I think you're conflating two different expectations. I think your post should read :

Let $y$ be the value of the chosen envelope and $z$ be the value of the unchosen envelope.

1. Let $y = x$ or $y = 2x$

2. $E[z | y ] = {5\over4}y$ and $E[y | x] = {3\over2}x$

MacTaggart's remarks concerning the A and B series are relevant here. The premises of the SB paradox are tenseless and so refer only to the B series , whereas Sleeping Beauty's question is tensed and so refers to her ability to correlate her A series to her B series.

The problem's premises can be written

$\begin{array} {|r|r|}\hline Sample\ Space & Awoken\ Mon & Awoken\ Tue & Joint\ Probability \\ \hline Heads & True & False & 1/2 \\ \hline Tails & True& True & 1/2 \\ \hline \end{array}$

According to this parameterisation, it doesn't make sense to ask SB "what Day is Today?" for "Today" isn't a random variable of the sample-space. (Thirders implicitly ask this question). But all that can be talked about, according to this parameterisation, is the state of the coin and whether SB is awoken on both monday and tuesday, and not "what day is today?".

To ring home the point, suppose that if the coin lands tails then the experimenters change SB's location from London on Monday to Paris on Tuesday. in which case, we can eliminate the notions of time and date from the premises of the the problem, by changing "Monday" to "London" and "Tuesday" to Paris.


Recall that MacTaggart decided the A series to be "unreal" and contradictory. From the B series point of view, "today" is subjective and refers to SB's mental state that comprises a variable that is separate and independent of the "calendar" events in the B series, such as wakening a subject up in London and then in Paris. Under the assumption of SB undergoing amnesia however, her mental state is uncorrelated with her understanding of B series events, and hence uncorrelated with the state of the coin.

No, the question has nothing whatsoever with her mental state.

She is being asked, given that she is awakened, what is the probability of heads. If she is awakened 1000 times for every tails and once with heads, given enough coin flips you can see that it is overwhelmingly likely to be tails, even though the probability of heads remains 50%. This is independent of her mental state. — hypericin

Her "mental state" M refers to her epistemic state upon waking. In the context of repeated trials, it would include her knowledge of previous trials. But the paradox doesn't assume repeated trials. If it did in the manner of your example, then

P ( C = tails |M) = 1000/1001

In which case, her mental state has proven to be a very reliable indicator of the state of the coin.

But this isn't part of the question's premises. We are told that in the context of a single trial, that her mental state is conditioned so as to have no previous memories of waking up. This condition implies that her mental state is statistically independent to the state of the coin, i.e.

P(C | M ) = P(C) = 1/2

Here i have assumed that she is told that the physical probability of the coin landing heads is 1/2. If she isn't told this, then she should refrain from assigning any distribution to P(C | M)

Good question: no. Solipsism is the idea that everything is in my mind, whereas analytical idealism is the idea that both our minds are in a universal mind. — Bob Ross

In my view :

Subjective Idealism and solipsism aren't ideas, but a tautological understanding that the meaning of all propositions is ultimately reducible to whatever is perceived or thought in the first-person at the end of the day.

Naturalism isn't an idea, but an understanding that the meaning of inter-subjectively valid propositions, such as those concerning the properties of natural kinds, cannot be identified with particular thoughts and experiences of the first person. For this reason, scientific naturalists talk about meaning in terms of potential experiences through the use of conditionals, counterfactual analysis , and perspectivally invariant abstract properties.

But this shouldn't be taken to imply that naturalism discounts experience as being it's semantic foundation - after all, naturalists pride themselves for judging the validity and soundness of their theories in terms of empirical evidence. And every naturalist must determine for themselves how they should privately cognize the inter-subjective propositions expressed by their fellow community. The assumption that naturalists can relate to their subject matter implicitly appeals to the existence of semantic "bridging" rules for converting the inter-subjective representations of naturalistic language into phenomenal first-person understanding.

So i don't consider Naturalism and solipsism or idealism to be incompatible per-se. I see them as comprising different semantic aspects of thought and language. Nonetheless, their logics are radically different, lending to the false impression of conflict.

E.g "Being is perception" is an unavoidable tautology of non-representational idealism that is necessarily appealed to whenever an observer interprets a physical proposition in terms of his personal experiences (regardless of whether he self-identifies as an idealist) .

On the other hand, "perception is representation" is an unavoidable tautology of naturalism for universalising intersubjective semantics in an abstract fashion that isn't dependent upon the perceptual judgements of any particular observer.

Taken together, "Being is Perception" and "Perception is Representation" don't necessarily imply that "Being is Representation", as is often naively assumed by materialists, if one understands these principles as referring to different and non-overlapping aspects of semantics.

Here's another analysis that only refers to credences , i.e subjective probabilities referring to the mental state of a believing agent - as opposed to physical probabilities referring to the physical tendencies of "mind-independent" reality.

