Then you should choose a mathematical model that's up to the task. An independent random variable with the sample space of {Real, Unreal] isn't it. — Dawnstorm

What is the main issue here? Whether the observation O is real/not real, right? What model do you propose other than that which has to do with the probability of O being real/not? As far as I can tell, P(real) lies at the heart of the issue where P(A) means the probability of A. :chin:

Not if you treat O as an independent random variable. If you do that the math forces you to consider those cases, lest the math be rendered useless. — Dawnstorm

We need to be careful here. What exactly do you mean by "treat O as an independent random variable? Why is it that the "math forces you [me] to consider those cases"? Please clarify on those points. Lastly why should I treat O as an independent random variable?

You <i>can</i> ignore those cases of coure. Let me show you:

RRR - 12.5 %
RUU - 12.5 %
RUR - 12.5 %
RRU - 12.5 %
URR - 12.5 %
UUR - 12.5 %
URU - 12.5 %
UUU - 12.5 %

Turns into:

RRR - 12.5 %
RUU - 12.5 %
RUR - 12.5 %
RRU - 12.5 %
URR - 12.5 %
UUR - 12.5 %
URU - 12.5 %
UUU - 12.5 %

And your probability that O is real remains 50 %, because 12.5 % are 50 % of 25 %. — Dawnstorm

You calculated 50% of 25% = 12.5%. Yes, there's nothing wrong with the arithmetic but what does it mean? For example 75% of 200 = 150 can mean I took three-fourths of the 200 dollars you had and now I have 150 dollars. Can you express your calculation that "12.5% are 50% of 25%" in words that are meaningful in the context we're discussing?

You're talking experimental probability and it's a fallback measure when theoretical probability can't be calculated.

In our case, theoretical probability can be calculated. — TheMadFool

One of the reasons I like your posts is because you make me consider new avenues of an argument. Perhaps this will turn into a discussion between experimental and theoretical probability!
https://www.softschools.com/math/topics/theoretical_vs_experimental_probability/

As noted above, the theoretical probability of a coin flip is 50%. The experimental probability happens after we count the flips. So if we flip the coin 10 times, and it comes up heads 4 times, we have an experimental probablity of 60/40% for heads/tails.

Yet each probability is still based on a knowable. In the case of the theoretical probability, why did we say 50% versus lets say, 77//23%? It is because there is knowledge that there are only two sides. The coin doesn't appear to be weighted either. So it is reasonable to postulate a probability of 50% from the postulates we know.

But, we do not have two sides to the problem you've presented. We have identified two possibilities, but that does not mean we have two likelihoods. A better way to imagine this is a dice roll. Lets say you're on a game board and you have 1 space left to go to win. You must roll a 1 to enter, anything above will not let you. You have two options next turn. Win the game, or not win the game next turn. But there are 5 different outcomes we know of that you could lose the game, if we're using a six sided die. It would be unreasonable to say its a 50% chance, even with theoretical probability.

The situation you presented is a win the game/lose the game possibility scenario, but you haven't told us what dice you are using. It could be a 6 sided, 8 sided, 10 sided, or more! All we know is that we need to go one more space to win, but we don't know the rules of movement. You need to define when something would be real, and when it would not be real.

And perhaps I can help with this example so we can get back to your main point. Let us say you and a group of friends are in a cave that has harmless, but hallucinatory gas. The theoretical probability and experimental probability over the long term have been found to be that what you observe apart from the cave walls themselves has a 50% chance of being real. You and x number of people enter the cave. All of you see in the distance what appears to be a bat fly by a distant tunnel. Does your argument still hold now?

I'd prefer you stick to your lottery example because this new scenario with the die roll is identical to it. And I've already informed you that the probability for observation O being real can be broken down into three choices, mutually exclusive and jointly exhaustive:

P(real) = probability of observation O being real

1. P(real) < 50%
2. P(real) = 50%
3. P(real) > 50%

Notice I'm not making the mistake, a mistake you were so kind to point out, of saying that just because there are two options, real/not real, the probability is 50%.

