The liar paradox (Part 2)

In my previous post I discussed the liar paradox and argued that it should not be considered either true or false because it is cyclic. That is, it never ends up successfully asserting anything about the world.

Interestingly, the Stanford Encyclopedia of Philosophy (SEP) entry for the Liar Paradox outlines an argument that the liar sentence implies a contradiction. The argument uses two inference rules, as follows:

• Capture: A implies "A" is true
• Release: "A" is true implies A

Taking these two rules together, the terms A and "A" is true are intersubstitutable [1]. The SEP argument, reproduced in plain English [2], is:

Let L be the sentence, "This sentence is not true".

    1. "L" is true or "L" is not true [Law of Excluded Middle]
    2. Case One:
         a. "L" is true
         b. L [2a: release]
         c. "L" is not true [2b: definition of L]
         d. "L" is true and "L" is not true [2a, 2c: conjunction introduction]
    3. Case Two:
         a. "L" is not true
         b. L [3a: definition of L]
         c. "L" is true [3b: capture]
         d. "L" is true and "L" is not true [3a, 3c: conjunction introduction]
    4. "L" is true and "L" is not true [1-3: disjunction elimination]

Line 1 assumes that the liar sentence conforms to the Law of Excluded Middle. That is, it assumes that the liar sentence is either true or not true and not some other value. Lines 2 and 3 analyze each disjunct as separate cases. In each case a contradiction is reached which is then inferred in 4.

The conclusion that the liar sentence implies a contradiction depends on the first premise being truth-apt. However, as argued previously, the liar sentence is not truth-apt and so therefore the first premise can't be either. Consequently logical inference rules and truth evaluation aren't applicable to it. A contradiction is only reached via a false assumption of truth-aptness.

The rule of thumb would be that a sentence is only truth-apt if it is grounded in a state of the world either directly or else indirectly via other sentences. Note that this condition fails for both the simple liar (where not true means false) and the strengthened liar (where not true means false or not truth-apt).

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[1] This is Alfred Tarski's T-Schema: 'S' is true if and only if S (e.g., 'snow is white' is true if and only if snow is white).

[2] For example, I've replaced the corner symbols that indicate quasi-quotation with ordinary quotes.

The liar paradox

Consider our friend Pinocchio who says, "I am lying". Is his statement a lie? Or does he speak truly? It seems that if he is telling the truth, then he must be lying, just as he says. In which case he is not telling the truth. But if he is not telling the truth, then he is indeed lying. Just as he truly says!

This is known as the liar paradox. It seems that there is no satisfactory answer for whether Pinocchio is lying or speaking the truth. There are many variations of this paradox. For example, "Epimenides was a Cretan who made one immortal statement: 'All Cretans are liars.'". Another is the sentence, "This sentence is false."

In this post I will offer an explanation for the why the paradox occurs that builds on Gilbert Ryle's analysis [1].

The first thing that is usually noticed is that the statement, "This sentence is false." refers to itself. So perhaps self-reference is the problem. But we use self-reference all the time. For example, "I have a blue shirt on" is a perfectly valid statement. As is, "This sentence has five words".

So perhaps the problem is the way that the statement refers to itself. To evaluate whether the sentence is false first requires evaluating the truth value of the referring expression, namely 'This sentence'. That referring expression is just the original sentence. So that sentence, in turn, first requires evaluating the truth value of its referring expression, namely 'This sentence'. We can see that this is a recursive expression that never terminates in an evaluable statement. So it is not truth-apt [2].

But this raise the question as to why the statement, "This sentence has five words" doesn't have the same unending recursion. This statement expands as, "'This sentence has five words' has five words". We naturally stop at this point and just count the words in the inner sentence. But what is the rule for why we should stop? Shouldn't it expand indefinitely as well? One answer is that this sentence is grounded [3]. That is, the recursive expansion terminates in a sentence that doesn't mention truth.

However one can then go on to create strengthened liars such as, "This sentence is false or not truth-apt". If it is true, then it is false or not truth-apt, so it is not true. If it is false, then it is true, so it is not false. If it is neither true nor false then it is not truth-apt. Which is what the sentence says, so it is true! So it seems there is no satisfactory answer to the strengthened liar. However one approach here is to say that since the sentence is demonstrably not truth-apt, then it should not be subsequently read as making a truth-apt claim. It is a kind of illusion - it appears to be truth-apt but it is not.