According to this interpretation of the paradox, the paradox is only psychological and concerns the mental state of an agent who derives contradictory credence assignments that conflict with his understanding of his mental state. So this interpretation isn't adequately analysed by appealing to a physical model.

Suppose the participant called Bob, before opening either envelope, tells himself that he knows absolutely nothing regarding the smallest quantity of dollars S that has been inserted into one of the two envelopes:

Before opening either envelope, Bob reasons that since he knows absolutely nothing about the value of S, that he should appeal to Laplace's principle of indifference (PoI) by assigning equal credence to any of the permissible values for S. He justifies this to himself by arguing that if he truly knows nothing about the value for S, then he doesn't even know the currency denominations that is used to describe S. So he assigns

P(S = s) = P(S = 2s) = P(S = 3s) = P(S = 4s) ..... for every positive number s.

There is only one "distribution" satisfying those constraints, namely the constant function P(S) = c ,
that cannot be normalised, where c is any positive number which can therefore be set to c = 1. This is called an 'improper prior', and it's use often results in conflicting credence estimates, as shown by other paradoxes, such as Bertrand's Paradox.

Having chosen this so-called "prior", Bob reasons that when conditioned on the unknown quantity S, the unknown quantity X in his unopened envelope has the value S with a subjective probability p, else the value 2S with subjective probability (1 - p):

P(X | S) = p Ind (X,S) + (1- p) Ind (X, 2S) (where Ind is the indicator function)

He again appeals to PoI and assigns p = 1/2 (which merely a non-informative proper prior)

Substituting his choices for P(S) and p, Bob realises that the unnormalised joint distribution P(S,X) describing his joint credences for S and X is

P (S , X) is proportional to 0. 5 Ind (X ,S) + 0.5 Ind (X , 2S)

Summing over S, he derives his credences for X, namely P(X) that he realises is also an improper prior.

P(X) is proportional to 1

Consequently, his subjective 'unnormalized' posterior distribution (which does in fact sum to 1, but is nevertheless the ratio of the two unnormalised distributions P(S,X) and P(X) ) is described by

P (S | X) 'is proportioanal to' 0. 5 Ind (X ,S) + 0.5 Ind (X , 2S)

Bob wonders what would happen if he were to naively compute expectations over this 'unnormalised' distribution. He decides to compute the implied expectation value for the unopened envelope V conditioned on the value of his unopened envelope:

P (V = 2x | X = x ) = P(S = x | X = x) = 0.5
P(V = 0.5 x | X = x) = 1 - P(S = x | X = x) = 0.5

E [V | X ] = 5/4 X

Bob decides that he cannot accept this expectation value, because it contradicts his earlier credences that are totally agnostic with regards to the states of S and X. However, Bob also knows that this conditional expectation value is a fallacious value, due to the fact that his subjective probability distribution P(S | X) isn't really normalised, in the sense of it being the ratio of two unnormalised distributions P(S,X) and P(X).

Bob therefore knows how to avoid the paradox, without needing to revise his earlier credences.

Crucially, Bob realises that his 'unnormalised' subjective distribution P(S | X) should only be used when calculating ratios of P(S | X) .

So instead of strongly concluding that E [ V | X ] = 5/4 X that involved averaging with respect to an unnormalised posterior distribution P(S | X), he reasons more weakly to only conclude

P(V= 2x | X = x) / P( V = 0.5x | X = x) = 1

Which merely states that his credences for V=2X and V=0.5X should be the same.

So if Bob is mad enough to reason with subjective probability distributions (which IMO should never be used in science, and which can be avoided even when discussing credences by using imprecise probabilities), Bob can nevertheless avoid self-contradiction without revising his earlier credences, simply by recognising the distinction between legitimate and non-legitimate expectation values.

Since you're an R user, you might find it interesting to define a model in RStan, using different choices for the prior P(S) for the smallest amount S put into a envelope. Provided the chosen prior P(S) is proper, a sample from the posterior distribution P( S | X) , where X is the observed quantity of one of the envelopes, will not be uniform, resulting in consistent and intuitive conditional expectations for E [ Y | X] (where Y refers to the quantity in the other envelope)

No, the question is what is the probability SB experiences an awakening with the coin being heads — hypericin

Unless additional premises are included in the problem, I cannot tell the difference between how I phrased the question and your phrasing of the question.

Upon being awoken, SB is asked: "What is your credence now for the proposition that the coin landed heads?" "

She is therefore being asked "What is P(C | M) , where M is your current mental state?"

But the premises don't specify anything regarding the correlation of SB's mental state M to either the state of the coin C or the day of the week D.

So we can only interpret P (C | M ) as referring to P(C) .

Yes, we're in full agreement. By "non-informative" I was referring to the distributional conditions of both uniformity of probability mass and of infinite support . But you're right in pointing out that a "non informative prior" is often used by Bayesians to refer only to uniformity of probability mass, in which the range of the support is considered to be a separate independent hyper-parameter.

