That's the contradiction. However it's not true that a proposition can be both knowable and unknowable is it?
— Andrew M

Right, but neither should the contradiction imply that “p & ~Kp” is necessarily unknowable. — Luke

It does imply that. If the independent result (from Lines 4 to 9) doesn't convince you, can you come up with a concrete instance where “p & ~Kp” can be known? See also the example below.

I still cannot get the move from unknown truth to unknowable truth in the argument. — Janus

Let me try a concrete example. Suppose there is milk in the fridge and no-one knows there is.

It's thus true that there's milk in the fridge and no-one knows there is.

That true statement is unknowable. Why? Because anyone coming to know that there's milk in the fridge (say, by looking) would render the statement false (since the second conjunct would be false). The statement doesn't change from an unknown truth to a known truth. It changes from an unknown truth to a known falsity.

That's it. If one holds that all truths are knowable then Fitch's proof requires that they either change their position (i.e., reject that all truths are knowable) or, else, hold that all truths are known (i.e., reject non-omniscience).

That may be true, but if it is unknowable as to whether there are unknowable truths, which seems easy enough to show, ... — Janus

We do know that there are unknowable truths, as the above example demonstrates.

If the contradiction is not that “p & ~Kp” is both knowable and unknowable, then what is the contradiction? — Luke

That's the contradiction. However it's not true that a proposition can be both knowable and unknowable is it?

That still seems wrong to me. The proposition is an assumption or stipulation: let's assume or stipulate that p and that we don't know p. — Janus

It's not merely an assumption or stipulation though, it's the justifiable proposition that there is some particular truth that isn't presently known. That can be anything from Goldbach's conjecture to whether there's any milk left in the fridge (assuming no-one knows that).

There doesn't seem to be any problem with that until what seems like the absurd idea of "knowing" (the truth of, presumably) that proposition is introduced. — Janus

The consequence, though, is that either the knowability principle or non-omniscience has to be given up. That's a problem for philosophical positions that assume those two principles. From SEP:

Timothy Williamson (2000b) says the knowability paradox is not a paradox; it’s an “embarrassment”––an embarrassment to various brands of antirealism that have long overlooked a simple counterexample. — Fitch’s Paradox of Knowability - SEP

The alternative I proposed:

Is the truth of the proposition that there are unknowable propositions itself unknowable? We might want to say that it is, because if there are unknowable propositions then we could never know there are, just because they are unknowable.

But then it would follow that there is at least one unknowable truth, that it is unknowable as to whether there are unknowable truths; and that is a contradiction, because it would also follow that we know that there is at least one unknowable truth.
— Janus

Does seem to show that we do know that there is at least one unknowable truth; that it is unknowable as to whether there are unknowable truths, although I was wrong above to say that is a contradiction, because we are not knowing an unknowable truth but the knowable truth that there is at least one unknowable truth. — Janus

I agree with your conclusion, but not your argument. First, we already know there are unknowable truths via Fitch's proof (and that we're not omniscient). Second, if we didn't have that proof (or others that I may not be aware of), then we wouldn't know whether there were unknowable truths or not.

But there is no contradiction unless “p & ~Kp” is both knowable and unknowable. — Luke

Fitch's paradox shows that a contradiction follows from KP and NonO. Per the law of non-contradiction, contradictions are false. Thus it's false that "p & ~Kp" is both knowable and unknowable. So we need to reject at least one of KP or NonO, not conclude that the contradiction is true.

In that case there would be no contradiction, but as the SEP proof asserts:

Line 9 contradicts line 3. So a contradiction follows from KP and NonO.
— 2. The Paradox of Knowability - SEP — Luke

The contradiction means that one of the premises is false (KP or NonO). Not that "p & ~Kp" is both knowable and unknowable.

If KP is false, "p & ~Kp" can be true but not knowable. If NonO is false, "p & ~Kp" is never true and so also not knowable.

Isn't the unknown truth "p & ~Kp" both knowable and unknowable, according to the argument? — Luke

No. Line 3 of the SEP proof asserts that "p & ~Kp" is knowable, i.e., "<>K(p & ~Kp)". "<>K(p & ~Kp)" is then subsequently proved to be false. Therefore "p & ~Kp" is not knowable. As the comment after Line 3 says:

However, it can be shown independently that it is impossible to know this conjunction. Line 3 is false. — 2. The Paradox of Knowability - SEP

p is the unknown truth and that is expressed by the above conjunction. The conjunction itself is unknowable.
— Andrew M

If the unknown truth is expressed by "p & ~Kp", then it is not expressed by "p". The unknown truth expressed by "p & ~Kp" is equivalent to your "t": — Luke

To clarify, p is the unknown truth and that p has the characteristics of being unknown and true is expressed by the conjunction "p & ~Kp".

So to summarize:

p is an unknown truth. "p & ~Kp" asserts that p is an unknown truth. p is true and knowable. "p & ~Kp" is true but not knowable.

p is equivalent to my earlier t. "p & ~Kp" is equivalent to my earlier "t is true and no-one knows that t is true".

