The trouble is that 'today is Monday', for which you have used the label 'M', is not an Event in the Kolmogorov sense of being a well-defined subset of the sample space and, in the absence of its being such an Event, it cannot be used in probability statements, as a probability statement is a probability measure of an Event.

The reason it can't be an event is that, so far as I can see, the definition of an event in the Kolmogorov framework, which is used, to the best of my knowledge, in all modern probability theory, is timeless. It cannot be relative to a particular time. So 'Beauty is woken on a Monday' and 'Beauty is woken on a Tuesday' are events because the time references they make are absolute, but 'today is Tuesday' is not an event because the reference 'today' is relative.

The same applies to 'I am awake', which is what I think you mean by the label 'A'. Because the 'am' is a relative time reference, it cannot be an Event in a probability space, and so cannot have a probability in the usual sense, that allows use of things like conditional probability rules.

To use a philosophy of time analogy, the question 'is today Tuesday', in the context of this experiment, has as much meaning in probability theory as it does when asked to a pan-dimensional being that is looking at our 4D spacetime from the outside, if McTaggart's B theory of time is the case.

Perhaps a probability space can be constructed in which 'today is Monday' or 'I am awake' is an event but so far I have not managed to construct one. I thought I had a lead, but it turned out to be a dud.

Unless we can construct a probability space in which 'today is Monday' is an event, and which also contains all the other information about the experiment, it cannot be meaningful to talk of the probability - conditional or unconditional - of the event 'today is Monday' or 'I am awake', so we cannot use conditional probabilities to calculate with them.

One way to attack the question of what 'credence' or 'degree of belief' means is to interpret it in terms of 'what would you bet, if you were Beauty'? The answer to that depends on the rules of the betting game that is offered.

Consider two different betting games, and we assume Beauty wants to maximise her expected profit.

Game 1: Beauty places her bet before the experiment starts, paying $1, and at the end she is paid $2 if in all her interviews she guessed the coin outcome correctly, otherwise she is paid nothing.

Under this game, her expected winnings are maximised at zero, whichever she chooses. But she must decide before going to sleep the first time which side she is going to guess, because her expected profit becomes negative if it is tails and she makes one guess of heads and another of tails.

Under this game, interpreting the betting strategy as 'degree of belief', we could say her 'degree of belief', at the time of being interviewed, that the coin has landed on Tails, is 1/2.

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.

Under this game, Beauty's expected winnings are maximised at 50 cents if she guesses Tails.

Under this game, interpreting the betting strategy as 'degree of belief', we could say her 'degree of belief', at the time of being interviewed, that the coin has landed on Tails, is 3/4.

I find it interesting that Game 2, which seems perfectly natural interpretation to me, is consistent with a 'degree of belief' of 3/4 rather than 1/2 or 1/3.

I wonder what betting game would be consistent with a probability of 1/3. I expect there must be a pretty natural one, since most people answer either 1/2 or 1/3 to this question.

I'm off for a jog on the beach now. Maybe it will come to me.

So we want the conditional P(H|A). — Srap Tasmaner

The trouble is that we cannot use conditional probabilities.

A conditional probability P(H|A) is the probability of event H given the probability of event A, and A will be an event that is known to be true. But the only event that Beauty knows to be true is that she has been woken at least once - because she has just been woken, so the known event is not a proper (ie smaller) subset of the entire sample space, call it S. So if H is heads and A is 'I am woken at least once' then P(H|A) = P(H|S) = P(H) = 1/2. The conditional probability is the same as the unconditional one.

To use conditional probabilities à la Bayes' Rule, we must have some information that narrows down the set of possibilities - that tells us we are in a proper subset of the sample space. But Beauty has no such information. All she knows is something she already knew before the experiment started.

I think it's to avoid that straightforward solution that the word 'credence', or sometimes 'degree of belief' is used instead of 'probability'.

I'll add that statements like P(Tails and Monday) or P(Tails | Monday) are ambiguous, because the 'Monday' could be the event 'I get woken on a Monday', for which the probability is 1, or it could mean 'today is Monday', and it's not immediately obvious how to set up the probability space so that 'today is Monday' is a well-defined event, without losing the connection to the coin toss.