Have you ever tried experimenting with psychedelics?

You might be suffering from a biologically rare type of depression that isn't treatable by usual methods.

Maybe a different example. I have a red ball hidden in one hand and a blue ball hidden in my other hand. You point to one of my hands at random. What is the probability that you pointed to the hand holding the red ball? It's 1/2. — Michael

It depends on what interpretation of probability you are appealing to. For those of us who reject Laplace's principle of indifference, the answer is to refrain from asserting a subjective probability.

In any case, it isn't relevant to the two envelopes problem, for It can be reproduced by appealing to a causal interpretation of probability in which a person observes the contents of an envelope he is given, without him making any decisions. A Bayesian analysis reveals that the culprit of the paradox is the assignment of a non-informative prior to the distribution that generates the envelopes contents.

Without that assumption, the conditional expectations involved behave sensibly and the paradox dissolves.

That’s not what happens in this example. I am shown two envelopes, one containing £10 and one containing £20, and I freely choose one at random. I don’t open it. The probability that I picked the one with £10 is 1/2. — Michael

I'll agree for sake of argument . I think the problem is how we are fitting our shared understanding of the problem to probability calculus.

In my preferred description, one of the envelopes is opened to reveal a quantity A, but It isn't known as to whether the other envelope is more than or less than A.

In your preferred description, the quantities of both envelopes is known a priori, but neither of the envelopes are opened.

The problem with your description, is that it runs contrary to how conditional probabilities and expectations are normally interpreted. For the information upon which a probability or expectation is conditioned, is normally interpreted as observed information, rather than possessed information that isn't revealed, compensated by assumed knowledge of other quantities.

The paradox is premised on not knowing the value of any. — Michael

That is flat out contradicted by the switching argument. Furthermore, without the premise of knowing the value of one of the envelopes, the paradox disappears.

To have an expectation value containing A on the one hand, and insisting that the envelope isn't opened on the other, is a bait and switch. Wikipedia's article isn't written very well, which might be part of the confusion.

But do you agree that the probability in my example situation is 1/2? — Michael

It is a half if you assume it to be 1/2, but not necessarily. Consider for instance someone sending you the smaller of two envelopes through the post, according to a probability that they have decided. You open the letter and are informed that if you return the envelope and it's contents, you will receive another envelope that has half as much or twice as much.

My argument with you is over the assigned probabilities. So ignore the expected value. I just want to know an answer to this:

1. One envelope contains £10
2. One envelope contains £20
3. I pick an envelope at random
4. I don't open my envelope
5. What is the probability that I picked the envelope containing £10?

My answer is 1/2. What is yours? — Michael

The paradox doesn't apply in that scenario, since the values of both envelopes are given.

To my understanding , the paradox requires,

1) Knowledge of the value of only one of the envelopes.

2) The assumption of a non-informative prior for P(M) , where M is the smallest (or greatest) money in the envelopes.

In particular, the puzzle is not solved by finding another way to calculate the probabilities that does not lead to a contradiction. — Michael

The contradiction vanishes when a normalised prior is used for P(M), since in that instance knowledge of the value of one envelope is indicative of the value of the other.

There is no opened envelope: — Michael

The switching argument begs to differ :

" 1. Denote by A the amount in the player's selected envelope."

The argument's computed expectation value of (5/4) A is a conditional expectation with respect to the variable A denoting the amount of money in the player's selected envelope.

Not opening the envelope and taking an expectation over the envelope's value means that A cannot be mentioned in the expectation value, even as a variable with an unspecified value.

The unconditional expectation of the players envelope value is 0.5 x M + 0.5 x 2M = 1.5M , where M is the mean of the unspecified distribution F for the smallest amount of money in an envelope. No paradox arises from this calculation.

I've subsequently come to realise however, that my initial analysis wasn't quite right. If my calculations are correct (that are too tedious to post here), the switching dilemma is caused by assuming an un-normalised "non-informative prior" for F(M), which causes the conditional expectations for each envelope to diverge.

Intuitively, this can be explained as follows:

if there could be any amount of money in the two envelopes, and if all monetary values are equally likely, then whatever value you observe in one envelope, the other envelope is likely to have an even higher value.

To remove this bug, one has to replace the physically implausible non-informative prior for F with a normalised distribution, so that the conditonal expectation for the unopened envelope gets lower as the observed money in one's opened envelope gets higher.

Thanks.

The premises of the Sleeping Beauty, at least in some popular incarnations of the problem, specify an experimental design, but they don't specify or assume how temporal self-location is related to that design. The design only specifies a few constraints relating the physical outcome of a coin toss C to the actual days of the week D that a subject is awoken. But an awoken subject's mental state M comprises yet another variable that has no specified relationships to C or D in the question's premises.

The question is literally asking, on the basis of a partial definition of (C,D) what is the marginal distribution of P(C), which it's premises already state is 1/2. So if it is intending to probe the valid philosophical question pertaining to self-location it needs to be more candid and ask

What is the probability P (C = h | M) ?