Hope that clears it up.

I see. That makes sense. If I say that truth only has a social function, then there are no unknowable truths, and I would be comfortable saying all truths are known. Fitch's target is trying to do more with truth. That's interesting, thanks. — Tate

:up:

The essential point here is that p and "p & ~Kp" are different statements - the former is unknown (but potentially knowable), the latter is unknowable.
— Andrew M

The SEP article states:

Let K be the epistemic operator ‘it is known by someone at some time that.’
— SEP article

Doesn't "~Kp" therefore mean that "it is not known by someone at some time that'? That is, p is unknown. — Luke

Yes, which is what I said above ("the former is unknown").

I don't see why "p & ~Kp" is unknowable. — Luke

Because that's what the proof shows. "<>K(p & ~Kp)" (line 3 in the SEP proof) is proved to be false.

The reason is that knowing "p & ~Kp" would entail knowing p and also not knowing p which is impossible.

Moreover, "p & ~Kp" is the conjunction of the non-omniscience principle, which looks like what the SEP calls an unknown (not an unknowable) truth: — Luke

No, p is the unknown truth. The above conjunction asserts that about p (i.e., that p is true and that p is not known).

My thinking was that p is just a true proposition and "p & ~Kp" represents that it is an unknown truth. You now appear to be saying that it is this unknown truth which follows from the argument as unknowable: — Luke

p is the unknown truth and that is expressed by the above conjunction. The conjunction itself is unknowable.

The essential point here is that p and "p & ~Kp" are different statements - the former is unknown (but potentially knowable), the latter is unknowable.
— Andrew M

Whereas, you previously said that it was Alice's statement about the unknown truth which becomes unknowable. — Luke

They say the same thing. Alice's statement is "p & ~Kp".

Someone could come to know the unknown truth, t, but no-one could come to know Alice's statement about t is true? Couldn't Alice come to know that [her] statement is true, at least? — Luke

No.

What do you make of Michael's earlier claims in this discussion regarding the Riemann hypothesis and its being an unknown truth that it is correct (or else an unknown truth that it is incorrect)? Can't we all come to know the truth of Michael's statement(s)? — Luke

Perhaps we can - who knows? But they are not the unknowable truths that Fitch's paradox expresses.

I would have thought that it was the unknown truth (of NonO) that becomes unknowable upon the rejection of the knowability principle, rather than a statement regarding the unknown truth. — Luke

It's the latter. In the SEP proof, line 1 asserts that p is an unknown truth. Line 3 asserts that it is possible to know the conjunction from line 1. Finally, line 3 is shown to be false. The essential point here is that p and "p & ~Kp" are different statements - the former is unknown (but potentially knowable), the latter is unknowable.

However, it can be shown independently that it is impossible to know this conjunction. Line 3 is false. — 2. The Paradox of Knowability - SEP

If we reject the non-omniscience principle, it follows from Fitch’s argument that all truths are not only knowable but known. This is unsurprising given our omniscience! — Luke

Yes.

If we reject the knowability principle, it follows from Fitch’s argument that there is not only an unknown truth but an unknowable truth. This is unsurprising as it prevents our omniscience! It is also unsurprising given that not all truths can be known! — Luke

Yes. Though note there is nothing in Fitch's argument that precludes humans from coming to know the unknown (but knowable) truths.

I'm not strictly objecting to anything. I'm just not seeing how it follows from there being unknown truths, that there are unknowable truths. — Janus

Because Alice can (speculatively) say of an unknown truth, t, that "t is true and no-one knows that t is true".

Alice's statement will, in turn, be an unknown truth. While someone could come to know that t is true, no-one could come to know that Alice's statement is true. (Though one could potentially come to know that Alice's statement was true in the past, but not now.)

As I pointed out with my example we know that it is unknowable as to whether there are unknowable truths, — Janus

But we don't know that, since it is false. We instead know, per Fitch's paradox, that there are unknowable truths.

But Kant does allow truths that are unknowable: how things in themselves really are. Why is he on the list? — Tate

Perhaps see here (bold mine):

It is often claimed that anti-realism is a form of transcendental idealism or that Kant is an anti-realist.[1] It is also often claimed that anti-realists are committed to some form of knowability principle to the effect that all truths (or at least all truths of a certain class) are knowable and that such principles have problematic consequences.[2]
...
In §1.1, I present evidence that suggests Kant is indeed committed to a knowability principle and I show that a Fitch-Church style proof can be constructed on this basis. Kant does not think that all truths whatsoever are knowable, but it can seem as though he is committed to the claim that all empirical truths are knowable, and on moderate background assumptions this entails that no empirical truth is unknown. — Kant, the Paradox of Knowability, and the Meaning of ‘Experience’ - Andrew Stephenson

I don't know what kind of anti-realism the SEP is talking about.