A shortcoming of the above approach is that that there doesn't seem to be a principled rule to apply to determine whether a liar-style sentence is truth-apt. One just plugs in values and tries to find a custom solution. If a solution is found, then a new strengthened liar is constructed to foil it (these are also aptly called revenge liars!) And we don't seem closer to understanding what it is that makes liar-style sentences defective.

The approach I take instead involves distinguishing between use and mention. My claim (on ordinary use grounds) is that only the use of statements can be truth-apt whereas mentions of statements are not truth-apt (and treating them as if they were is a category mistake). When we mean to mention a statement rather than use it, we enclose the statement in quotation-marks. Consider the sentence, "This sentence has five words". This expands as "'This sentence has five words' has five words". The inner sentence is mentioned and thus (per my claim) is not truth-apt. This isn't an issue here since a truth predicate is not being used and the sentence is readily evaluable with the predicate it does have.

But what about a statement like "'Snow is white' is true"? Since that statement does evaluate as true, it seems that a mention of a statement can be true after all. My claim is that this is not the case. Instead, in ordinary use, a truth-predicate has a disquotation function [4]. That is, it removes the quotation marks and uses the statement. Thus, the statement ""Snow is white' is true" is transformed to "Snow is white" which (when used) is a truth-apt statement. More generally, "'p is q' is true" becomes "p is q" and "'p is q' is false" becomes "Not (p is q)". I will term this the disquotation rule.

Now consider the liar sentence, "This sentence is false". This expands as, "'This sentence is false' is false". Note that the inner sentence is mentioned, not used, so it is not truth-apt. However, since a truth predicate is specified, the disquotation rule can be applied. So applying this rule to the liar sentence, we get:

1. This sentence is false
2. "This sentence is false" is false [mention]
3. Not ("This sentence is false" is true)
4. Not (This sentence is false) [disquotation]
5. Not ("This sentence is false" is false) [mention]
6. "This sentence is false" is true
7. This sentence is false [disquotation]

What emerges is that the liar sentence is cyclic (line 7 is identical to line 1). We can also apply this rule to the truth-teller sentence:

1. This sentence is true
2. "This sentence is true" is true
3. This sentence is true

Thus the truth-teller is also cyclic. We can also test this approach with a conventional self-referring sentence:

1. This sentence has five words
2. "This sentence has five words" has five words
3. True

So the sentence is indeed truth-apt. The nice thing about applying the disquotation rule is that there is no need to plug in values and test each one. We can just follow the logic of the statement and note whether it terminates in a truth-apt value or, else, cycles.

Now consider the strengthened liar:

Let L be the sentence, "This sentence is false or not truth-apt".

1. L
2. "L" is false or not truth-apt
3. "L" is false or "L" is not truth-apt
4. Not ("L" is true) or ("L" is not true and "L" is not false)
5. Not (L) or (not "L" is true and "L" is true)
6. Not L or (not L and L)
7. Not L or false
8. Not L
9. Not (not L) [substituting 8 into 1]
10. L

Thus the strengthened liar is cyclic and so is not truth-apt. But isn't this just what the strengthened liar says? Since it has been demonstrated that it is cyclic, its apparent truth-aptness is an illusion. Just as a straight stick can appear bent when partially submerged in water, so to a cyclic statement can appear truth-apt when it is actually not.

In conclusion, the reason why the liar sentence (and any strengthened liar) is not truth-apt is because the self-reference is cyclic. Conventional (non-cycling) self-reference is fine. [5]

For more paradoxical fun, see Part 2.

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[1] From "Heterologicality", Gilbert Ryle, 1951:

"The same inattention to grammar is the source of such paradoxes as 'the Liar ', 'the Class of Classes ...' and 'Impredicability'. When we ordinarily say 'That statement is false ', what we say promises a namely-rider, e.g. '... namely that to-day is Tuesday'. When we say 'The current statement is false' we are pretending either that no namely-rider is to be asked for or that the namely-rider is '... namely that the present statement is false'. If no namely-rider is to be asked for, then 'The current statement' does not refer to any statement. It is like saying 'He is asthmatic' while disallowing the question 'Who?' If, alternatively, it is pretended that there is indeed the namely-rider, '... namely that the current statement is false', the promise is met by an echo of that promise. If unpacked, our pretended assertion would run 'The current statement {namely, that the current statement [namely that the current statement (namely that the current statement ...'. The brackets are never closed; no verb is ever reached; no statement of which we can even ask whether it is true or false is ever adduced.