But if nothing about M and it's relationships to C and D are specified or assumed, one isn't in a position to assign conditional probabilities, and can only respond with the vacuous answer " P(C = h | M) is between 0 and 1. "

Always start by writing down the probability of everything :

Variables :

C = Tossed coin (Binary variable in {head, tail} )
D = Awoken Day (Binary variable in {mon, tue} )

Unknown Probability Measure :

P (C, D) = P (C | D) P(D) = P(D | C) P(C)

Constraints given on the Probability Measure :

P (C = head) = 1/2
P (D = mon | C = head) = 1
P (C = tail | D = tue) = 1

Task :

Determine the marginal distribution P (C = head ) from the above premises

Answer :

P(C = head) = 1/2 (by premise)



Why was the scientific american wasting time on this?

What is the probability that your envelope contains twice as much money as the other? It's 1/2. — Michael

Again, that's an additional subjective premise that isn't objectively implied by the 'physics' of the two-envelope premise. Nevertheless we can assume it for sake of argument and continue the discussion as an exercise in subjective Bayesian probability estimation in which we hope for self-consistency.

There's no reason that the probability in the second case should be different to the probability in the first case. — Michael

Yes there is. That's why conditional probabilities should be used. The conditional probability

P ( B = b | A = a)

that refers to the amount of money in the unopened envelope B when conditioned on the amount of money in opened envelope A, is generally unequal to

P (A =a | B = b)

that refers to the amount of money in unopened envelope A when conditioned on the amount of money in opened envelope B.

Knowing the former conditional distribution upon opening envelope A generally says nothing about the latter distribution unless the ratios of the priors P(A=a) and P(B=b) is assumed to be 1, as indicated by Bayes Theorem.

Only if you take the ratio to be 1 do the calculated subjective conditional expectations come into conflict with respect to decision making.

Assuming your subjective premise:

- Recall the fact that the implied subjective expectation regarding the amount of money gained upon opening either envelope is 1.5x, where x is the smallest amount of money (as i showed in my first analysis).

- Also recall that unlike in my first analysis that referred to the envelopes by labels that were assigned to them prior to them being opened, here we are defining A to refer to the opened envelope. In which case the probability that we will open A is 1!

Doesn't the fact that you will definitely open A first, together with the fact that your expected gain is 1.5x imply that your subjective prior for P(B) should be different to your subjective prior for P(A)?

It just assumes that:

P(A = the smaller envelope) = P(B = the smaller envelope) = 1/2 — Michael

That expression is used to represent the same set of initial assumptions, but is less explicit with regards to its premises, such as the fact that some distribution is responsible for placing a certain amount of money in each envelope.

For debugging probabilistic arguments and resolving paradoxes it is better to use conditionals so that every premise is made explicit, so that problematic premises are more easily identified.

Therefore, within the same equation, A is referring to two different amounts. Am I correct in thinking that this is why the equation gives a false result. — RussellA

I would say that is a potential cause of the paradox, but isn't the paradox itself. The paradox is the fact that the switching argument consists of a logically inconsistent set of probabilistic assumptions.

The switching argument, which produces a contradictory strategy for solving the two-envelope problem, starts by subjectively assuming, without evidence, the following conditional distribution, with respect to envelopes A and B whose values are a and b respectively :

P (B = (1/2) a | A = a) = P(B = 2a | A = a) = 1/2 For all values a

which yields the conditional expectation value

E [ B | A = a] = (5/4) a For all values a

So far, terrible reasoning with respect to the two-envelope problem, but no inconsistency.

Next, the switching argument further assumes, without evidence, that

P (A = a | B = (1/2) a) = 1 / 2 for all values a

However, using Bayes Theorem gives

P (A = a | B = (1/2) a) = P (B = (1/2) a | A = a) x P(A = a) / P(B = (1/2) a)

Hence the switching argument assumes that ratio on the right hand side is always 1 :

P(A = a) / P(B = (1/2) a) = 1 for all values a

This is the most deranged part of the argument. It amounts to asserting "If we know nothing about the prior distribution of A and nothing about the prior distribution of B, then we can be certain that the ratio of their probabilities is 1 - for our ignorance cancels out! "

We then derive the contradiction

E [ A | B = b ] = (5/4) b For all values b

Relative to your use of probabilities, how do you distinguish knowing that an outcome has probability 1/2 from not knowing the likelihood of an outcome?

So by your logic we can't even talk about the probability of a coin toss landing heads being 12
1
2
? — Michael

In my view , epistemic probabilities are derived from causal knowledge or assumptions on the basis abductive reasoning and repeated trials. One cannot derive causal knowledge from mathematical concepts as is demanded in the two-envelopes problem.

In my strong opinion, Laplace's principle of Indifference and the principle of maximum entropy are grave misuses of probability calculus that only lead to erroneous inferences and the conflation of ignorance with information.