Unlike the others, this one is not a misunderstanding of a well-understood and resolved problem. There is an interesting discussion to be had about it.

What is the correct answer depends on what meaning one attaches to the word 'credence'. The use of that unusual word rather than 'probability' in posing the problem is deliberate.

There's a long discussion about it here on physicsforums.

I can't remember, off-hand, whether I was a thirder or a halfer.

A direct contradiction that doesn't result from a compelling argument (e.g. some bachelor is married) is not the same as a paradox, which does show up as the conclusion of a compelling argument. — MindForged

The bachelor statement is not a direct contradiction. One has to deduce the contradiction by a series of steps, so the only difference between that and the assertion of the existence of a Russell set is the length of the deduction by which one arrives at a contradiction from the statement.

To see this, note that a contradiction is a statement of the form

P and not P

Now a person is said to be a bachelor at time t if at that time they are an adult, male human that has never been married. We can write this as:

bachelor(x, t) <-> adult(x, t) and male(x, t) and human(x, t) and for all t' (t' <= t -> not married(x, t'))

Then the statement 'Paul is a married bachelor at time t' is formalised as:

married(Paul, t) and bachelor(Paul, t)

which is the same as

married(Paul, t) and adult(Paul, t) and male(Paul, t) and human(Paul, t) and for all t' (t' <= t -> not married(Paul, t'))

Now intuitively we feel confident that we will be able to deduce a contradiction of the form

married(Paul, t) and not married(Paul, t)

from this.

But that deduction will take quite a number of steps. For a start we need to get rid of two other conjuncts adult(Paul, t) and male(Paul, t) using 'AND elimination'. We also somehow need to deduce

not married(Paul, t)

from

for all t' (t <= t -> not married(Paul, t'))

That is going to involve using an instance of the axiom schema of substitution as well as the axiom that A -> A or B ('OR introduction').

I expect that deducing the contradiction will require a proof of at least ten steps. SO the contradiction is certainly not direct.

For the contradiction to be direct, we would need to define bachelor(x,t) to simply mean x is married at time t. But that is not how bachelor is defined. Under that definition a divorcee, a newborn, a widow, a frog, a rock and the number 3.45 would all be bachelors.

The same analysis can demonstrate that the statement that x is a square circle is not a direct contradiction, but rather a statement from which a contradiction can be deduced by a series of steps - just like the assertion of the existence of a Russell set.

No

No

No. We won't.

No. Russell's Paradox is about the set of all sets that do not contain themselves. It does not mention barbers or shaving.

The town barber, who is a man, shaves exactly every man in the town who does not shave himself. — Jeremiah

This statement is false.

If we are working with a logic that allows proof by contradiction, it is false because it allows deduction of contradictory propositions.

If we are not, we can in any case deduce the negation of the sentence.

This case is not the same as Russell's Paradox. Russell's Paradox arises from the inconsistency of the axioms of Naïve Set Theory, whereas this statement is meaningful and false in any reasonable logic.

Since the barber statement is false, the answer to the question 'who shaves the barber' is 'anybody, including the barber, could shave the barber, but we don't know who does it'.

My concept is the same as yours. I reached the conclusion a few years ago that the only concept of 'physical' that made sense to me is an epistemological one. Thus, what is physical will change over time as our scientific theories change. Further, the concept of physical that we have in 500 years time may not even be intelligible to us at present, if it uses concepts we do not currently have.

A problem with using a definition based on spatiotemporal locations is that the wave function of quantum mechanics does not have a spatiotemporal location, so we then get embroiled in a common debate within QM about whether the wave function is 'physical'. This becomes even more stressed when we consider non-local interpretations of QM such as Bohm's with its pilot wave.

Another thing I like about the epistemological definition is that it makes it possible for everybody - materialists and supernaturalists alike - to agree that there are probably non-physical things.