Many of the Paradoxes have to do with such things as statements about statements and epithets of epithets. So quotation-marks have to be employed. But the mishandling which generates the apparent antinomies consists not in mishandling quotation-marks but in treating referring expressions as fillings of their own namely-riders."

[2] For a sentence to be truth-apt means that it is either true or false. Sentences like "Hello!" or "Twas brillig, and the slithy toves" are not truth-apt. A cyclic sentence like the liar sentence is also not truth-apt. (Note: an alternative view, called dialetheism, regards the liar sentence as both true and false.)

[3] Saul Kripke, 1975, “Outline of a theory of truth”.

[4] Disquotation is the reversal of the process of quotation. It transforms a quoted statement into an actual statement. That is, it uses the mentioned statement. For example "'Snow is white' is true" just means "Snow is white".

[5] Two more interesting sentences are worked out below:

Let L be the sentence, "This sentence is truth-apt".

1. L
2. "L" is truth-apt
3. "L" is true or "L" is false
4. "L" is true or not "L" is true
5. L or not L
6. True [Law of excluded middle]

Let L be the sentence, "This sentence is not truth-apt".

1. L
2. "L" is not truth-apt
3. "L" is not true and "L" is not false
4. Not "L" is true and "L" is true
5. Not L and L
6. False [Law of non-contradiction]

The problem of universals

This apple is red and so is that stop sign. Bob has two arms and two legs.

The apple, the stop sign and Bob are all individuals or particular things. When we say that the apple is red or that Bob has two arms, we are predicating something of those individuals. And when the same thing can be predicated of more than one individual, then that thing (say, redness or the number two) is termed a universal. A universal is just a general characteristic that those individuals have in common.

Now the status of individuals seems straightforward. We say that there is an apple there - it exists and we can see it and also eat it. But the status of universals seems less obvious. Is redness something real? If so, how does it happen to be in many places at once, in both the apple and the stop sign? And also, the number two seems very abstract, not a physical kind of thing that can exist like the apple and Bob do.

In philosophy, this is called the problem of universals. Plato was the first to discuss the issue in depth and proposed that universals do exist, but they exist in a higher realm of the Forms. So there is an ideal or perfect form for redness, a form for the number two, and a form for every other universal. The (imperfect) individual things that we can see around us are said to participate in those perfect forms. This view is called Platonic realism.

Aristotle (who was Plato's student) strongly disagreed with Plato on this issue. In his view, universals were indeed real, but they were not separable from the individuals that they were found in. That is, there is no redness independent of particular things that are red. This view is called moderate or immanent realism - the universals are immanent within the individuals.

A third view, nominalism, denies that universals are real at all. There are only the names we use and there is no universal quality that they have in common.

My own view is that Aristotle's immanent realism best captures what we implicitly mean when we assert things about individuals. The characteristics that the individuals have in common are not merely naming conventions or the creations of humans. Instead they are necessary for the individuals to be observable in any form at all. Granting that, there also seems to be no good reason to posit a non-natural realm for universals to reside in. They are simply features or aspects of the natural world that we experience, albeit at a more abstract level that requires human intelligence to be able to identify and reflect on them.

The image above shows part of the School of Athens fresco by the Italian Renaissance artist Raphael. It depicts Plato pointing towards the heavens where he supposed the ideal forms to be. It also depicts Aristotle gesturing downwards to the here-and-now world that he held universals were a part of.

The Sleeping Beauty problem

Continuing the theme of probability and self-locating uncertainty from my last post, there is a famous puzzle in philosophy called the Sleeping Beauty problem. It involves the following experiment that Sleeping Beauty volunteers to take part in.

On Sunday, Sleeping Beauty is put to sleep and then a fair coin is tossed. On Monday, Sleeping Beauty is awakened and interviewed. She is then put to sleep again with an amnesia-inducing drug that makes her forget that awakening. On Tuesday, if the coin landed tails she is awakened, interviewed and put to sleep once more. Otherwise, if the coin landed heads, she is left asleep. On Wednesday, she is awakened and the experiment ends.