I don't think that's at all reasonable, or even relevant when we consider puzzles like this. For the sake of puzzles like this we assume a coin toss landing heads has a probability of 12
1
2
, and we assume that my choice of envelope is truly random. — Michael

It is fair to make probabilistic assumptions, but they shouldn't be arbitrary (unless for the sake of philosophical argument) and they must be explicitly stated, and any conclusion cannot go farther than what is explicitly assumed from the outset. The less one assumes, the less one can conclude.

Since the two-envelopes problem isn't well posed, more must be assumed and/or imprecise probabilities are needed.

The only rational response to the two-envelopes problem as it is traditionally stated without additional assumptions, is to reply

"The probability of getting a greater or lesser prize when opening the other envelope, is between 0 and 1"

Why not?

I know that one envelope contains twice as much as the other. I pick one at random. What is the probability that I picked the smaller envelope? It seems perfectly correct to say 12
1
2
. — Michael

By "meaningful probabilities" i am referring to epistemically meaningful probabilities that quantify how the real world is expected to behave on the basis of past experience, as opposed to purely mathematical probabilities that merely quantify mathematical properties such as combinations and symmetry.

For example, suppose that the surface area of "heads" on an unknown but symmetric coin is roughly half of it's total surface area. If nothing else is known and assumed about the coin , including how and where it is to be thrown, then one cannot deduce solely only on the basis of the coin's mathematical properties that it's likelihood of landing heads is 50/50. Nevertheless, probabilities are often assigned to such unknown objects purely on the basis of their mathematical properties, leading to the conflation of "mathematical probability" with "physical probability" or "epistemic probability".

The two-envelopes question is epistemic because it concerns decision making in the real-world; so more than mathematics must be appealed to when deciding whether or not a strategy exists for decision-making in that context.

Our conclusions might agree. I am saying that only the statements 1,3 4 and 5 are valid in the "switching argument". For any prior probability distribution over envelope choices or envelope contents is consistent with the premise

"Imagine you are given two identical envelopes, each containing money. One contains twice as much as the other. You may pick one envelope and keep the money it contains. "

And a chosen prior probability distribution mustn't refer to a person's subjective beliefs about the envelopes, but to a logically consistent set of causal hypotheses concerning the generation of the envelopes and their contents, by which a contradiction isn't derivable.

All that the two-envelope premise concretely specifies is a relation from the cartesian product of booleans and naturals to the naturals

r :: B x N ---> N
r ( 0, a) = a
r ( 1 ,a) = 2a

From which we trivially obtain the inverse relation

ir :: N --> N x N
ir a = { (0,a) , (1 , a/2) }

There is literally nothing else that can be said, deductively or inductively, from the two envelope premise alone. One cannot extract a meaningful notion of probabilities, let alone expectations and averages in relation to that premise when it is stated without additional causal assumptions, which are necessarily to give physical meaning to probabilities in that context.

An optimal decision doesn't exist on the basis of the information provided, because the premises fail to specify a well-posed problem :

Let P ( r | x , e) denote the probability of obtaining a value r when opening an envelope labelled e, where x represents the smallest amount of money in the two envelopes. Both e and x are assumed to be hidden variables in the sense that they aren't deducible from a drawn value of r.

The premises of the problem allow the following physical characterisation of P:

P ( r | x , e = 0) := Ind (r ; x)
P ( r | x, e = 1) := Ind (r ; 2x)

Here Ind(r; y) is the indicator function that outputs 1 when r equals y and is otherwise 0. (i.e. we get whatever is in the chosen envelope).

To "objectify" the problem, suppose that instead of choosing an envelope, an envelope is automatically drawn from a prior probability distribution p(e) and then opened. Then we could use Bayes theorem to compute a physically interpretable probability that a given value of r is "caused" by a given envelope:

P (e | r , x) =

P( r | e, x) P( e)
---------------------
P (r | x)

where P(r | x) is obtained by summing over e in the numerator, i.e

P(r|x) = P(r | e= 0, x) P(e = 0) + P(r | e = 1 , x) P(e=1)

But the problem doesn't specify the prior probability P(e) , which implies that P (e | r , x) doesn't have a unique solution, which in turn implies that the decision problem isn't well-posed.

Instead, one might follow the OP and a number of authors, and make the arbitrary selection
P (e =1) = P(e =2) = 1/2. In which case Bayes rule gives

P (r | x) = 0.5 Ind (x ; r +2r)

whose expected value is E [ r | x] = 1.5x

However, the problem is still ill-posed due to the fact that P(r | x) has zero mass at it's expected value of 1.5x :

(r = x) => (r < 1.5x )
(r = 2x) => (r > 1.5x )

Unless we make additional assumptions about P(x) , such that we can gain knowledge of x from knowledge of r, we cannot know which of these cases is the most likely, and so cannot deduce anything about the envelope e from the assumption of a uniform prior for P(e).