What choice among 1, 2, and 3, is the correct value for P(real)?

And I've already informed you that the probability for observation O being real can be broken down into three choices, mutually exclusive and jointly exhaustive: — TheMadFool

Yes, and I have countered that by stating that doesn't work, and an alternative. So currently, I have stated this that no, it cannot be broken down in a probability, because you do not have a die.
If you so like, you do not have the number of power balls in a lottery. Is it 3,4,5? How many numbers are on those power balls? These things are what allow us to determine the theoretical probability.
But you have none of this information in regards to what is real, and what is not. I've even given you an example in the cave that can solve this problem so you can address your original argument.
You have decided not to address that.

Repeating the same thing in light of this does not further the discussion, it ignores the discussion. If there is something you don't understand in what I'm putting forward, feel free to ask. If you do understand, but are not addressing it, then I have said my peace and will leave here with the knowledge that you were unable to refute that your argument is incorrect. If this is fine by you, then so be it. If not, then you'll have to do more then simply repeat what you've already stated.

Yes, and I have countered that by stating that doesn't work, and an alternative. — Philosophim

What do you mean by "that doesn't work"? In what sense doesn't work. There are three possibilities here:

P(real) = probability that O is real

1. P(real) < 50%
2. P(real) = 50%
3. P(real) > 50%

A better, more rigorous way, to express it would be:

1. 0 <= P(real) < 50%
2. P(real) = 50%
3. 50% < P(real) <= 100%

We know that 0% <= P(real) <= 100%

In other words, I've provided you choices that represent every possible value P(real) can assume. So, "that doesn't work" is essentially nonsense.

In other words, I've provided you choices that represent every possible value P(real) can assume. So, "that doesn't work" is essentially nonsense. — TheMadFool

Ok, you're ignoring the discussion of theoretical probability, the cave with the bat scenario, and everything else I've put forward and simply keep repeating the same thing without addressing them. You have made your choice. Enjoy the discussion with others, there is nothing more to say at this point.

Ok, you're ignoring the discussion of theoretical probability, the cave with the bat scenario, and everything else I've put forward and simply keep repeating the same thing without addressing them. You have made your choice. Enjoy the discussion with others, there is nothing more to say at this point — Philosophim

I'm not ignoring anything. You're unable to crtique my claim in a way that sheds light on the issue. Sorry, if you feel that way because I responded to your concern about how the existence of only two options e.g. like win/lose in a lottery, doesn't imply the odds of winning/losing is 50%. After that you haven't said anything substantial. Sorry. :smile:

What is the main issue here? Whether the observation O is real/not real, right? What model do you propose other than that which has to do with the probability of O being real/not? As far as I can tell, P(real) lies at the heart of the issue where P(A) means the probability of A. :chin: — TheMadFool

You haven't quite made clear what "observation O is real/not real" means.

Let's say you see a unicorn, and you ask others if they can see it, too. How would you describe this in terms of probability-prone variables?

This is what I would do:

There are three observations: O(x, y, z). Variable O can have two values:

"sees unicorn"/"does not see unicorn".

We know the values of the variable. The input comes straight from our experience. So O is not a random variable. No probability. It's either "sees" or "doesn't see", and we get the values by asking.

The second variable is the event E, which also a binary: "there is a unicorn" and "there is no unicorn".

What you're doing is basically estimating the likelihood of E with a coinflip: you set E as a random variable. It's random, because we have no way of knowing the value because observation doesn't count.

But what we're doing when we ask others if they're seeing the unicorn, too, is not asking whether the unicorn is real.

Your case is this:

O(x) = sees Unicorn
O(y) = sees Unicorn
O(z) = sees unicorn

E = ?