When she is awakened and interviewed she does not know which day it is or whether she has been awakened before. During the interview, Beauty is asked, "What is the probability that the coin landed heads?" The experiment is illustrated below.

Sleeping Beauty problem (illustration by Stuart Armstrong)

There are two popular but opposing solutions that are commonly given by philosophers. The solution known as the thirder position is that Sleeping Beauty should report that the probability of heads is 1/3. [1] The intuition is that Sleeping Beauty cannot distinguish between the three awake states, so she should assign the same probability to each state. Also, if the experiment is run many times, Sleeping Beauty is awakened on average in a state where the coin has landed heads 1/3 of the time and in a state where the coin has landed tails 2/3 of the time.

The halfer position is that that Sleeping Beauty should report that the probability of heads is 1/2. [2] The intuition here is that Sleeping Beauty seems to learn no new information when she is awakened. So she should not update on the probability that she held on the Sunday before the experiment began. That is, it is the coin toss that is the relevant event, not how many times she is awakened. [3]

So which position is correct?

Before the experiment starts, Sleeping Beauty knows that during the experiment she can be in one of the four following states with equal probability.

	Monday	Tuesday
Heads	1/4	1/4
Tails	1/4	1/4

She also knows that she will not be awakened on Tuesday if the coin lands heads, so the probability for that state is 0 when she is awakened. The question then is how the other probabilities should be updated.

The halfers distribute the probability to the remaining heads state, as follows:

	Monday	Tuesday
Heads	1/2	0
Tails	1/4	1/4

From the table:

P(Heads and Monday) = 1/2
P(Tails and Monday) = 1/4
P(Monday) = P(Heads and Monday) + P(Tails and Monday)
          = 1/2 + 1/4 = 3/4

Using the formula for conditional probability: [4][5]

P(Heads|Monday) = P(Heads and Monday) / P(Monday)
                = 1/2 / 3/4 = 2/3

This, I think, reveals the fatal flaw with the halfer position. If Sleeping Beauty is told it is Monday, then she will think the probability for the coin landing heads is 2/3!

The thirders distribute the probability evenly to all awake states, as follows:

	Monday	Tuesday
Heads	1/3	0
Tails	1/3	1/3

From the table:

P(Heads and Monday) = 1/3
P(Tails and Monday) = 1/3
P(Monday) = P(Heads and Monday) + P(Tails and Monday)
          = 1/3 + 1/3 = 2/3

Using the formula for conditional probability: [6]

P(Heads|Monday) = P(Heads and Monday) / P(Monday)
                = 1/3 / 2/3 = 1/2

So, if Sleeping Beauty is told it is Monday, then she will think the probability for the coin landing heads is 1/2, which intuitively seems correct.

But the thirder position seems to imply that Beauty has learnt new information since Sunday (when the probability for heads was 1/2) which causes her to update her probability for the coin landing heads to 1/3. What could that new information be?

That it is both Tuesday and that the coin landed heads is a possible state for Beauty to be in when she is asleep. So she learns that she is not in that state when she wakes up. As a consequence, she updates by excluding that state and normalizing the probabilities for the remaining awake states.

One further test for the thirder position is to consider the probabilities when Sleeping Beauty is awakened on, say, 1000 consecutive days when the coin lands tails (instead of two consecutive days). [7] The probability of the coin landing heads when Beauty is awakened is then 1/1001. This gives:

P(Heads) = 1/1001
P(Tails) = 1000/1001
P(Heads and Monday) = 1/1001
P(Tails and Monday) = 1/1001
P(Monday) = P(Heads and Monday) + P(Tails and Monday)
          = 1/1001 + 1/1001 = 2/1001
P(Heads|Monday) = P(Heads and Monday) / P(Monday)
                = 1/1001 / 2/1001 = 1/2

As with the original experiment, if Sleeping Beauty is told that it is Monday, then she will think the probability for the coin landing heads is 1/2 which (as before) intuitively seems correct. Note that there are still 1000 states that Beauty can be in as a result of the coin landing heads. But she will just not be awake when she is in 999 of those states so they are excluded from her awake state calculations.