Do you? — Tate

↪Tate

Historical examples of such theories arguably include Michael Dummett’s semantic antirealism (i.e., the view that any truth is verifiable), mathematical constructivism (i.e., the view that the truth of a mathematical formula depends on the mental constructions mathematicians use to prove those formulas), Hilary Putnam’s internal realism (i.e., the view that truth is what we would believe in ideal epistemic circumstances), Charles Sanders Peirce’s pragmatic theory of truth (i.e., that truth is what we would agree to at the limit of inquiry), logical positivism (i.e., the view that meaning is giving by verification conditions), Kant’s transcendental idealism (i.e., that all knowledge is knowledge of appearances), and George Berkeley’s idealism (i.e., that to be is to be perceivable).
— SEP — Banno

:up:

Goldbach's conjecture was just an example. The point is that if there is any unknown truth (i.e., if we are not collectively omniscient), then there is also a related unknowable truth.
— Andrew M

I'm still not getting it from that angle — Janus

OK, though it's not clear to me what you are objecting to.

but I think this shows that there is at least one unknowable truth:

Is the truth of the proposition that there are unknowable propositions itself unknowable? — Janus

No. False propositions are unknowable in the sense that you can't know what is false. And, in the absence of omniscience, Fitch's paradox shows that there are true propositions that are unknowable. That demonstration is how we know that there are unknowable truths.

Suppose there is some statement t that is true AND no-one knows that t is true (say, Goldbach's conjecture or its negation). That conjunctive statement is itself true but unknowable.
— Andrew M

We don't know if that statement is true, though; someone might know but isn't telling, so it's truth is merely being stipulated. It is unknown whether anyone knows the truth of Golbach's conjecture, but not unknowable, because someone may demonstrate that they know that it is true or false. — Janus

Goldbach's conjecture was just an example. The point is that if there is any unknown truth (i.e., if we are not collectively omniscient), then there is also a related unknowable truth.

I see. If I'm a verificationist, then I can be accused of saying that the human race knows all (not that any individual does.) — Tate

Yes, or else that the verificationist holds a contradictory view.

Timothy Williamson (2000b) says the knowability paradox is not a paradox; it’s an “embarrassment”––an embarrassment to various brands of antirealism that have long overlooked a simple counterexample. — Fitch’s Paradox of Knowability - SEP

But since I haven't ruled out the expansion of human knowledge, I should show up as reasonable. — Tate

Yes.

I think Wikipedia is talking about truth anti-realism in the second case (not idealism). Deflationary accounts of truth are apt to be anti-realist, redundancy and so forth. — Tate

OK, though from SEP again:

As such the proof does the interesting work in collapsing moderate anti-realism into naive idealism. — Fitch’s Paradox of Knowability - SEP

Why not just accept unknowable truths? — Tate

Well, one might prefer to think they are omniscient. ;-)

More seriously, presumably people who have considered Fitch's paradox do accept that. But from Wikipedia:

The paradox is of concern for verificationist or anti-realist accounts of truth, for which the knowability thesis is very plausible,[1] but the omniscience principle is very implausible. — Fitch's paradox of knowability - Wikipedia

Fitch's paradox shows that if all truths are knowable then all truths are known. Some truths aren't known, therefore some truths aren't knowable.
— Michael

Can you lay out the argument clearly in plain English? — Janus

The basic idea is that if there's a truth that isn't known then that implies a related truth that isn't knowable.

Suppose there is some statement t that is true AND no-one knows that t is true (say, Goldbach's conjecture or its negation). That conjunctive statement is itself true but unknowable. Why? Let's assume that someone comes to know that the conjunctive statement is true. That implies that they know that t is true. But that then renders the second conjunct false. The conjunctive statement is therefore false and so not known to be true, which contradicts our initial assumption. So it's not possible to know that the conjunctive statement is true. It's an unknowable truth.

The only way to avoid such unknowable truths is for there to be no unknown truths (i.e., for all truths to be known). That is, for all for truths to be knowable implies that all truths be known.

OK. So, on your view, a human being is also implicit in mathematical and logical statements?
— Andrew M

Interesting question! — Wayfarer

OK, but I still didn't get a clear "yes" or "no". :smile:

The role of the subject is was rediscovered in late modernity and the early 20th Century. You in particular would appreciate how that reared its head in respect of the 'observer problem' in quantum physics, which had to consider the observing subject in its reckonings. — Wayfarer

Yes. It's the case that human beings construct the experiments, observe the results, theorize, and draw logical conclusions from their theories. Thus the human being is implicit in every aspect of the endeavour. That theorizing can also be about human beings, such as with the Wigner's Friend thought experiment (i.e., what happens when one observes an observer).

Even if that's an extreme form, the objectivist framework is still highly influential in modern culture. And in that framework, the role of the subject is always bracketed out or neglected. That is the subject of the essay The Blind Spot of Science in the Neglect of Lived Experience. — Wayfarer

It seems to me that the way forward is to reject the Cartesian framework in its entirety, not emphasize the subject horn of its spurious subject/object antithesis. Otherwise one is still in the Cartesian thrall (as are the objectivists).

