## Is Logic Empirical?

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It seems that disjunction in quantum logic has a different meaning than in classical logic. In classical logic, A or B means either A is true or B is true. In quantum logic, A or B means either A is true or B is true or it is indefinite whether A or B is true. You pointed out that this indefiniteness is not merely epistemic (at least according to the Copenhagen interpretation). It might be epistemically indefinite i.e. uncertain, whether a coin landed on heads or tails, but we know that it actually did land on one or the other side. But in the case of the photon, it is metaphysically indeterminate whether it went through slit A1 or A2. Disjunction in quantum logic can express this state of metaphysical indeterminacy.

Yes, that's right. But note Putnam's analogy with non-Euclidean geometry. Geodesic has a different meaning to straight line. However a geodesic on a flat surface is a straight line. Similarly, quantum disjunction has a different meaning to classical disjunction. However a quantum disjunction of measured quantities is a classical disjunction.

If P is indeterminate, then the proposition "A or not A" does not make sense, for the same reason that the proposition "the present king of France is bald or not bald" does not make sense. There is no present king of France, so it's neither quite correct to say he is bald nor that he is not bald. Likewise, there is no determinate outcome of P, so it is neither quite correct to say A obtains nor not A obtains. Neither example proves that the law of excluded middle has exceptions. All existing subjects either have or lack a given predicate, but if the subject does not exist, then it does not make sense to assert that the subject lacks the predicate. It does not make sense to assert that the present king of France lacks baldness because this implies that he has hair, which he does not because he doesn't exist. Likewise, it does not make sense to assert that the outcome of P is not A, because this implies that P has a determinate outcome. For a more concrete example of indeterminacy, take the statement "Bob will leave his house tomorrow". Assuming that Bob has free will and the future does not yet exist, it is undetermined whether he will leave his house tomorrow. So it is neither quite true to say that he will leave his house nor that he won't leave his house because both statements falsely imply that his future is already determined.

Agreed. Along the same lines, consider Aristotle's future sea battle scenario:

One of the two propositions in such instances must be true and the other false, but we cannot say determinately that this or that is false, but must leave the alternative undecided. One may indeed be more likely to be true than the other, but it cannot be either actually true or actually false. It is therefore plain that it is not necessary that of an affirmation and a denial, one should be true and the other false. For in the case of that which exists potentially, but not actually, the rule which applies to that which exists actually does not hold good.

Aristotle's proposal was that the sea battle propositions had the potential to be true or false, but weren't actually true or false until the event occurred. Similarly, "the present King of France is bald" has the potential to be true or false, but isn't actually true or false without a present King of France. (In Peter Strawson's terminology, "the present King of France is wise" represents a presuppositional failure and therefore isn't truth-apt.)

So the same idea can be applied to QM. There is a potential answer to the question of which slit the photon goes through, but no actual (or definite) answer in the absence of a measurement. As physicist Asher Peres put it, "unperformed experiments have no results".

Just because classical disjunctions don't express indeterminacy doesn't mean that indeterminacy defies the laws of classical logic. We can still reason about indeterminate states of affairs using classical logic. For instance, we can conclude that, if A obtains, then P is not indeterminate.

So I don't think that we should think of quantum logic as a deeper form of logic and classical logic as merely a special case. It may be true that the physical world is fundamentally indeterminate, meaning that determinate processes such as coin flips are a special case in relation to the indeterminate subatomic processes which underlie them. And it does seem to be true that quantum logic is often more useful than classical logic when it comes to describing quantum phenomena. But this is only because quantum logic is specifically designed to express indeterminacy, not because classical logic is violated by indeterminacy.

Yes, one approach here is to say that classical logic applies when things are definite, e.g., when a measurement has been performed, or the subject being predicated exists, or the contingent event has occurred. But it does not apply outside that context. So it's not that classical logic is violated by indeterminacy, it's that the preconditions for its use have not been met. Garbage in, garbage out.

In a similar way, Euclidean geometry is applicable when a surface is flat. We know that the angles of triangles will sum to 180°, the Pythagorean Theorem will hold, and so on. But it is not applicable (or needs to be applied in a different way) to curved surfaces.
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Yes, one approach here is to say that classical logic applies when things are definite, e.g., when a measurement has been performed, or the subject being predicated exists, or the contingent event has occurred. But it does not apply outside that context. So it's not that classical logic is violated by indeterminacy, it's that the preconditions for its use have not been met. Garbage in, garbage out.

If classical logic doesn't apply to indefinite scenarios, then this would seem to be due to a limitation of the applicability of the law of excluded middle. A statement is indefinite if neither it nor its negation is true. For instance, it is neither true nor false that the photon passed through A1.