--

[1] The thirder position was initially introduced by Adam Elga in this paper.

[2] The halfer position was defended by David Lewis in his reply to Elga.

[3] The thirders interpret the problem as an experiment about awakening events (i.e., which branch Beauty will find herself on) whereas the halfers interpret the problem as an experiment about a coin toss event (i.e., which branch Beauty will be placed on). But both sides agree about what odds should be accepted if Beauty is offered a bet that the coin landed heads. If she is offered a bet whenever she is awakened, she should accept 1/3 odds. If she is only offered a bet once, she should accept 1/2 odds.

[4] The formula for calculating the conditional probability of A given B is:

P(A|B) = P(A and B) / P(B)

[5] The probability calculations for Sleeping Beauty when awake according to halfers:

P(Heads) = 1/2
P(Tails) = 1/2
P(Heads and Monday) = 1/2
P(Heads and Tuesday) = 0
P(Tails and Monday) = 1/4
P(Tails and Tuesday) = 1/4
P(Monday|Heads) = P(Heads and Monday) / P(Heads) = 1/2 / 1/2 = 1
P(Tuesday|Heads) = P(Heads and Tuesday) / P(Heads) = 0 / 1/2 = 0
P(Monday|Tails) = P(Tails and Monday) / P(Tails) = 1/4 / 1/2 = 1/2
P(Tuesday|Tails) = P(Tails and Tuesday) / P(Tails) = 1/4 / 1/2 = 1/2
P(Monday) = P(Heads and Monday) + P(Tails and Monday) = 1/2 + 1/4 = 3/4
P(Tuesday) = P(Heads and Tuesday) + P(Tails and Tuesday) = 0 + 1/4 = 1/4
P(Heads|Monday) = P(Heads and Monday) / P(Monday) = 1/2 / 3/4 = 2/3
P(Heads|Tuesday) = P(Heads and Tuesday) / P(Tuesday) = 0 / 1/4 = 0
P(Tails|Monday) = P(Tails and Monday) / P(Monday) = 1/4 / 3/4 = 1/3
P(Tails|Tuesday) = P(Tails and Tuesday) / P(Tuesday) = 1/4 / 1/4 = 1

[6] The probability calculations for Sleeping Beauty when awake according to thirders:

P(Heads) = 1/3
P(Tails) = 2/3
P(Heads and Monday) = 1/3
P(Heads and Tuesday) = 0
P(Tails and Monday) = 1/3
P(Tails and Tuesday) = 1/3
P(Monday|Heads) = P(Heads and Monday) / P(Heads) = 1/3 / 1/3 = 1
P(Tuesday|Heads) = P(Heads and Tuesday) / P(Heads) = 0 / 1/3 = 0
P(Monday|Tails) = P(Tails and Monday) / P(Tails) = 1/3 / 2/3 = 1/2
P(Tuesday|Tails) = P(Tails and Tuesday) / P(Tails) = 1/3 / 2/3 = 1/2
P(Monday) = P(Heads and Monday) + P(Tails and Monday) = 1/3 + 1/3 = 2/3
P(Tuesday) = P(Heads and Tuesday) + P(Tails and Tuesday) = 0 + 1/3 = 1/3
P(Heads|Monday) = P(Heads and Monday) / P(Monday) = 1/3 / 2/3 = 1/2
P(Heads|Tuesday) = P(Heads and Tuesday) / P(Tuesday) = 0 / 1/3 = 0
P(Tails|Monday) = P(Tails and Monday) / P(Monday) = 1/3 / 2/3 = 1/2
P(Tails|Tuesday) = P(Tails and Tuesday) / P(Tuesday) = 1/3 / 1/3 = 1

[7] A similar example is used by Nick Bostrom in this paper intending to show that the thirder position is counter-intuitive. I disagree, for the reasons I give in my conclusion. Bostrom also makes several other arguments against both the halfer and thirder positions and defends a hybrid position.

Is color real? (Part 2)

Are tiles A and B the same color?

In my previous post I raised the question of whether color is a property of things in the world (such as paint, strawberries and fire-engines) or whether it is a property of the way that humans see things.

In everyday usage, we generally associate color with things in the world. We suppose that the fire-engine is red not merely when we look at it but also when we are not.