You calculate the probablity for E the following way:

First you assume that

P(E) = 50 %
P(~E) = 50 %

And then you calculate P(E(x))*P(E(y))*P(E(z)) = 12.5 %

However it's far from clear what it means to cross E with person X, Y, Z. There's no reason to bring in the observations at all. P(E) is always 50 %, as per your assumption.

Even in the case:

O(x) = does not see Unicorn
O(y) = does not see Unicorn
O(z) = does not see Unicorn

P(E) would still be 50 %, since you derived the likelihood simply from the two logical possibilities.

O(x) simply inspires the question, but E is an independent random variable. Basically, if there is a unicorn (E), then there is a unicorn no matter what values O take. There is no E(x) that corresponds to O(x), for example. There's only E.

What you propose looks like a game:

No matter who sees the unicorn, they all flip a coin, and if the coin comes up heads they say the unicorn is there, and if the coin comes up tails they say it's not there. But they'll only accept that the unicorn is there by full consensus, so they keep flipping coins until it's all heads or tails. In that case the likelihood that the unicorn exists or doesn't exist, as per consensus, would be equal, but only because there are exactly 2 ways the game can end. With a different likelihood the probability changes according to how many constellations end the game, and how many of those constellations are dis/favourable.

You haven't quite made clear what "observation O is real/not real" means. — Dawnstorm

The observation O is real in the same sense that we consider, for instance, that the Eiffel tower is real and the observation O is not real in the same sense that, for example, flying dragons are not real. If you have any issues, please think of two things which you, for you, are real and not real and apply the same criteria you applied to make that distinction to our discussion.

There are three observations: O(x, y, z). Variable O can have two values:

"sees unicorn"/"does not see unicorn".

We know the values of the variable. The input comes straight from our experience. So O is not a random variable. No probability. It's either "sees" or "doesn't see", and we get the values by asking. — Dawnstorm

Don't you see we've already gone beyond this point? We're not debating whether X observes or doesn't observe, whether he sees/doesn't see? X has made an observation, X has seen. What concerns us from thereon is whether what X has observed, O, is real/not real.

No matter who sees the unicorn, they all flip a coin, and if the coin comes up heads they say the unicorn is there, and if the coin comes up tails they say it's not there. But they'll only accept that the unicorn is there by full consensus, so they keep flipping coins until it's all heads or tails. In that case the likelihood that the unicorn exists or doesn't exist, as per consensus, would be equal, but only because there are exactly 2 ways the game can end. With a different likelihood the probability changes according to how many constellations end the game, and how many of those constellations are dis/favourable. — Dawnstorm

X, Y, and Z are not flipping coins to decide whether O is real or not. You've been misled to think in those terms because of the fact that the probability here is 50% - looks like the chances of heads or tails in a coin flip which is 50%.

The last part of your post makes sense, the part where you said "with a different likelihood the probability changes..."

If it's likely that the observation O is real, the calculations vindicate the repeatability principle. For instance, if the probability that the observation O is real, say, 90% then the probability of O being real if all 3, X, Y, and Z observe O is 90% * 90% * 90* = 72.9% and the probability that O is not real = 27.1%. However, to assign a likelihood of greater than 50% to O being real on a single observation is a problem. Why? What reasons can you give to think O, based on a single observation of O, is likely i.e. has a probability of greater than 50%?

Going the other way, if you begin by saying that O is unlikely to be real (assign a value less than 50% to the probability of O being real), you need to give an explanation for why you think so. Can you?

The only value for the probability that O is real from a single observation of O that makes sense is 50% because you're not committing to anything regarding the reality of O. It could be real, it could be not real, 50:50 odds.

For instance, if the probability that the observation O is real, say, 90% then the probability of O being real if all 3, X, Y, and Z observe O is 90% * 90% * 90* = 72.9% and the probability that O is not real = 27.1%. — TheMadFool

Probability that O is real: 90 %
Probability that O is not real: 10 %

If all three observe O, the probability that O is not real is 10%*10%*10% = 0.1%. So the likelihood that O is real must be 99.9% right. But wait. 90%*90%*90% is 72.9% (as you correctly calculated). 72.9%+0.1%=73%. We're missing 27%.