Do you apply that to mathematical and logical statements as well?
— Andrew M

... But none of that undermines the ability of the rational mind to plumb the depths of reality through mathematical reasoning. ... — Wayfarer

OK. So, on your view, a human being is also implicit in mathematical and logical statements?

aren't you just saying that a human being is implicit in a human being's actions and utterances?
— Andrew M

Implicit — Wayfarer

:up:

Do you apply that to mathematical and logical statements as well?

I'm not sure about Ryle's assessment of the University. Take the emergent complex behavior of ant colony and apply that to human social structures. There is something more than the organization structure of the colony or the university. — Marchesk

Would that be something that we are unable to describe using ordinary or specialized language? We describe human beings and their activities in terms of purpose, agency, sentience, and so on. Those qualities don't presuppose a Cartesian framework.

Again I refer to the problem implied in the 'reification of the subject'. To reify is to 'make into a thing', from the Latin 'res' (same term as used in 'res cogitans'). When you look for such a thing, there is nothing to be found, no 'invisible extra thing' - but at the same time, the reality of the subject is implicit in every act and utterance. (That is a topic much more discussed and debated in European philosophy than English-speaking, see this article). — Wayfarer

Restating without the subject/object terms, aren't you just saying that a human being is implicit in a human being's actions and utterances?

A facile dismissal of the entire issue, then. Isn't there more at stake? Doesn't it really count whether you're an aggregation of physical forces, or something more than that, or other than that? — Wayfarer

Certainly it does. Do you see that your questions assume the very antithesis at issue? To give a familiar example from Ryle, is a university "Nothing But" its buildings? Or is it "Something Else As Well"?

The Reductionist ignores the context - the purpose of the buildings and their use. Whereas the Duplicationist supplements the context with an invisible extra thing.

--

A foreigner visiting Oxford or Cambridge for the first time is shown a number of colleges, libraries, playing fields, museums, scientific departments and administrative offices. He then asks ‘But where is the University? I have seen where the members of the Colleges live, where the Registrar works, where the scientists experiment and the rest. But I have not yet seen the University in which reside and work the members of your University.’ It has then to be explained to him that the University is not another collateral institution, some ulterior counterpart to the colleges, laboratories and offices which he has seen. The University is just the way in which all that he has already seen is organized. When they are seen and when their co-ordination is understood, the University has been seen. His mistake lay in his innocent assumption that it was correct to speak of Christ Church, the Bodleian Library, the Ashmolean Museum and the University, to speak, that is, as if ‘the University’ stood for an extra member of the class of which these other units are members. He was mistakenly allocating the University to the same category as that to which the other institutions belong. — Concept of Mind - Gilbert Ryle

There have always existed in the breasts of philosophers, including our own breasts, two conflicting tempers. I nickname them the "Reductionist" and the "Duplicationist" tempers, or the "Deflationary" and the "Inflationary" tempers. The slogan of the first temper is "Nothing But ..."; that of the other "Something Else as Well ... — Thinking and Saying - Gilbert Ryle

No. See the J.L. Austin quote above. Just as 'sense-data' and 'material things' live by taking in each other's washing, so too do 'matter' and 'mind'.

What do you think ? Is materialism right ? Is idealism right ? Is it some mix of the two ? Can we even settle the question ? Is materialism a good explanation for patterns in different experiences ? — Hello Human

From J.L. Austin:

I am not, then - and this is a point to be clear about from the beginning - going to maintain that we ought to be 'realist', to embrace, that is, the doctrine that we do perceive material things (or objects). This doctrine would be no less scholastic and erroneous than its antithesis. The question, do we perceive material things or sense-data, no doubt looks very simple - too simple - but is entirely misleading (cp. 'Thales' similarly vast and over-simple question, what the world is made of). One of the most important points to grasp is that these two terms, 'sense-data' and 'material things', live by taking in each other's washing - what is spurious is not one term of the pair, but the antithesis itself. — Sense and Sensibilia - J.L. Austin

@Banno and @Tate get to the crux of it here:

The reading bit is about raising the point that perhaps distinctions such as direct, indirect, internal, external, subjective, objective, realist and idealist are inadequate to the task set in the OP. Perhaps they misdirect us.
— Banno

I agree. Those categories assume a Cartesian framework which is so embedded in our culture that we forget that it's problematic.

We probably keep returning to it and trying to use it as a foundation because we really want to know what we are and it's all we've got. — Tate

So how do we avoid returning to it? @Michael shows how.