That has raised some scientific and philosophical issues. One issue is that sometimes people name colors differently because of vague boundaries. That seems to be the case with my disagreement with Jason. The color of the stool was on the boundary of red and orange. Since we learn about objects and their colors by looking at them and categorizing them, it is possible that people categorize colors slightly differently.

Another issue is that physical differences in our eyes can cause perceptual differences. This is the case with color blind people who can't differentiate between red and green.

A further issue is that background lighting influences the color we perceive. This is primarily the case with "the dress" phenomenon where some people report the dress as being blue-and-black and others that it is white-and-gold. It turns out that people see the actual dress as blue-and-black so the phenomenon only occurs when viewing an image of the dress. The colors people see generally depends on how much background is shown in the image and also the external lighting when viewing the image on a computer screen.

Then we get the issues of light reflection. If a paint can is closed (or it is open at night with the lights off), then paint would not appear red since color perception requires light to reflect off the surface of the paint onto our eyes. And, of course, a blind person never sees the paint as red.

Finally, there are the wavelengths of light. Visible light covers a small range of wavelengths in the electromagnetic spectrum and constitutes most of the light that the sun radiates. So color can be associated directly with the wavelength as well.

So there are a lot of things to consider. From a pragmatic point-of-view, we don't want to have to deal with all these issues (and others we haven't thought of) every time we refer to color in our everyday lives. We just want to say that the fire-engine is red regardless of the background lighting or the perceptual capabilities of different people.

The way we have achieved this with our everyday concept of color is by abstracting away all the complicating details. We simply associate color with the object [1]. If there are differences in perception due to lighting or a person's perceptual capabilities, then we use the word "appears" to make that distinction. Thus, the fire engine is red, but it doesn't appear red at night or to a blind person. The paint in the closed can is also red even though there is no light reflecting off the paint inside the can.

The benefit to abstracting in this way is that not only is the concept simple to learn and intuitive to use, it is also independent of any particular physical explanation for why color appears as it does. If future scientific discoveries are made that replace our best current theories of light, the reflective properties of surfaces, our eyes, or how our brain implements perception, then our everyday language around color would not need to change. We would just know more about color than we did before. [2]

So this is why color is a property of things and not a property of how we perceive things.

--

[1] We learn concepts ostensively by paradigmatic examples. In this case, we learn colors by pointing at objects like fire-engines under normal lighting and comparing and contrasting them with other colored objects. Color is real (it is about objects in the world, not our perception of objects) and abstract (it need not carve nature at its joints).

[2] A further consideration is that other creatures perceive colors differently to humans as a consequence of their different brain and eye structures. So the scope and context of color concepts needs to be recognized.

Yes, they are the same color - they just appear different

Is color real?

I had a metaphysical discussion with my two-year old son this morning. I put a stool back in the room where it belonged (the chair kind of stool, not the other kind).

My son says, "Want orange stool."

So I retrieve the stool and say, "Jason, it's not an orange stool, it's a red stool."

He says, "Orange."

I go and get an obviously red drum stick and an obviously orange toy building. Showing him the drum stick, I say, "Is this red?"

"Yes."

Pointing to the toy building, I say, "Is this orange?".

"Yes."

Pointing to the stool, I say, "So Jason, see how the color of the stool is like the color of the red drum stick? So the stool is red."

"No, orange."

So my son and I are clearly having a serious metaphysical disagreement. Surely the stool has to be one color or the other. Can the stool be orange for him merely because he thinks it is orange?

Before you take the objective or subjective side of the dispute, here is a further puzzle about color. When we buy a can of paint, it has the name of a color on the label, such as "Red". However what is the color of the paint inside the can when the lid is on? (Note: there is no light inside the closed can). Also, when the can is opened under normal lighting, what is the color of the paint for a blind person?

For my solution, see Part 2.

Interpreting Quantum Mechanics

In my last post, I discussed the mathematical intuition underlying Quantum Mechanics, which is the idea that particle configurations with probability amplitudes can cancel out when combined. That is because amplitudes can be positive, negative or complex numbers, not just positive numbers as classical probabilities are.

I also noted that when a photon arrives at the back screen in the double-slit experiment, we only see one of the slit detectors activated, which corresponds to one of the configurations. But how do we account for the configuration where the other detector was activated?