In your way of calculating you're missing 75% when you assume a probability of 50%, and 27% when you're using a probability of 90%.

Instead of realising we're missing cases, you proclaim a paradox. That's not good maths.

First, if I observe x, that is presumptive evidence that x happened. There is no a priori reason to suppose that x did not happen.

Second, the purpose of repeatability in science is not to confirm or dispute what you observed, but to see if you have correctly identified the factors causing x. Perhaps x was caused by some extraneous factor you have not identified. If I can set up my experiment using all the factors you identified, and observe x, that is good evidence that you have correctly identified the relevant factors.

Third, there is no rational basis for assigning numbers to things we cannot count or measure. Among these innumerables is the "probability" subjectively assigned to beliefs, and the "utility" of acts and decisions. Bayesian probability is simply transvestite prejudice -- prejudice in mathematical garb. Putting lipstick on it does not make it rational.

First, if I observe x, that is presumptive evidence that x happened. There is no a priori reason to suppose that x did not happen. — Dfpolis

:ok:

Second, the purpose of repeatability in science is not to confirm or dispute what you observed, but to see if you have correctly identified the factors causing x. Perhaps x was caused by some extraneous factor you have not identified. If I can set up my experiment using all the factors you identified, and observe x, that is good evidence that you have correctly identified the relevant factors. — Dfpolis

We're not on the same page on this. The very idea of repeatability is to either confirm or disconfirm an observation. There's no need to bring up the issue of causality because at the end of the day it's about an observation - whether it can be observed by different people in different settings.

Third, there is no rational basis for assigning numbers to things we cannot count or measure. Among these innumerables is the "probability" subjectively assigned to beliefs, and the "utility" of acts and decisions. Bayesian probability is simply transvestite prejudice -- prejudice in mathematical garb. Putting lipstick on it does not make it rational. — Dfpolis

What concerns us is whether a given observation is real or not. Either it's real or it's not. If one person makes an observation then the odds of that being real are 50:50. Any other value you assign would amount to begging the question - assuming the very thing you're unsure of.

You have hit the nail in the head. This is a serious problem.

Have you read "The Black Swan"? It's a notorious no-BS book regarding this exact subject.

You have hit the nail in the head. This is a serious problem.

Have you read "The Black Swan"? It's a notorious no-BS book regarding this exact subject. — dussias

I was actually hoping to be proven wrong because the entire scientific community can't be wrong about this.

We're not on the same page on this. The very idea of repeatability is to either confirm or disconfirm an observation. — TheMadFool

And its interpretation. Perhaps you observed x because your electronics failed, not because of what you believed was the experimental arrangement. Perhaps your sample was contaminated or unrepresentative. I can think of many possible scenarios, none of which call your experience or veracity into question, only the adequacy of its description.

There's no need to bring up the issue of causality because at the end of the day it's about an observation - whether it can be observed by different people in different settings. — TheMadFool

Every observation of the same supposed type is a different token. None is exactly the same. You report, "I did x, and observed y." Someone else does x, but does not observe y. Does that mean that you lied? Or that y was a miracle? Or does it mean that factors not included in x lead to the observation of y? All are possible, but statistically, the last is most common.

What concerns us is whether a given observation is real or not. Either it's real or it's not. If one person makes an observation then the odds of that being real are 50:50. — TheMadFool

You will have to define "real" in some non-standard way to make sense of this. You agreed with my first point, which means we have a presumption of reality.

On what basis are you calculating the probabilities? The existence of two possibilities does not justify the assumption that they are equal. By your logic, if I flip a coin it may either land balanced on an edge or not, so the probability of a flipped coin ending on an edge is 50%. Care to place a bet?