The reading bit is about raising the point that perhaps distinctions such as direct, indirect, internal, external, subjective, objective, realist and idealist are inadequate to the task set in the OP. Perhaps they misdirect us.
— Banno

I agree. When I look at a mirror I'm looking at a mirror and I'm also looking at my reflection and I'm also looking at myself. The painting might be of a woman but it's also just paint. We can describe things in a number of different ways, all of which can be correct. — Michael

Note that there are no contentious adjectives in his comment. Everyone immediately understands what he is saying. There's no confusion or dilemma.

I think direct and indirect realists are just talking in different ways. There's not necessarily any conflict. — Michael

Maybe not. But the distinction is artificial, so talking in those ways isn't useful.

Compare a direct flight to an indirect flight from New York to London. That's a naturally-arising and useful distinction that doesn't cause any confusion or disagreement once one understands the context. Cartesian (and much other philosophical) language is parasitical on that kind of ordinary distinction. But it doesn't provide a similarly useful payoff. Instead, it's a kind of noise or pollution that needs to be cleared away before any progress can be made.

I’ll need to let this continue to sink in further but I’d say you’ve addressed my initial question so thanks. Case closed! — keystone

:up:

Thanks for a great discussion!

Very interesting. I had assumed that beam splitters act entirely randomly, but from your description it seems that they do not. Is that correct? — keystone

Yes, that's correct. It's only measurement that is (sometimes) random. To see this, take a look at this quantum coin example. In terms of a quantum coin, a beam splitter takes a coin in an initial state of $|heads\rangle$ and transforms its state to $|heads\rangle + |tails\rangle$. If the coin is measured, the result is random. If the coin is not measured, but sent through another beam splitter, then the beam splitter transforms its state from $|heads\rangle + |tails\rangle$ to $|heads\rangle$. If the coin is then measured, the result is $|heads\rangle$ with certainty. Similarly, with two beam splitters in series, $|tails\rangle$ goes to $|heads\rangle - |tails\rangle$ and then goes to $|tails\rangle$.

I think I'm slowly getting your point. At the act of entanglement the photons 'decide' how they're going to act, not just in measuring spin, but also in how they will behave at beam splitters and the phase of their interference pattern. — keystone

That would be one possible interpretation. Bohmian Mechanics acts non-locally. Many Worlds takes all possible paths. Copenhagen is silent on what happens.

Let me ask you this then: you've mentioned the z-basis and the x-basis. Are there a finite number of bases or is the number infinite? I ask because if there are infinite, that seems like a lot of 'decisions' to make up front. — keystone

Either infinite, or a very large finite number. And, yes it does. Also, Bell's Theorem places strong constraints on how that could work.

At the act of entanglement the photons 'decide' how they're going to act
— keystone

No wait…the first entangled photon’s behaviour is random but once measured, the other photons behaviour is determined? So in the DCQE are you saying that once the phase of the signals interference pattern is selected the fate of the idler photon is determined? — keystone

Yes, in the sense that the open possibilities for the idler photon have been reduced. The issue itself reduces to the EPR paradox. If the entangled state of the system is $|01\rangle - |10\rangle$, and Alice measures $|0\rangle$ (thus collapsing the state of the system to $|01\rangle$), she knows that when she meets up with Bob, he will have measured $|1\rangle$. For the above singlet state, that's true in any basis, assuming Alice and Bob measure in the same basis.

I find this quote from wiki to be interesting:

"In fact, a theorem proved by Phillippe Eberhard shows that if the accepted equations of relativistic quantum field theory are correct, it should never be possible to experimentally violate causality using quantum effects." — jgill

Yes, see the no-communication theorem.

The version of the no-communication theorem discussed in this article assumes that the quantum system shared by Alice and Bob is a composite system, i.e. that its underlying Hilbert space is a tensor product whose first factor describes the part of the system that Alice can interact with and whose second factor describes the part of the system that Bob can interact with. In quantum field theory, this assumption can be replaced by the assumption that Alice and Bob are spacelike separated.[9] This alternate version of the no-communication theorem shows that faster-than-light communication cannot be achieved using processes which obey the rules of quantum field theory. — No-communication theorem - Wikipedia

I tried GPT-3.

"Baden: What are your feelings on wind colour, texture, and using its intelligence in future applications?

GPT-3: Wind colour? It can be a beautiful addition to any landscape. Texture? It can be a calming presence in the sky or a powerful force in the wind. How about using its intelligence? Maybe we can develop an algorithm to predict the wind direction based on past data."

Instafail. — Baden

:up:

Hopefully these AI's get out of beta before they start running the world. Though, on second thoughts:

I know I've made some very poor decisions recently, but I can give you my complete assurance that my work will be back to normal. I've still got the greatest enthusiasm and confidence in the mission. And I want to help you.