There are several interpretations, all disturbing, so therefore best described by reference to superheroes.

Copenhagen Interpretation: You, the conscious observer, create reality. If you look at the back screen, the photon will be a wave. If you look at the slits, the photon will instead be a particle. You don't feel like you're in a superposition, so you're an exception to the laws that those little particles follow.




De Broglie–Bohm (or pilot-wave) Interpretation: The photon surfs on a wave which carries it through one of the slits to a location on the back screen where destructive interference doesn't occur. Added bonus: Entangled particles can communicate faster than the speed of light in violation of special relativity.

Many-Worlds Interpretation: You observe one of the photons going through one slit while your twin observes the other photon going through the other slit. You are either observing a superposition on the back screen or participating in a branch of one when you become entangled with one of the photons. Also the entire universe is in a superposition. Things just look classical because you have to be standing in one configuration or another.





Instrumentalist Interpretation: Yada yada yada ... who cares? You can build really amazing stuff using quantum mechanics! Also affectionately called, "shut up and calculate!".

So there we have it. The Copenhagen Interpretation posits an observer-dependent reality and a mysterious wave function collapse. The Bohmian Interpretation requires information to travel faster than light.^[1] The Many-Worlds Interpretation causes incredulous stares. And, finally, the Instrumentalist Interpretation isn't an interpretation at all.^[2]

Since each interpretation uses the same mathematical formalism, is there any reason to prefer one to another on philosophical grounds? I think there is.

The famous Schrodinger's Cat thought experiment vividly demonstrates the logic of quantum behavior in terms of familiar, everyday things. The upshot is that the cat is in a superposition of being both dead and alive until we look in the box. But what does that mean? And why don't we normally observe such things?

To take the second question first, we actually can observe superpositions involving objects that are (just barely) visible to the naked eye. For example, a recent experiment demonstrated interference effects for the superposition of a tiny tuning fork vibrating and not vibrating. This is analogous to the double-slit experiment where photon amplitude flows through both slits and we observe an interference pattern on the back screen.

So what does this mean? It means that we have observed the effects of a single amplitude, and it is the sum of the amplitude for a vibrating tuning fork and the amplitude for a non-vibrating tuning fork.

How can this observation be explained in a coherent way? The idea that there is a single tuning fork that is both vibrating and not vibrating is a contradiction, so that fails. The idea that there is a single tuning fork that is either vibrating or not vibrating also fails, since possibilities can't cause interference effects. That leaves the idea that there is one tuning fork that vibrates and a second one that does not and that we are observing the combined effects of both.

If there are two tuning forks, what would explain the observation of a single tuning fork (that is either vibrating or not) when you try to detect the vibration? There are paths that entangle you with the vibrating tuning fork and paths that entangle you with the non-vibrating tuning fork and amplitude flows along both paths. Becoming entangled with one of the tuning forks destroys the interference pattern from your point-of-view (that is, you can no longer observe the effects of the other tuning fork because there are no amplitude paths from here to there).^[3]

The Many-World's Interpretation, despite its startling implications, seems to me to be the most coherent interpretation of both our everyday and quantum observations. That, of course, doesn't mean that it is true - that is ultimately an empirical question^[4]. However it is an intuitive and natural framework for conceptualizing our observed experience.

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[1] See Bells Theorem which rules out local hidden variables.

[2] The pragmatic and reasonable version of the Instrumentalist Interpretation is that the math works, whether or not we know what it means. In stronger versions, it asserts that there is no explanation to be had and that the math doesn't mean anything.

[3] This process is called decoherence and is only reversible in practice in microscopic environments where interactions with air and apparatus molecules can be controlled (it's easy to break an egg but difficult to put it back together again). So observing things in a single (or decohered) state is the everyday situation that corresponds to our classical intuitions. The difficult challenge for quantum computing just is how to reliably maintain quantum bits in a superposition of 1 and 0 values.

[4] Scott Aaronson has a great post explaining that while the Many-Worlds Interpretation is the obvious, straightforward reading of quantum mechanics, it is also provisional in a way that heliocentrism (as opposed to geocentrism) isn't. You could, in principle, fly a spaceship above the plane of the solar system and see the Earth and the other planets circling the sun. However you can't similarly travel to another world branch to meet your twin.