And its interpretation. Perhaps you observed x because your electronics failed, not because of what you believed was the experimental arrangement. Perhaps your sample was contaminated or unrepresentative. I can think of many possible scenarios, none of which call your experience or veracity into question, only the adequacy of its description — Dfpolis

Exactly. All what you said boils down to the issue of whether a single individual's observation is real or not.

Every observation of the same supposed type is a different token. None is exactly the same. You report, "I did x, and observed y." Someone else does x, but does not observe y. Does that mean that you lied? Or that y was a miracle? Or does it mean that factors not included in x lead to the observation of y? All are possible, but statistically, the last is most common. — Dfpolis

:ok: However, I feel introducing causality at this point is entirely unhelpful and distracting. The issue is plain and simple: One person observes something. Is it real or not? Scientific repeatability requires other people to make the same observation and that being successful validates the observation. There's a flaw in this as I've pointed out.

On what basis are you calculating the probabilities? — Dfpolis

It's very simple actually. If I happen to make a single observation I'm uncertain whether it's real or not which, phrased differently, means I have no clue whether it's likely to be real or unlikely to be real. Now, I must assign a numerical value to the probability of my observation being real. If I assign a value greater than 50% to the probability that means I think it's likely but this contradicts my assertion that I'm uncertain - I don't know whether the observation is likely or unlikely to be real. The same logic applies if I assign a value to the probability less than 50% in which case I would be saying the observation is unlikely to be real and that again contradicts my state of uncertainty. The only probability value that fits my epistemic state - uncertainty (not knowing whether likely/unlikely) - is 50%.

All what you said boils down to the issue of whether a single individual's observation is real or not. — TheMadFool

Not by my definition of "real." If your meter read 17, for whatever reason, you really observed 17.

I have no clue whether it's likely to be real or unlikely to be real — TheMadFool

Of course you do. Unless you have a sensory, neural or cognitive disorder, all the clues point to the fact that what you observed what was really there. As I said earlier, your use of "real" is non-standard.

If I assign a value greater than 50% to the probability that means I think it's likely but this contradicts my assertion that I'm uncertain — TheMadFool

Your subjective certainty is more likely to reflect your childhood experience than your observation. If you are only 50% sure that what you saw is real, that says your self-confidence has been harmed -- not that there is any question involving reality.

The only probability value that fits my epistemic state - uncertainty (not knowing whether likely/unlikely) - is 50%. — TheMadFool

So the odds of a coin landing on edge is 50%.

Not by my definition of "real." If your meter read 17, for whatever reason, you really observed 17. — Dfpolis

Does that single measurement suffice in, say, writing a paper that's to be submitted in a peer-reviewed journal? I don't think so.

Of course you do. Unless you have a sensory, neural or cognitive disorder, all the clues point to the fact that what you observed what was really there. As I said earlier, your use of "real" is non-standard. — Dfpolis

That could be it but I'm mainly interested in the distinction between real and hallucination - this has priority over whatever may follow, right?

Your subjective certainty is more likely to reflect your childhood experience than your observation. If you are only 50% sure that what you saw is real, that says your self-confidence has been harmed -- not that there is any question involving reality. — Dfpolis

Indeed it's an issue of confidence. How confident are we that a certain observation is real or not? By the way, do you mean that you would assign a value other than 50% to the probability that a single observation is real? What are your reasons for that?

So the odds of a coin landing on edge is 50%. — Dfpolis

There is no third option between being real and not being real.

1. Not everyone has picked up on it, but the "real" and "not real" thing is a bad framing, right from the start. As Mww correctly pointed out, we know the observation is real, the question is whether it points to some new phenomenon or just, say, noise in a cable or whatever.

2. The idea that if we have two options then those two options must have 50% probability, is a logical fallacy. I can't seem to find the name of the fallacy right now, but it is a known, named fallacy. Probability does not work like that.

3. Multiplying the "50% chance of being real" at the end is also wrong. The whole point of repeating the experiment is to raise our confidence level that it wasn't just experimental error. To the extent these numbers make any sense at all, the 50% would need to be raised after each positive result.