Any idea how "idealism" can be used to solve "the hard problem"? Do share, Rogue. — 180 Proof

Deflationism, that rejects representational accounts of semantics , shares much in common with subjective idealism and logical positivism that both considered every proposition to be reducible to sense-data.

In so far as the hard-problem is considered to be a metaphysical problem that is an artifact of representationalism, idealism can be considered to be a metaphysical strategy for dissolving the hard-problem, even if such a strategy is regarded to be epistemically impractical for the inter-subjective purposes of science , as the positivists discovered.

Putting it cynically, your proposition and methodology of divine revelation isn't qualitatively different to the thoughts of the average physicist, who uses commonsense to argue for both the big-bang theory whilst simultaneously denying a beginning of time.

Yet there are alternative theories, such as the Hawking-Hartle proposal that time is finite but lacks a boundary, that consider the question as to whether the universe truly has a beginning or not to be a topological question whose answer is relative to perspective.

Also, the subjective nature of phenomenological time concerning the intimate world of experience, is a distinct question whose relationship to the theoretical time of physics isn't decided. So a person could conceivably be a psychological presentist who denies the existence of a beginning/end with respect to their world of experience, who nevertheless believes physical time to have a beginning/end.

Perhaps the hard problem is inconceivable for phenomenolgists, but I'm not a phenomenologist. — Luke

So what is your definition of unconsciousness? Is it a pure postulate, or something that reduces to empirical criteria?

Granting this, how does it imply that the hard problem is inconceivable? — Luke

We can take the hard-problem in it's broadest sense, as asking what grounds the existence of first-personal phenomenological criteria that are used to understand propositions?

For phenomenologists who consider first-personal phenomenological criteria to be the very essence of meaning, the question is circular and makes no sense from their perspective. Which is what i was getting at above.

On the other hand, scientists working in the natural sciences will either side with the phenomenologist or not, depending on whether they believe the inter-subjective empirical criteria that they use to understand scientific theories to be ultimately grounded in first-person phenomenology or in pure reason. (e.g whether they are ultimately empiricists equipped with a deflationary understanding of truth and an anti-representationalist understanding of their own minds, or whether they are ultimately rationalists equipped with a correspondence understanding of truth and a representational understanding of themselves).

As for Dennett, he sometimes sounds like a rationalist who agrees with the phenomenologist that the question is meaningless, but for opposite reasons, namely due to a narrow interpretation of the natural sciences as denying the question on the basis of it being materially inconsequential (as opposed to be phenomenologically inconsequential)

Why not? — Luke

By definition, there does not exist empirical criteria for asserting self-unconsciousness in the present. So the proposition "I am presently unconscious" is presumably meaningless when taken in the fullest possible sense. In which case, an assertion of self-unconsciousness can only amount to a speculative hypothesis regarding an absence of a previous mental state (or equivalently, of the presence of an unconscious past mental state).

Typically, a person appeals to a present state of amnesia to infer that they were unconscious in the past. Whether or not one accepts the validity of this inference depends on one's conception of memory. An empiricist isn't likely to regard an inability to remember the past as saying anything literal about the past.

If there were no experiential dimension then there would be no hard problem, but since there is, there is. — Luke

Consider what it would mean to say that there is no experiential dimension. Unless that possibility is conceivable, then the hard problem isn't conceivable. Can you really conceive an absence of experience?

Consider the empirical criteria we might use when we assert that a sleeping person is unconscious. Then consider the rational arguments the the sleeping person uses after waking up, when they infer on the basis of amnesia to have been unconscious during sleep.

Is our empirical criteria regarding the present unconsciousness of a sleeping person the same as, or even comparable to, the amnesia that the awoken person appeals to when inferring "self unconsciousness" in the past?

You are asking basic questions that concern the topic of "Denotational Semantics", which use partially ordered sets (more specifically, Scott Domains) to denote partial states of evaluation with respect to the computation of a term such as a number. Terms of any type are represented as having a totally undefined value prior to evaluation, a partially defined value during the course of evaluation, and in the case of finite terms that can be fully evaluated, a totally defined value after evaluation known as a "normal form".

In denotational semantics, the Type corresponding to 'Computable real numbers' refers to the set of fix-point equations that if iteratively applied on a given rational number, generates a sequence of prefixes that are Cauchy convergent. To obtain an extensional value for a term of 'computable real number type' requires iteratively evaluating the term and then terminating the iterative evaluation abruptly after an arbitrary number of finite iterations, to produce a finite prefix representing a rational number that is very misleadingly said , to "approximate" the real number concerned (it is misleading since we are comparing apples, namely fix-point equations that are defined intensionally in terms of equations and that refer to types, to oranges that are observable states of computation that refer to terms.

A question remains as to who gets to decide when to terminate the iterative evaluation : the interpreter/compiler, or the user of the program? In programming languages with strict semantics, their respective interpreters and compilers always evaluate the term of every type to the fullest extent possible, meaning that real numbers cannot exist as types in such languages, since their terms have no "normal form" and would cause programs to loop endlessly if evaluated. In such languages, real number constants tend to be denoted by rational numbers with a priori fixed values decided at compile time.