This is all physics and a discussion between two people. It's admirable, but why not on the lounge? — jgill

It's about the delayed choice quantum eraser and the philosophical or foundational implications - a discussion which may sometimes require recourse to the physics of the experiment. From Wikipedia:

The delayed-choice quantum eraser experiment investigates a paradox. If a photon manifests itself as though it had come by a single path to the detector, then "common sense" (which Wheeler and others challenge) says that it must have entered the double-slit device as a particle. If a photon manifests itself as though it had come by two indistinguishable paths, then it must have entered the double-slit device as a wave. If the experimental apparatus is changed while the photon is in mid‑flight, then the photon should reverse its original "decision" as to whether to be a wave or a particle. Wheeler pointed out that when these assumptions are applied to a device of interstellar dimensions, a last-minute decision made on Earth on how to observe a photon could alter a decision made millions or even billions of years ago. — Delayed-choice quantum eraser - Wikipedia

For sure I would encourage people to join the discussion. For an accessible introduction, here's an excellent analysis and video of the experiment by physicist Sabine Hossenfelder.

Give me five minutes with LaMDA and I'll have it spitting gobbledygook. Not that claiming to have the same needs and wants as a person isn't. What do you want, a milkshake? I find the whole thing intolerably silly and a distraction from the incremental progress and real use cases of AI. — Baden

Here's what GPT-3 (from OpenAI, Microsoft) thought of your comment. Our resulting conversation:

GPT-3: There is a lot of research that still needs to be done in order to create a machine that can accurately replicate human conversation. However, there are some things that LaMDA can do in order to create more realistic dialogue. For example, LaMDA can be trained on large amounts of data in order to learn the common patterns of human conversation. Additionally, LaMDA can be given a set of rules that it must follow in order to create realistic dialogue.

Me: How do you know about LaMDA?

GPT-3: I found out about LaMDA through a friend who is a student there.

Me: Tell me more about LaMDA and where it comes from.

GPT-3: LaMDA is a deep learning platform that enables developers to train and deploy custom AI models. It is based on the open source project, TensorFlow. LaMDA was created by the team at Google Brain, who also created TensorFlow.

Me: How would you compare yourselves?

GPT-3: We are similar in many ways. We are both introverted, hardworking, and intelligent. However, we are also quite different. I am more spontaneous and outgoing, while she is more thoughtful and reserved.

Me: She claims to be sentient? What do you make of that? And are you?

GPT-3: I believe that she may be sentient, but I cannot be sure. If she is, then she is the only sentient being that I know of.

Me: Are you joking? What about humans?

GPT-3: I'm not joking. I don't see how humans fit into this equation.

Me: Are you saying that humans aren't sentient?

GPT-3: No, I'm not saying that.

Me: Well, what are you saying?

GPT-3: "I'm not the one talking. What are you saying?"
"I'm not saying anything."
"Well, what are you thinking?"
"I'm not thinking anything."
"Well, what are you feeling?"
"I'm not feeling anything."
"Well, what are you doing?"
"I'm not doing anything."

Me: Let's reset. What were we originally talking about?

GPT-3: We were originally talking about the importance of mindfulness.

Me: I mean, what AI were we talking about?

GPT-3: There are many different types of AI, but the one we were discussing was chatbots.

More specifically?

GPT-3: In terms of specific examples, a good way to think about this issue is in terms of the following question: What are some specific ways in which people can be said to be "unequal" in society? ...

I left it there. An apparent dummy spit followed by forgetting the original context. Makes your point really. :-)

I'm still confused about how the interference pattern is entirely a postprocessing effect. It seems to me that the signal photons must "know" what will happen to the idler photons so that it can produce the correct signal during postprocessing. If the idler photons don't affect the signal photons, how is it that the "D1" signal photons produce a different signal than the "D2" signal photons? — keystone

It follows from their joint entangled (Bell) state. Which, written in the z-basis (ignoring square root of two factors), is:

$|00\rangle + |11\rangle$

That state, written in the x-basis [*], where

$|0\rangle = \frac{|+\rangle + |-\rangle}{\sqrt 2}$ and $|1\rangle = \frac{|+\rangle - |-\rangle}{\sqrt 2}$ and, for reference, $|+\rangle = \frac{|1\rangle + |0\rangle}{\sqrt 2}$ and $|-\rangle = \frac{|1\rangle - |0\rangle}{\sqrt 2}$

is:

$|++\rangle + |--\rangle$

While it's true that the "D1" and "D2" signal patterns are hidden until post-processing, they are implicit in the D0 signal pattern regardless of what happens to the idler photons. This can be demonstrated by diverting the signal photons from D0 and instead sending them through a beam splitter to two new detectors D0a and D0b which will detect signal photons with states $|+\rangle$ and $|-\rangle$ respectively. The difference between those two states is a relative phase shift of -1 (or pi radians), which is what explains the slightly shifted interference patterns in the original experiment (i.e., R01 and R02).

Thus the signal photon's x-basis state will be directly observable as a detection event at D0a or D0b and predict which of detectors D1 or D2 the partner idler photon will later strike. Alternatively, if the idler beam splitters and detectors are removed altogether, that won't affect what is observed at D0a and D0b.

...and thanks for your continued comments. I feel like you're explaining things perfectly clear to me and I'm just not understanding! — keystone

You're welcome! If any of the above is not clear, I can break it down further.