1. Not everyone has picked up on it, but the "real" and "not real" thing is a bad framing right away. As Mww correctly pointed out, we know the observation is real, the question is just whether it points to some new phenomenon or just, say, noise in a cable or something. — Mijin

I beg to differ. When you observe something, say a reading on weighing scale that reads 12 kg, what's the checklist you have to go through before you come to the conclusion that there is a mass that's 12 kg?

2. The idea that if we have two options then those two options must have 50% probability is a logical fallacy. I can't seem to find the name of the fallacy right now, but it is a known, named fallacy. Probability does not work like that.

That raises a lot more questions than answers, friend.

3. Multiplying the "50% chance of being real" is also wrong. The whole point of repeating the experiment is to raise our confidence level that it wasn't just experimental error. To the extent these numbers make any sense at all, the 50% needs to be updated after each positive result.

It's being updated - probabilities are being multiplied. What other mathematical operation would you say is the correct method of updating to the final probability?

I beg to differ. When you observe something, say a reading on weighing scale that reads 12 kg, what's the checklist you have to go through before you come to the conclusion that what there is a mass that's 12 kg? — TheMadFool

What wouldn't be on the checklist is the words "real" or "not real".
The checklist would be on what I could infer from my observation and what further data I need to collect to raise confidence in such inferences.
It's nothing to do with real or not real.

That raises a lot more questions than answers, friend. — TheMadFool

The question for me is whether you're interested in understanding this, or if the whole thread is just for you to proselytize. Numerous examples have been given as to why the number of alternatives has nothing to do with their probability. Ergo there is no reason to assign a probability of 50%.

It's being updated - probabilities are being multiplied. What other mathematical operation would you say is the correct method of updating to the final probability? — TheMadFool

It's not as simple as one operation; the actual calculation of a P value depends on the specifics of how much data is being collected, what the noise range is for that data and so on. But yes, as we gather more data our confidence in a proposition goes up.

Multiplying this number, as you've done, comes up with obviously absurd results. Imagine I am trying to figure out if you ate my cookie. If there are cookie crumbs on your shirt, I'm 80% confident you did it. If your fingerprints are on the cookie jar, 50%. But if I see both things, then by your logic, it's somehow less likely than either individual piece.
(and note, even if you quibble with the actual numbers, the point is, as long as they are less than 100% this will always be the case; multiplying them will decrease our confidence, by your logic.

What wouldn't be on the checklist is the words "real" or "not real". — Mijin

That's odd. The first thing that needs to be established is whether an observation is real or not. If you don't have that on your checklist, you could end up, well, chasing mirages to, say, measure the amount of water in it.

Also consider a scenario where you observe a gold ingot on a table. Before you start measuring its weight, you must first determine whether the ingot is actually real or not, right? You couldn't measure the weight if it weren't real.

The question for me is whether you're interested in understanding this, or if the whole thread is just for you to proselytize. Numerous examples have been given as to why the number of alternatives has nothing to do with their probability. Ergo there is no reason to assign a probability of 50%. — Mijin

Ok. What you say implies that the probability has to be something other than 50%. Two possibilities - either less than 50% or more than 50% - make your choice and explain why.

It's not as simple as one operation; the actual calculation of a P value depends on the specifics of how much data is being collected, what the noise range is for that data and so on. But yes, as we gather more data our confidence in a proposition goes up.

Multiplying this number, as you've done, comes up with obviously absurd results. Imagine I am trying to figure out if you ate my cookie. If there are cookie crumbs on your shirt, I'm 80% confident you did it. If your fingerprints are on the cookie jar, 50%. But if I see both things, then by your logic, it's somehow less likely than either individual piece.
(and note, even if you quibble with the actual numbers, the point is, as long as they are less than 100% this will always be the case; multiplying them will decrease our confidence, by your logic. — Mijin

If three people are involved, the probability that each one's observation being real is 50%. The probability that all three of them are observing something real is calculated thus: 50% * 50% * 50%

P(A & B) = P(A) * P(B/A) but since these are independent events P(B/A) = P(B) and so we get the following: P(A & B) = P(A) * P(B)

Also consider a scenario where you observe a gold ingot on a table. Before you start measuring its weight, you must first determine whether the ingot is actually real or not, right? You couldn't measure the weight if it weren't real. — TheMadFool

No; there is clearly something being weighed here, the observation is "real".
However, the proposition that I have N kilograms of gold needs further investigation to confirm.