The analogy is in construction. You need a lot of scaffolding to help support some structures, so it looks self standing. However, it is only standing because of the scaffolding. — wellwisher

This analogy doesn't fit. Scaffolding does not support the structure being built. Scaffolding is erected to enable the workers to stand safely next to the structure while they build, paint or renovate it.

What paradox? You have described a mathematical structure. If you think there's a paradox in it that needs to be resolved, you need to explain what it is that is paradoxical about the structure and what you would regard as a resolution.

Paradoxes are either (1) logical contradictions, or (2) logically consistent but surprising to some.

Contradictions require resolution, but there is nothing contradictory about the structure, so it is not the first type. So if you see the structure as a paradox it must be the second type - surprising to some. A surprise does not need resolution.

So there's your answer - there is nothing to resolve.

That's the fourth dodge.

I imagine there are plenty of Star Trek discussion boards and that on those boards, each thread has a point, that is generally posed as a question, eg:

- Do you think we will ever have teleporters?
- Do you think Spock has emotions but just doesn't show them?
- Who do you think would win in a fight between a Klingon and a Sontaran?
- Who is your favourite commander of the Enterprise?

or sometimes they might be propositions put out as challenges, and seeking opinions for or against, eg:

- I think Captain Kirk is really evil, and here's why
- I think it's unrealistic that nearly all aliens are bipeds with only superficial differences from humans

What is the proposition you'd like to put out for challenge, or the question you want to ask, in relation to the mathematical construct in the OP?

That's the third time you dodged the question - which was originally put to you in post #2. Are you going to answer the question? What was your aim?

It's not for me to figure out. You started the thread. What was your aim?

Such objects should not exist — Jeremiah

That is a feeling. The 18th century British invaders of Australia had a similar feeling when they first saw a platypus. When they found that the object in question was undeniably there in front of them, their 'should not exist' transformed to 'well, I am very surprised'.

Is the aim of this thread then to muse over the nature of the emotion we call Surprise?

My thought is, don't sit on it. — Baden

Wise advice.

Fortunately, it is impossible to sit on it, because it has no tip. The pointy bit just recedes endlessly, never culminating in a spike. The ultimate in child-safety mathematical structures.

As for getting it off the ground, that would be impossible because, even though it has finite volume, and hence finite mass (if we assume constant density), its moment of inertia would be infinite because of its being infinitely long. So it would require an infinite torque to rotate it to an erect position.

Short version - funny things happen with infinity. (one reason why maths is so much fun)

I read someone in the previous posts said that Hume opposed to Newtonian Science? Is that justified comment?

No. It's just people trying to put words in Hume's mouth that he never spoke or wrote.

If Hume had been arguing against Newton, he would have been written off as a loony, rather than being as respected as he was in his time. Newton had godlike status in 18th century Britain and nobody could have publicly said anything against his physical theories without being ridiculed.

Metaphysics yes. Sophistry no.

You assert that each world is real — Posty McPostface

I didn't say that. I wouldn't, because I don't know what it means.

The closest I came is to echo your use of the term 'a reality', which I did in the interests of following your terminology. I assumed by 'a reality' you meant 'one of the worlds in the many-worlds collection'. At any rate, that is what I meant by it, and such a use provides no basis for asserting that the sentences 'that world is real' or 'that world is not real' mean anything.

I think you will find it easier to understand MWI if you drop all attempts to divide things into the 'real' and 'not real'.

Then, which version of 'me' is the real one? — Posty McPostface

I don't know what you mean by 'the real one' and I suspect that you don't either.

So, why would this reality seem real to me, rather than some other? — Posty McPostface

This was answered in post 2, and has been answered again in the post immediately above this. You say you find the response 'circular' but you have not explained what you mean by that, or why you think that.

I think, as Deep Thought pointed out so long ago, the problem is that you don't really understand what the question is.

I think you might have a misunderstanding of MWI. There is nothing in it, so far as I know, that says anything about a relationship between consciousness and decoherence or wave function collapse.