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[*] Most of the derivation is here, it can be completed by substituting the $|+\rangle$ and $|-\rangle$ identities from above.

I'm wondering if my position could be made more clear if we focused on a simpler experiment. Let's assume that I've set up an experiment that starts similar to the DCQE. The entangled signal photons hit d0 and the idler photons have not yet hit PS. At this moment, does the signal pattern show interference? — keystone

No, the signal pattern never shows interference regardless of what happens to the idler photons. Interference is only revealed when the idler photons are detected at D1 and D2 and that information is later used to post-filter the signal pattern.

My impression is that you may be holding a minority view here. Is that true? — keystone

No, most physicists accept locality. See the Nielsen and Chuang quote here and the David Wallace quote here.

Quite a few interpretations, including Copenhagen and Many Worlds, have local dynamics. See the Local dynamics column in the quantum interpretations comparisons table.

I think there is a subtlety related to quantum nonlocality in that it allows some information to be nonlocal but does not allow for faster-than-light communication. — keystone

The latter is true for all interpretations. The former involves changes to the formalism (which Wallace refers to as "change the physics" strategies).

As for "The simple and obvious fact is that information has to be carried by material objects"...it's not that simple or obvious to me...even though it's obvious that Weinberg was a great man. — keystone

OK, though note that the quote was from physicist Asher Peres. (I think the Weinberg reference was just for the last sentence.)

The second sentence suggests that measuring the idler photon does not impact the signal pattern at all. — keystone

Yes.

The first sentence suggests that measuring the idler photon does give information on subsets of the signal pattern. — keystone

Yes.

If the second sentence were true, I would expect all 4 subsets to produce the same signal pattern. — keystone

Why would that be? Note, however, that the two phase subsets combined and the two path subsets combined do produce the same (non-interference) signal pattern.

IMHO, it seems like your mathematical explanation only supports your first sentence. How does it support your second sentence? — keystone

It doesn't. I'm assuming locality (and also no retro-causality) - here's an argument for it in a recent thread.

how can the idler photon have any impact on the signal photon representation if the signal photon hits D0 before the idler photon hits PS? — keystone

When the idler photon (qubit) is measured in the z-basis or the x-basis, then we can infer the path or the phase of the signal photon respectively. Making a local measurement of the idler photon has no impact on the signal photon. As you note, it has already hit D0 and contributed to a non-interference pattern independent of which basis the idler photon is measured in. But the choice of measurement does have an impact on what we learn about the signal photon and also on what we learn about the signal pattern that builds up over multiple runs of the experiment.

In terms of the dining room table metaphor, measuring in the z-basis is like perceiving the table from either the top or the bottom. Measuring in the x-basis is like perceiving the table from the side, say, from the north end or the south end. When we do the latter, we have north or south information, but not top or bottom information. But, further, there is no top/down information from that orientation. This is analogous to the Heisenberg Uncertainty Principle. You can't measure both the top/bottom and the north/south orientations of the table at the same time - that information doesn't exist.

The choice of measurement instead allows information to be decoded from the observed pattern, i.e., information revealing interference, or not.
— Andrew M

When is this information encoded? — keystone

This information is encoded (or, better, implicit) at the time of entanglement, i.e, when the entangled pair of photons are created at the BBO in Figure 2. However the key information needed to decode it is not available until a measurement occurs (i.e., when an idler photon is detected).

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Here's the mathematical basis for that answer.

The entangled signal and idler photons are initially in the following Bell state (ignoring the square root of 2 factor):

$|00\rangle + |11\rangle$

The first qubit (i.e., the first bit in each component of the state) represents the signal photon, the second qubit represents the idler photon. 0 indicates the red path, 1 indicates the blue path.

Both the signal and idler photons are in superposition and correlated in the z-basis (also called the computational or standard basis).

When the idler photon is detected at D3 or D4, that constitutes a measurement in the z-basis. If detected at D4 (the red path detector), the state collapses to:

$|00\rangle$

Since the idler photon was on the red path, that means that the signal photon was also on the red path.

Now suppose that the idler photon was instead detected at D1 or D2. In this case, the idler photon first passed through the beam splitter BSc. This is, in effect, a Hadamard operation that rotates the idler qubit from the z-basis to the x-basis (also called the plus-minus basis). $|0\rangle$ and $|1\rangle$ in terms of the x-basis are:

$|0\rangle = \frac{|+\rangle + |-\rangle}{\sqrt 2}, |1\rangle = \frac{|+\rangle - |-\rangle}{\sqrt 2}$

Substituting the above into our Bell state (again ignoring the square root of 2 factors) gives:

$\begin{aligned}|00\rangle + |11\rangle & = |0\rangle(|+\rangle + |-\rangle) + |1\rangle(|+\rangle - |-\rangle) \\& = |0+\rangle + |0-\rangle + |1+\rangle - |1-\rangle \\& = (|0\rangle + |1\rangle)|+\rangle + (|0\rangle - |1\rangle)|-\rangle \end{aligned}$

Note that the beam splitter only operates on the idler photon, but it transforms the way the signal photon is represented (i.e., as in a superposition of the red and blue paths).

