equivalent it would be easy to conduct enormous numbers of simulands. That corresponds to the Thirder argument in the Beauty problem because it says that every consciousness, whether simulated or not, is equally likely to be the one I am experiencing, so each one has probability 1/(pN + qM).

In contrast, the Halfer position says that the probability of being a particular simuland is p/N, and the probability of being a particular non-simulated consciousness is q/M. So the probability of being a simulated consciousness is

N(p/N) / (N(p/N) + q(q/M)) = p / (p+q)

That will be much lower that Bostrom's estimate because it is not affected by N being much bigger than M. It makes no difference how many simulands the simulators create.

Hence, the Halfer position provides support to those that don't like Bostrom's suggestion that they are probably a simuland.

What do you think?

This looks correct, except that:

- on line 1 you have written P => Q => R but it should be P => (Q => R)
- on line 8 you have written P => Q => R => R but it should be (P => Q) => (P => R)

Here's some Monty Hall code in R
Under this code the prob of winning under no-switch is about 1/3 and 2/3 if the contestant switches.

The Monty Hall problem and its analysis is identical to choosing 1 door, or choosing 2 doors. — tom

Yes, if you switch you are effectively choosing two doors, which leads to the accepted solution that it is optimal to switch. That's what I said above.

Perhaps I misunderstood your post. It appeared to me from your statement that you've never met anybody that agrees with you on this, that you were rejecting the accepted solution. If you were, I'd be interested to know why. If not, there's nothing to disagree on.

Ah, so now you're claiming that whatever god had to say, it may or may not be correct. — Txastopher

I read it as chatterbears feeling that, if God made it clear that She wanted to punish homosexual acts, chatterbears would conclude that God was immoral according to chatterbear's ethical system (eg see Stephen Law's 'Evil God' hypothesis, or this saying that is wrongly attributed to Marcus Aurelius) and would refuse to serve Her.

If we accept the RC church's teachings then we have to accept that it is possible to know with certainty of God's existence and power and yet refuse to serve Her, because that's what the teachings say Satan did.

One positive thing that I have noticed, though, is that Trump didn't continue the legacy of Bush and Obama in regards to the war on terror. — Posty McPostface

I think that people from majority Muslim countries that had their requests to travel to the US denied by the new administration, on the sole grounds that they are from a Muslim country, would disagree.

Line 6 looks wrong. It looks like you are trying to close a subproof commenced on line 3 but you can't do that until you've closed the subproof opened by the assumption on line 5. You can't close a subproof while any subproofs of it are still open.

You can fix that by deducing R on line 5 (maybe needing one or two additional lines), using lines 1, 3 and 4.

re the Monty Hall problem

You, as the contestant, know for certain that one of the other two doors is empty, once the door is opened, you still know for certain that one of the doors is empty. OK, you now now which one is empty, and that IS information of sorts, but is it relevant information? — tom

If the door you had selected has the car behind it (call this event C) then it is not relevant information.

But if your selected door does not have the car behind it then it is very relevant information, because it means the car is behind the remaining closed door that you did not select.

Since P(C)=1/3 there is a 1/3 probability of the information being irrrelevant and a 2/3 probability of it being relevant. So the probability of a win is increased by assuming the information is relevant, and using it, which means switching. The calcs are:

We have P(C) = 1/3. If you retain your selection then the probability of a win is:

1/3 x P(win | C & not switch) + 2/3 x P(Win | ~C & not switch) = 1/3 x 1 + 2/3 x 0 = 1/3.

On the other hand if you switch the probability of a win is:

1/3 x P(win | C & switch) + 2/3 x P(Win | ~C & switch) = 1/3 x 0 + 2/3 x 1 = 2/3.

We could call the new information 'conditionally relevant information', since its helpfulness is conditional on an unknown, which is whether C is the case.

Another approach, for those that don't find the above persuasive, is to say that there is no new information, but reject the dictum that one should not change one's guess in the absence of new information, replacing it with a dictum that one should not change one's strategy unless one receives new information. The best strategy is to change one's guess, after Monty has opened a door. That strategy does not change when Monty opens the door.

Consider a slightly different game in which Monty doesn't open a door but you can still switch your guess and, if you do so, you will win if the car is behind either of the two doors you did not originally select. The probabilities in this game are identical to those in the original Monty Hall problem, but in this case you clearly get no new info, since no door is opened until the final reveal-all. But it is in your interests to switch. You will have changed your guess, but not your strategy.