What you say implies that the probability has to be something other than 50%. Two possibilities - either less than 50% or more than 50% - make your choice and explain why. — TheMadFool

The probability will depend on the specifics of what's being measured and how. It's something calculated, not something known apriori from just saying "well either it is or it ain't".
If you look up how to calculate p value you find it's not a trivial topic.

If three people are involved, the probability that each one's observation being real is 50%. The probability that all three of them are observing something real is calculated thus: 50% * 50% * 50% — TheMadFool

No it's not calculated like that. Can you respond to the argument I just made, refuting this (with the cookie example)? Or indeed, any of the other examples that others have posted?

No; there is clearly something being weighed here, the observation is "real".
However, the proposition that I have N kilograms of gold needs further investigation to confirm.
Is the difference clear now? — Mijin

So you would immediately run toward a shimmering image of water in a desert? Every such instance is for 100% a desert oasis? What about hallucinations?

The probability will depend on the specifics of what's being measured and how. It's something calculated, not something known apriori from the number of alternatives. — Mijin

Well, what are the "specifics"? As far as I can tell an observation is either real or not real - two mutually exclusive and jointly exhaustive possibilities. I'd be more than happy to see your version of the mathematical model appropriate for the issue at hand.

No it's not calculated like that. Can you respond to the argument I just made, refuting this (with the cookie example) — Mijin

How is it calculated then? The cookie example is an unnecessary diversion. Anyway, here goes...

If there are cookie crumbs on your shirt, I'm 80% confident you did it. If your fingerprints are on the cookie jar, 50%. But if I see both things, then by your logic, it's somehow less likely than either individual piece. — Mijin

I admit that assigning initial probability values is rather arbitrary BUT not in the case of the scenario that I presented in the OP - the value 50% is fully justifiable. However, once this is done, probability calculus takes over - there are fixed rules on how to calculate combined probabilities and that's that.

All I can say is Bayes' theorem seems applicable but that, from preliminary examination, doesn't solve the problem.

Yeah I'm done here.
None of this is complicated to someone who genuinely wants to learn and understand why their intuition seems contrary to scientific convention. This is clearly not the case here.

Does that single measurement suffice in, say, writing a paper that's to be submitted in a peer-reviewed journal? I don't think so. — TheMadFool

That is a totally different question than asking if the meter reading was real. The question of reality is ontological, that of what suffices for publication is methodological.

I'm mainly interested in the distinction between real and hallucination - this has priority over whatever may follow, right? — TheMadFool

No, it does not have priority. The presumption is that unless you have a medical history of hallucinations, what you see is really there. Priority goes to relevant questions, not to vague and unsupported possibilities. In the first quotation above, you posed the standard of publication in a peer reviewed journal. No such journal has ever asked me to submit medical records showing I have no history of hallucination or mental illness.

How confident are we that a certain observation is real or not? By the way, do you mean that you would assign a value other than 50% to the probability that a single observation is real? What are your reasons for that? — TheMadFool

We are morally certain that our careful observations are correct. Moral certitude means that we can rely on a proposition in good conscience. It does not mean that our belief in it is infallible.

I assign no numerical values to what cannot be counted or measured, because, strictly speaking, it is meaningless to do so. Of course, people do assign probability numbers to their beliefs. One might interpret such probabilities in terms of the odds of a fair bet, but such numbers are not a measure of the probability of a proposition being true, because there is no such probability. If the proposition is meaningful, by which I mean that it asserts some determinable fact, then it is either true or false relative to a determined context.