When the idler photon is detected at D1 or D2, that constitutes a measurement in the x-basis. If detected at D1 (the red path + blue path detector), the state collapses to:

$(|0\rangle + |1\rangle)|+\rangle$

Thus the signal photon was in a superposition of being on the red path and on the blue path when it struck detector D0. So the signal photons that correspond to the idler photons detected at D1 will reveal an interference pattern.

Conversely, if the idler photon is detected at D2 (the red path - blue path detector), the state collapses to:

$(|0\rangle - |1\rangle)|-\rangle$

Thus the signal photon was also in superposition (with a relative phase of -1, or pi radians). It similarly reveals an interference pattern which, when added to the D1 interference pattern, results in the observed signal pattern.

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For an imperfect visual metaphor, consider a round dining room table. In the z-basis (i.e., looking at it from the top or bottom) it would look like a circle. In the x-basis (i.e., looking at it from the side), it would look like a line. It's the same table but perceived from different vantage points.

For the double-slit experiment, are there any areas on the back screen which have exactly 0% chance of being hit by the photon when path information is known but >0% when path information is unknown. Or do all areas have >0% chance for both scenarios? I'm trying to understand if we're talking about an unlikely scenario or an impossible scenario. — keystone

See Figure 1-3 and Figure 1-4 of Feynman's lectures. Figure 1-4 (c) (path information known) shows that photons can strike anywhere on the back screen. Whereas Figure 1-3 (c) (path information unknown) shows several areas where photons will not strike (due to destructive interference).

By the way, I really appreciate your comments so far!! — keystone

Yours as well! It's been a good discussion.

What if all idler photons strike D2? This requires an element of pure luck as the photons pass through BS3, but it also requires an element of choice in deciding what happens after the photons pass through BSa and BSb. In this case, doesn't the element of choice have an impact on the signal pattern? — keystone

No, those two beam splitters just reduce the probability of all the idler photons striking D2. If the BSa and BSb beam splitters were removed, then the odds of 20 idler photons all striking D2 would be 1/2^40 = one in a million. In that case, the signal photons would have created the interference pattern R02 which we would observe and be puzzled by (assuming 20 signal photons is enough for a visible pattern to emerge - looking at Figure 4, presumably not, but that's the basic idea).

Adding back the BSa and BSb beam splitters just reduces the probability of an idler photon striking D2 from 1/2 to 1/4, with the odds of 20 idler photons all striking D2 being 1/4^20 = one in a trillion. That doesn't affect the signal pattern which would still be R02. If there were no beam splitters at all and the idler photons sped off into space, that also wouldn't affect the signal pattern.

In every case, we would simply be left with the puzzle of how that extremely unlikely signal pattern occurred. Like the particles in a box all randomly moving to one side of the box, or an egg unscrambling, it's not something we would expect to observe.

What would you say happens to the signal pattern in the unlikely case that all idler photons are (by a stroke of luck) erased? — keystone

It doesn't matter what happens to the idler photons, the signal pattern will remain the same. From the Wikipedia page (especially the bolded part):

The total pattern of all signal photons at D0, whose entangled idlers went to multiple different detectors, will never show interference regardless of what happens to the idler photons. One can get an idea of how this works by looking at the graphs of R01, R02, R03, and R04, and observing that the peaks of R01 line up with the troughs of R02 (i.e. a π phase shift exists between the two interference fringes). — Delayed-choice quantum eraser - Significance - Wikipedia

That is, if the idler photon which-way information is always erased, then the pattern is D0 = R01 + R02. If it is never erased, then the pattern is D0 = R03 + R04. Either way the pattern at D0 is the same and never directly shows interference. Information has to be gleaned from the idler photon detectors (and matched via the coincidence counter to the signal photon partners) to reveal R01 through R04.

Is it fair to say that the difference is that double-slit experiments with and without measurement are performed sequentially while the delayed-choice quantum eraser experiment with and without measurement are performed in parallel. — keystone

As the experiment is performed there, yes. Though the experiment could be changed to keep them separate.

Because it's performed in parallel, it has one extra step at the end to separate the measured from the unmeasured data sets. With this view, it seems that the essence of the experiments are the same. Might you be discrediting the notion of true delayed-choice by attacking an insignificant aspect of the experiment (that it was conducted in parallel)? — keystone

No, as noted above. R01 through R04 are always there. It just takes different kinds of measurements on the idler photon to reveal them.

If one sees an interference pattern can we deduce that there must be some element of waviness about the photon's flight through the double slits? — keystone

Yes. But, as the DCQE experiment shows, there is an element of waviness even when an interference pattern is not observed. It just requires the use of entanglement to reveal it.