Imagine a stock market strategy of buying stocks of companies that have had a certain event, say a change of CEO. Say our research has showed that on average there is an upwards surge in stock price on appointment of a new CEO, that peaks on average three days after the announcement, and then often subsides. Then we could set up a hedge fund that buys stocks that have had such announcements, and sells them after three days. In most cases there will be no important new info about the company in those three days, but we will still sell the stock. We've had no new info. We've changed our stock-pick (our guess) but not our strategy.

First let me say that my inclination is to being a halfer, and I argued for that 'cause' on physicsforums last year, so here I'm arguing not against halfer being a preferred position (IMHO) but rather against the idea that it can be proven that it is preferable.

With that preface, perhaps I misunderstood your post here. In the fifth line of the reasoning, you assert, seemingly as an axiom, that P(Heads)=0.5. While - as indicated above - I sympathise with that position, it does look like assuming one's conclusion, if the object of the exercise is to determine what probability Beauty assigns to the coin having landed Heads.

If we do not assume our conclusion, we need to make P(Heads) an unknown, which I called p, and see if we can assert other, less controversial relationships that enable us to deduce that. But I can't see how we can do that without assuming something that is essentially equivalent to P(Heads)=0.5.

To me it looks irresolvable, which is why I like to turn to betting games to make the question concrete.

Nevertheless, if it's Tuesday only tails will save her. — Benkei

We can assume without loss of generality that she decides before the experiment begins what she is going to guess when awoken.

Say she chose heads. Then when awoken she can know with certainty that it is not Tuesday because if it were, the coin must have landed Tails, so she would be dead, having already guessed Heads on Monday.

So if she chose Heads she will never get to Tuesday.

A variant that one might take in seeking to avoid this objection is where the execution occurs on Wednesday, so she will still be woken and asked to guess on Tuesday even if she guessed heads on Monday. But on examination we see that this doesn't change the preferences because she knows on waking that if it is Tuesday and her chosen strategy was Heads then she is already condemned to death so there is nothing at stake in today's answer.

In line 4 it assumes that P(Heads) is 0.5, That would be reasonable for the probability space of an independent observer. But this probability space is the epistemological probability space of Beauty, and we have no axioms that justify our assuming that the probabilities will be the same in that probability space as in the independent observer's one.

Under Kolmogorov's framework, an event that occurs in more than one probability space can have different probabilities in different spaces. Indeed, that freedom is central to the mathematics of derivative pricing in finance, where alternative probability spaces are employed that use what are called 'risk-neutral probabilities'.

I liked the solution I read on the physics forum from PeroK. Let's change the story shall we?

*** Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice during the experiment, Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening.

A fair coin will be tossed to determine which experimental procedure to undertake: if the coin comes up heads, Beauty will be awakened and interviewed on Monday only. If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday. The interview consists of her being asked whether the coin toss was heads or tails, if she guesses wrong, she'll be executed. If she still lives, she will be awakened on Wednesday without interview and the experiment ends. ***


Given these facts, what answer will Sleeping Beauty give?

If the toss is heads, she will awaken Monday. If she'd then say heads, she lives. If she'd say tails she'd be dead.

If the toss is tails, she will awaken first on Monday. If she then says heads, she dies. If she says tails, she'll live.

If she said tails, she will awaken again on Tuesday. If she then says heads, she dies. If she says tails, she'll live.

In the above we see there's only one event where saying tails gets you killed and two events where saying heads gets you killed. Sleeping Beauty would be smartest to state tails. — Benkei

I don't feel that this changes the situation, because although there are two events at which saying Heads can get Beauty killed, if she decides to say Heads, she will never make it to the second event, so the event of being killed at the second waking has a zero probability of occurring.

This assumes that Beauty doesn't randomly choose what to say, by tossing a coin herself after being woken. But so far that possibility has not been canvassed (and I don't think it leads anywhere productive). If the choice is not random then whatever reasoning she uses on Monday will be used again on Tuesday, since the days are indistinguishable to her, so she will give the same answer on both days.

Is that what I've done above?

P(Tails and Monday) = P(Tails) * P(Monday|Tails) = 0.5 * 0.5 = 0.25
P(Heads and Monday) = P(Heads) * P(Monday|Heads) = 0.5 * 1 = 0.5

Therefore, P(Monday) = 0.75 — Michael

A simple, discrete probability space consists of two things - a sample space, which is the set of all possible outcomes, called Events, and probabilities of each Event.

It seems from what you are doing here that your sample space consists of four Events:

1. Coin landed Heads and today is Monday
2. Coin landed Heads and today is Tuesday
3. Coin landed Tails and today is Monday
4. Coin landed Tails and today is Tuesday

I think most people would agree that 2 is impossible, so it must have probability zero. The other probabilities are at first unknown and must be inferred by other relationships we have been given.