By contrast, in a language with lazy semantics such as Haskell, terms can be used and passed around in partially evaluated form. This means that real numbers can exist in the sense of partially-evaluated "infinite lists" consisting of an evaluated prefix and an unevaluated tail. These lazy languages allow runtime conditions to decide what rational value is used in place of a term of real-number type, which is allowed to vary during the course of computation and which corresponds more closely to the notion of "potential infinity".

In mathematics, one has to distinguish intensional definitions from extensional interpretations. The former refers to analytic tautologies of an assumed convention, such as Pi being declared as the "exact" ratio of circumference to diameter. Such propositions say nothing of the actual world, because they are norms of linguistic representation.

On the other hand, extensional interpretations of a convention refer to worldly observations to which a mathematics convention is applied, such declaring a visible circle to represent or "approximate" Pi. Here the word "approximate" is misleading, for the visible circle isn't part of the convention but an imprecise application of it. (Any application of a convention is invariably imprecisely applied)

The premise of a shared reality is incompatible with the premise of non-representational perceptual access for all. For if I judge my own perception of the world to be direct, deflationary and non-representational, then I must judge everyone else's perception of the same world as being indirect, and representational according to truth-by-correspondence.

The only way I can reconcile everyone's claims to be non-representational direct realists, is to interpret each and every person as referring to a different world.

If we continue down the tree with this alternating pattern RLRLRLRLRLRL... we approach the Golden Ratio.

Is there anything wrong with completing this tree and saying that the infinite digit RL is the Golden Ratio? — keystone

Constructively speaking, there's nothing wrong with your identification of real numbers with "infinite" paths, i.e. the non-wellfounded sets known as "streams", provided such paths are finitely describable. For a computable real is equivalent to a circularly defined equation that can be lazily evaluated for any desired number of iterations to yield a finite prefix. In your case, that would be an impredicatively defined binary stream such as S, defined as the fixed point condition

S = 1 x ~S

where _x_ is the cartesian product and ~ is logical NOT (i.e. S is the liar sentence).

To faclitate the identification of streams with cauchy convergent sequences, S can be considered equivalent to other streams for which it shares a bisimulation with respect to some filter for deciding how streams should be compared. The stern-brocot tree can also be interpreted as a game-tree, such that a computable real number is identified with a "winning strategy" for converging towards an opponent's position who attempts to diverge from the player's path to some epsilon quantity.

Surreal Numbers also share a similar binary- tree construction, and their fabled ability to embed the real-numbers might be recalled. But this rests upon the assumption that transfinite induction is valid, which isn't constructively permissible due to it's reliance on the axiom of choice. Your indicated idea of using fixed-points to define real numbers, although not original is more promising.

I believe the non-standard identification of real-numbers with streams and more generally co-algebras, was originally due to Peter Aczel in the eighties, who became famous for inventing/popularising non-wellfounded set theory. For an alternative approach to non-standard analysis that is constructive and sticks to well-founded sets by merely augmenting them with additional axioms to denote terms at the fixed points, see Martin Lof's notes under "The Mathematics of infinity"

Concepts are public. Concepts are norms. How else could you even ask me that question with a sense of being entitled to an answer ? A tacit commitment to the philosophical situation is prior to every other issue. I touch on that in my new thread, if you want to join. — plaque flag

That concepts are norms isn't the same as saying that concepts are public. These are two distinct semantic claims.

I have only had a precursory glance at Brandom's introduction to inferentialism but I suspect you might be misreading, or at the very least dramatically oversimplifying his views, which to a large extent is understandable given this is an abbreviated public forum space where people speak with highly constrained time and space and without knowing of each others prior knowledge and agendas.


In Chapter 5, "A Social Route From Reasoning to Representing" , Brandom makes generally non-controversial arguments that language serves as a medium of 'representation' in the context of social norms.

Beliefs and claims that are propositionally contentful are neces-
sarily representationally contentful because their inferential ar-
ticulation essentially involves a social dimension. That social
dimension is unavoidable because the inferential significance of a
claim, the appropriate antecedents and consequences of a doxastic
commitment, depends on the background of collateral commit-
ments available for service as auxiliary hypotheses. Thus any speci-
fication of a propositional content must be made from the
perspective of some such set of commitments. One wants to say
that the correct inferential role is determined by the collateral
claims that are true. Just so; that is what each interlocutor wants to
say: each has an at least slightly different perspective from which
to evaluate inferential proprieties. Representational locutions
make explicit the sorting of commitments into those attributed
and those undertaken—without which communication would be
impossible, given those differences of perspective. The representa-
tional dimension of propositional contents reflects the social
structure of their inferential articulation in the game of giving and
asking for reasons.
.