As to the circularity you think you see in my explanation - why not point out where you think it lies and we can discuss it?

why is this reality real and apparent to my sensory apparatus and not any other, like one where I won the lotto — Posty McPostface

This reality is apparent to the sensory apparatus of P McP that is in this reality, and the reality in which P McP wins the lottery is apparent to the sensory apparatus of PMcP that is in the reality where PMcP wins the lottery.

Perhaps you are extrapolating from the quantum suicide thought experiment. The difference between that and this is that, in that thought experiment, in all the other realities there is no PMcP to experience that he has died, because he has died. So the only reality that can be experienced by a PMcP is the one in which he has survived. Those conditions do not apply in the lottery case, because neither winning nor losing the lottery causes your sensory apparatus to cease functioning.

One would think that with Newton's work in view that Hume would not make such silly mistakes. — Ron Cram

How fortunate for us then, that you have at last arrived at our forum to point them out to us.

http://alisongopnik.com/Papers_Alison/Gopnik_HumeStudies_withTOC.pdf

We can know that a baseball hit on a certain launch angle at a threshhold velocity will go over the fence for a homerun. Hume seems to be denying the possibility of exactly this kind of prediction.

Hume would doubt that we could know that. The doubt would rest on his observation of the problem of induction, which is related to this cause and effect issue, but not exactly the same.

Even if we sidestep the problem of induction by taking a principle of uniformity of nature as axiomatic, Hume's quote above stands because, without experience of bats and balls, we would have no reason to suppose that a bat hitting a ball would culminate in the ball travelling a long way. Newton's mechanics, which is the essential backdrop for an appreciation of Hume, was founded on empirical observation.

Hume was not denying the possibility of making predictions. Rather he highlighted that (1) predictions are based on past experience and (2) no matter how large a volume of past experience we have, the prediction cannot be made with certainty because of the problem of induction.

It's an interesting demonstration of the extent to which philosophy allows of different interpretations that while you and I both have an affinity for the mystical, you see Hume as discouraging of it and I see him as encouraging of it.

And if the answer derived from metaphysics is different from the answer derived from physics, which do you think we should accept?

We don't have to choose between them. They are different because they deal with different subject matter. Physics deals with predictions of future observations. Metaphysics deals with ontological claims.

We can accept neither, both or just one without contradiction. They cannot conflict.

However I would suggest that it is sensible to accept physics, as one cannot negotiate this world effectively without it. Whether to also accept metaphysical claims, and if so which ones, is much more individual.

Were people in his day truly unaware of kinetic energy and the cause and effect of one billiard ball hitting another? — Ron Cram

Hume was writing at a time shortly after Newton, when Newton and his theories had the status of the Beatles and their works - so Hume and the people for whom he was writing not only knew about Newton's mechanics but they were front of mind.

Hume's point was that people were mistaking a scientific theory that at base is simply a mathematical model for predicting future observations, for a metaphysical claim about the fundamental nature of things. The metaphysical claims, such as notions of 'cause', are interpretations of the theory of mechanics, not a part of it, and no end of confusion arises from confusing the two. Hume did his best to clarify the distinction, but human nature is incorrigible and people continued then, as they do now, to confuse metaphysics with physics.

I once heard a story of a young woman that was visited by an angel that told her she would bear a child that would save humanity from its suffering. I don't know if the story is true - it was based a long time ago in a land far away - but if the womit were true that sounds like a pretty good reason not to terminate the pregnancy.

"The only thing you can be certain of is that you cannot be certain of anything"

This sentence is clearly paradoxical and so cannot be true. — Yadoula

It is paradoxical because it should make an exception for that one thing. That is, it should say

"The only thing you can be certain of is that you cannot be certain of anything else"

Invoking the Principle of Charity, I think it is reasonable to presume that that non-paradoxical form is what was meant.

Whether it is true is a different discussion, but I contend that that form is not paradoxical.

OK, so I ask again. If it is so easy, why has it not been done already? — Sir2u

This question was answered in my antepenultimate post.

You responded with an opinion, based on a handful of expat Americans whose opinion you sought, that Americans don't want to pay for gun control. I suspect your opinion is mistaken.