So the odds of a coin landing on edge is 50%. — Dfpolis

There is no third option between being real and not being real. — TheMadFool

It depends on what you mean by "being real." Still, the existence of a third option is irrelevant to what I said.

Your claim is that "X is either y or not y" justifies assigning equal probabilities to y and not y. Since a flipped coin will either land balanced on its edge or not, then (by your logic) there is a 50% chance that it will end on edge. I do not see how you can escape this conclusion.

Maybe you should start someplace like this or this. There's lots of really solid material online about what you're trying to understand.

2. If all 3, X, Y, and Z observe O then the probability of O being real (call this R) = 50% * 50% * 50% = 12.5% — TheMadFool

1. If all 3, X, Y, and Z observe O then the probability of O not being real (call this NR) = 50% * 50 * 50% = 12.5% — TheMadFool

This is peculiar. Because the probability of reality of O is a subjective probability, therefore the mathematician has to consider the reality probability independent probability from each other.

Let me illustrate. Given a coin of heads and tails on the sides. Given that the coin is tossed, the probability of heads or tails in one toss are equal, at 50% each.

Now. X, Y, and Z each toss the coin once. You say that the probabily of tail is 12.5%, and the probability of heads is also 12.5% of any given ONE toss. That is simply absurd. The probability that the coin will land on heads (or else tails) in each one of the three times of the tosses, is 50% times three tosses, and averaged over three tosses.

If the observation decided to be true is 50-50 by each of X, Y, and Z, then the observation's probability is (50%+50%+50%)/3, just like in the coin toss.

As it stands, X, Y and Z will each have a belief: O was real, or O was unreal. They can't beleive both that it was real and unreal.

So the probability that all three think it's real is 12.5%. The probability that all three think O is unreal, is 12.5 percent. The probability that one will think it's real and two will thing it's unrea is 37.5 percent The probability that one will think O is unreal and two will think it's real is 37.5 percent.

That is a totally different question than asking if the meter reading was real. The question of reality is ontological, that of what suffices for publication is methodological. — Dfpolis

There are two layers to observational data. First concerns its reality and the second concerns its correctness. For both, we need multiple observers. The probability calculations are the same for both and the error commited is identical in both cases.

No, it does not have priority. The presumption is that unless you have a medical history of hallucinations, what you see is really there. Priority goes to relevant questions, not to vague and unsupported possibilities. In the first quotation above, you posed the standard of publication in a peer reviewed journal. No such journal has ever asked me to submit medical records showing I have no history of hallucination or mental illness. — Dfpolis

So, if I'm hallucinating myself conducting a high-precision experiment with hallucinated equipment and hallucinated colleagues, I can publish my findings in a scientific journal?

We are morally certain that our careful observations are correct. Moral certitude means that we can rely on a proposition in good conscience. It does not mean that our belief in it is infallible.

I assign no numerical values to what cannot be counted or measured, because, strictly speaking, it is meaningless to do so. Of course, people do assign probability numbers to their beliefs. One might interpret such probabilities in terms of the odds of a fair bet, but such numbers are not a measure of the probability of a proposition being true, because there is no such probability. If the proposition is meaningful, by which I mean that it asserts some determinable fact, then it is either true or false relative to a determined context. — Dfpolis

The choices available are simple: an observation is real or not. You don't know which. Probabilistically, 50% chance of it being real and 50% chance of it being not real. There's no two ways about it.

It depends on what you mean by "being real." Still, the existence of a third option is irrelevant to what I said.

Your claim is that "X is either y or not y" justifies assigning equal probabilities to y and not y. Since a flipped coin will either land balanced on its edge or not, then (by your logic) there is a 50% chance that it will end on edge. I do not see how you can escape this conclusion. — Dfpolis

There are two possibilities (real/not real) and either one is as likely as the other. 50% chance of being real and 50% chance being not real.