One such relationship on which I think most people would agree is that is that the probabilities of 3 and 4 must be the same.

That leaves one degree of freedom, which is the probability of 1 which, since the prob of 2 is zero, is also the probability of Heads in this probability space - which we note may not necessarily be the same as the probability space of an independent observer (who will of course say the probability of Heads is 1/2), since this is an epistemological probability space based on the knowledge state of Beauty just after being woken up.

Let the probability of 1 be p. Then the probabilities of 3 and 4 will each be (1-p)/2.

If we set p=1/2 then each of 3 and 4 have probability 1/4 and the probability of Heads is 1/2.
If we set p=1/3 then each of 3 and 4 have probability 1/3 and the probability of Heads is 1/3.
If we set p=1/4 then each of 3 and 4 have probability 3/8 and the probability of Heads is 1/4.

It seems to me that there is no indisputable way of removing the free parameter p. To do so, we need to make an assertion of probability, but it can't be derived without circularity.

We can say p=1/2, in which case we are asserting a principle that Beauty should have the same epistemological probabilities for Heads and Tails as a non-amnesified, independent observer. IN that case we conclude that the probability Beauty assigns to Heads is 1/2.

Alternatively, we can say p=1/3, in which case we are asserting a principle that all non-impossible events in the sample space should have the same probability. In that case we conclude that the probability Beauty assigns to Heads is 1/3.

Or we can say p=1/4, in which case we are basing Beauty's assigned probabilities on the expected values of winnings from a betting strategy.

I expect there are other arguments, with various degrees of convolutedness, for other values of p.

I am inclined to conclude that, if we try to use probabilities rather than betting to solve this, we are left with a degree of freedom - the value of p - that cannot be removed without making a controversial assertion about how Beauty should assign probabilities in her epistemological probability space.

In short, it seems that Beauty's degree of belief that the coin came up Heads can reasonably be whatever she wants it to be - subject to it being in the range [0,1].

Game 2: At each interview, Beauty bets $1 to guess what coin came up, and loses that dollar if wrong or wins $2 if right — andrewk

This is why I said I couldn't understand how you were setting the odds. Doing it this way is paying off 2-1 on both heads and tails, which is incoherent. (Unless you meant both tails interviews would taken together pay off $2, but that's still incoherent.) — Srap Tasmaner

I don't know betting terminology, so I may have used the wrong words. What I meant by 'wins $2' is that she gets her own dollar back, plus another dollar. Perhaps the correct betting terminology for that is 'wins $1'. My ignorance of gambling terminology is gargantuan.

Either way -- 2-1 or 3-1 -- offering even odds in this game loses money. We agree on that, right?

Yes I think we can agree that if a bookmaker offers Beauty Game 2, which has even odds, Beauty can select a strategy under which the expected value of the bookmaker's profit is negative.

Any takers on the challenge to define a probability space in which the statement 'today is Monday' corresponds to a well-defined Event?

My policy is to wager $1 on tails whenever I'm asked. You're paying even money.
If the toss is heads, I lose $1; this happens half the time, so my expected loss is $0.50.
If the toss is tails, I make $1 each time I'm asked; this scenario happens half the time, so I have an expected profit of $1.
So I make at least $0.50 on average each time I play, no matter how the toss goes.
I have made a Dutch book against you. — Srap Tasmaner

This is Betting Game 2 from this post.

My calculation is that expecting that profit is the same as if there were a single $1 bet at even odds that the result of a coin toss will be tails and the coin had a 3/4 probability of coming up tails. To see this, note that in that case the expected profit is

$1 x 3/4 + (-$1) x 1/4 = $0.50

Hence, in interpreting the term 'degree of belief' it seems reasonable to make it the probability of tails when betting on a single flip of an unfair coin, at which one would have the same expected profit. So under this interpretation of 'degree of belief', the answer is 3/4 for tails, and hence 1/4 for heads.

Examiner tosses a fair coin, and then tosses another. If the first toss was tails, she asks Beauty her credence that the first toss came up heads; if the first toss was heads she only asks for Beauty's credence if the second toss was heads as well, otherwise the round is over.

Done this way, Beauty will know that when she was not asked the first toss was heads, but she can do nothing with that knowledge. She's not asked and the round is over. What matters is that she's always asked when it was tails and asked half the time when it was heads. So her credence that it was heads should be 1/3. — Srap Tasmaner

I can't quite follow this. I think there are a couple of grammar glitches in it that make it hard to understand what it is saying. Can you please expand on it, maybe using bullet points for clarity, and explain in what way you find it equivalent to the original question?

thanks