So, when speaking in the context of language being a medium for representation , then qualia - which by definition is said to refer to only what an individual speaker could know - gets the chop.

But what Brandom doesn't do in that passage is insist that meaning is essentially representational or that meaning and knowledge are necessarily public affairs. Indeed, that interpretation of Brandom would contradict the very idea that Brandom was an non-representational semanticist at heart. I suspect that Brandom, much like Wittgenstein, makes no negative semantic, metaphysical or mentalistic claims regarding the meaning or existence of "private language". I suspect that all he means, is that private concepts aren't being used representationally and hence beetles in boxes aren't an extensional aspect of the social representations inculcated by social norms. Nevertheless Beetles do matter when it comes to the perspectival and idiosyncratic aspects of language that are relative to each individual who must individually adapt their mother tongue in a bespoke inferential fashion to match their own worlds; such beetles are necessary, but lie beyond the aperspectival limitations of social norms and communication.

Whenever I understand myself to be seeing the "same" object as a someone else, I am not making a literal comparison of mine and their experiences, nor of mine and their semantic conditions of assertibility.

All I am doing is interpolating from my own experiences whether or not the sentence "We are seeing the same object" meets my personal criteria of assertability.

I radically disagree.

Social norms govern inferences in the first place. The situation is liquid enough, however, that an individual philosopher can get a new inference accepted / treated as valid. --- typically by using inferences which are already so treated along with uncontroversial premises. — plaque flag

Given that society rarely agrees upon anything and constantly changes its mind, not to mention the ever-changing customs of isolated Robinson Crusoes who have no access to society, I can't see what "social norms governing inferences" amounts to, nor do I see the ultimate relevance of social norms with regards to inferential semantics.

Do you mean that remark descriptively in the non-controversial general sense that philosophers are often influenced by their society, or do you mean it in the controversial prescriptive sense that philosophers ought to align with the prejudices of their society, because society gets to define what truth is, or that society must know better?

I claim that meaning is public. Claims don't represent claimant's meaning-as-hidden-stuff. — plaque flag

Putting aside what privacy means, there are two very distinct ways of interpreting that claim.

A. Private Language is False.

This is a semantic claim . According to this interpretation, private language is a thinkable possibility that is nevertheless false in either theory or in practice. Often this interpretation assumes conventionalism about meaning, whereupon public convention is believed to undermine a speaker's ability to mean what he wants. Those who hold this view often attack a speaker for talking about "private language".

B. "Private Language" is Nonsense.

This is an ontological or meta-semantic claim. According to this interpretation, whatever might be called a "Private Language " is actually "public" as a matter of tautology. According to this interpretation, which makes no semantic claims, a speaker is free to say and mean anything he wants, because the act of speaking is always understood as referring to something that is happening in the world of the speaker, either via direct acquaintance with the speaker as in the case of "qualia", or indirectly with the speaker via some causal theory of reference. In both cases, the speaker is interpreted as referring to something true that is nevertheless "public", even in the case of "qualia".

Which claim are you making?

'Content' sounds representational again. The point is to look at which inferences tend to be accepted. Let me emphasize that these norms are 'liquid', unfinished, infinite task. — plaque flag

So do you agree that social norms are generally a terrible way of inferring anything about an individual's behaviour?

It'll be hard to understand me if you stick to a representationalist semantics. I like inferentialism, which I connect to something like neorationalism, (resource linked earlier in the thread if you are interested.) — plaque flag

That's odd, because my attacks on conventionalism are precisely an attack on representationalism, including the idea that conventions tell us about what speakers mean.

If meaning is inferential, then the references of a speakers utterances are strongly identified with the local and proximal causes of the speakers utterances, and only weakly identified with distal causes that perfuse the convention the speaker is using in an optional capacity.

How do you reconcile your commitment to inferential semantics with your apparent claim to know the propositional content of speakers utterances?

(When interpreted with empathy, do Flat-Earther's really exist?)
— sime

Yes. They make claims about our world in our language. Their claims have inferential purchase. If I believe them, I will also believe implications of their claims --- which may be why I can't believe them, for their claims imply others that are not consistent with other of my beliefs. — plaque flag

But do people really share the same belief objects whether agreeing or disagreeing about the truth of a proposition? For how can linguistic conventions decide what the object of a proposition is?

If you accept that the Earth isn't flat, then you presumably accept that a flat Earth cannot be the physical cause of a Flat-Earther's beliefs. In which case, how and in what sense can he be said to be referring to the Earth?

When you interpret a flat earther to be speaking about 'our earth', are you claiming to have knowledge about the speaker's beliefs, intentions, mental state, circumstances and so on? or are you merely referring to what convention says about the speaker's verbal behaviour?

The norms of linguistic convention are certainly correlated to facts about the world, for otherwise nobody would ever trust each other's remarks. But can this justify elevating the status of convention to the ground or justification of meaning? For don't our conventions often mislead and betray us about the facts of truth and meaning?