I continue to hope that Trump won't be impeached because, amongst other things, that would make him a martyr of the hard right and lessen or even reverse the electoral damage they are starting to accrue.

I was delighted to find that I have an ally in James Comey on that. I saw the following quotes in a news article this morning and was so struck by their freshness that I wrote them in my diary:

Asked whether he believed Mr Trump should be impeached, Mr Comey replied, "I hope not because I think impeaching and removing Donald Trump from office would let the American people off the hook and have something happen indirectly that I believe they're duty bound to do directly."

"People in this country need to stand up and go to the voting booth and vote their values."

Hey but maybe they could convince people to do the same with their guns. — Sir2u

They managed to convince them in Australia in 1996-7. In fact they convinced them to take the guns not to the roadside but all the way to the local police station or other designated local collection facility and hand them in.

As noted above, all the problems you mention have been solved in other countries. Sure the solutions cost money but spending money to provide security is a fundamental role of government, not an optional extra. IIRC for Hobbes, it was the only role of government. Given what the US spends on defence and on spying on its own citizens, that principle seems to be perfectly well accepted there.

So why has it not been done already, surely there are sufficient experts in the country to arrange for all of these problems to be resolved efficiently. This then, is obviously one of the reasons why legislature has not been passed. — Sir2u

My impression is that that is not the reason. If it were the reason, the debate would be about the details of draft legislative bills. But it is not. The debate is about whether there should be a bill at all, and the NRA seeks to stop the discussion ever getting beyond that point to issues like working through the practical details. They would fear that if it got to the stage of discussing implementation they would have lost, as it would indicate that the electorate was accepting gun regulation as potentially reasonable and practical, rather than some devil-inspired commie plot.

The reason it has not been done already is simply that the NRA is enormously powerful and does not want gun control legislation of any form, no matter how practical and affordable it may be.

You ask a number of valid technical questions about how a proposed gun control act would work. I don't see the questions as being important to the philosophical debate though, because we can observe that they have practical, satisfactory answers from the simple fact that most OECD countries have rules of this type and they work in an acceptable, cost-effective manner.

For any proposed piece of legislation, however uncontroversial, I could ask dozens of important questions about who implements it, who pays, how it is enforced, what is done to protect abuse and so on, but they don't really have any bearing on the determination of whether to do the legislation unless there is reason to suppose they do not have satisfactory answers.

Imagine a discussion between a couple over whether to have a child. One might ask - but what will we feed them, how will we toilet train them, how will we teach them road safety, at what age will we allow them to have a mobile phone etc etc etc. But those questions are not part of the decision about whether to have a child, because we know that there are acceptable answers to them [pax, ye antinatalists. I am not saying that means the answer is always 'yes, we should have a child'. I am saying that it will turn on, amongst other things, the issues that concern you, like what are the moral implications of creating another being that can suffer, rather than how many hours of screen time per day we will allow them].

I'm not an academic, but wonder if the complaints made regarding academic philosophy would also apply to other academic disciplines/studies. — Ciceronianus the White

I think they do. In particular, the observation:

Although there is already a growing mountain of philosophical research that’s impossible to keep up with, it’s common for journal referees to reject your paper because you didn’t engage with [X] paper/book, where often [X] is either written by the referee themselves or someone they’re chummy with. — Rachel Williams

seems to apply equally to any other discipline that has mountains of research, including natural and social sciences.

As regards that problem at least, and more generally the unhealthy culture of KPIs based on publications and citations rather than meaningful innovation, the illness pervades all of academia, not just philosophy. Those people that have a strong enough vocation to persist in academia despite this repellant feature, and the many other drawbacks listed by @Hanover, have my admiration. But I can't help hoping none of my children take such a path.

I don't think alienation from aspects of production - be they factors or products - is intrinsically a problem. What is a problem is alienation from other people and from a sense of purpose, and it is easy for both of those to occur in modern, large-scale production processes. But with care that can be avoided and it is in the interests of employers as well as employees that that care be taken. Some employers do this well. Others do not.