But the purpose of the present king of France analogy was to expose what I think may be a confusion on the part of some (perhaps not Putnam) who assert that quantum mechanics proves that the laws of classical logic are limited in their application and open to empirically informed revision. The reason the law of excluded middle doesn't apply to the photon's position is the same as the reason it doesn't apply to the present king of France. Neither the photon's (definite) position nor the present king of France exist, and the absence or presence of a property can only be meaningfully predicated of something which exists. No empirically motivated insight is required to grasp this principle. As you brought up, it was grasped at least to some extent by Aristotle. The empirically motivated insights of quantum theories have certainly led us to revise our understanding of physics. Whereas it was previously believed that all spatial entities had definite positions, it is now widely held that subatomic particles exist in superpositions. This notion, although well supported, is highly counterintuitive. I think the confusion occurs when people make something along the lines of this argument:

(1) QM has led us to revise our intuitive understanding of space.
(2) QM has led us to develop new logical languages (with different connectives).
(3) Therefore, QM has led us to revise our intuitive understanding of logic.

This syllogism is obviously unsophisticated and invalid, so it might be fair to call it a straw man, but I think it illustrates the way people often think about this topic. "If quantum mechanics can defy our intuitions about space, then why not our intuitions about logic," the reasoning goes. I think the conclusion is false because the idea that excluded middle doesn't always apply didn't originate with quantum logic. The fact that quantum logic is non-distributive, as I far as I can tell, follows directly from the fact that excluded middle doesn't apply to quantum superpositions i.e. we can neither say that the photon passes through A1 nor that it does not pass through A1 because it doesn't have a definite position.

Although the law of excluded middle may not be universal in as obvious a sense as the law of non-contradiction, I still think we can truthfully call it universal. Reality consists of things which exist, and as long as excluded middle applies to all things which exist, it applies to all things in reality. Therefore, it is universal. Since no definite position of the photon exists, the law of excluded middle does not apply to it. But the photon itself exists, so the law applies to the photon. For instance, it is either true or false that the photon has a definite position.

I'm not sure if we have a substantial disagreement at this point. I agree that the law of excluded middle cannot be carelessly applied to all propositions, and you've shown that quantum disjunctions can describe superposition in a way which classical disjunctions can't. But I don't see this as evidence that the law of excluded middle is contingent or that logic is open to empirical revision (I'm not sure if you ever intended to make that argument, but my original purpose in this post was to argue against it). We develop new logical languages to more effectively describe new facets of reality which we discover. But the fundamental logical laws of reality, or at leasts the laws of our capacity to think about reality, remain the same regardless of what we observe. One point I will concede is that quantum disjunctions are no less valid than classical disjunctions. They are equally applicable to reality, or at least to the parts of reality for which they are intended.
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In my view, which is based on Aristotle, Aquinas and John of St. Thomas, the rules of logic are not given a priori, but are abstracted from experience.

To understand this, we must distinguish what Henry Veatch calls "intentional logic" from modern logic, which is quite different. The kind of logic I am discussing is the art and science of correct thinking, not a set of rules for symbolic manipulation.

The rules of modern logic cannot be applied without thinking an Aristotelian syllogism:
Every case with these characteristics is a case in which this rule applies.
This case (the one I am thinking about) has these characteristics.
Therefore, this is a case in which this rule applies.
This is simply Aristotle syllogism in Barbara, and is what we must think to apply any scientific knowledge we have. Hence, despite the vigorous protestations of modern logicians, they have not done away with Aristotle's logic, but rely on it whenever they apply the rules of manipulation they have developed.

So, we need to consider how it is that we think when we think correctly. Robert Boole, the founder of Boolean logic, entitled his masterwork The Laws of Thought, but it you reflect, there is no law preventing us from thinking "square circle," or "triangles have four sides." It is only if we want our thought to apply to reality, to what is, that we should not think these kinds of thoughts.

So, let me suggest that we abstract from our experience an understanding of what it means to be -- an understanding of the nature of existence. And, implicit in this a posteriori understanding are laws of being that must be reflected in our thought, if our thought is to apply to what is. For example, we come to understand that nothing can both be and not be at one and the same time in one and the same way, and, from this we derive the logical rule that we cannot both assert and deny the same thing at the same time. So we come to grasp the logical principles of Identity, Contradiction and Excluded Middle from the corresponding ontological principles -- and we abstract those from our experience of being.

In counting different kinds of things (pennies, apples, Legos), abstraction allows us to see that counting, and so the relationship between numbers, does not depend on what we count. In the same way, we can see from our experience with different kinds of being, that some relations (the ontological principles above) do not depend on what kind of being we are dealing with, but are true of being per se.
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Hence, despite the vigorous protestations of modern logicians, they have not done away with Aristotle's logic, but rely on it whenever they apply the rules of manipulation they have developed.

Agreed.

there is no law preventing us from thinking "square circle," or "triangles have four sides." It is only if we want our thought to apply to reality, to what is, that we should not think these kinds of thoughts.

There's no law preventing us from thinking the words square circle, but we can't form a concept corresponding to these words. We can form some coherent concepts which don't seem to apply to reality e.g. wizards and unicorns. So I think it's correct to call the laws of logic laws of thought, since they constrain not only what is possible in reality but what is possible for us to conceive in our own minds.

So, let me suggest that we abstract from our experience an understanding of what it means to be -- an understanding of the nature of existence. And, implicit in this a posteriori understanding are laws of being that must be reflected in our thought, if our thought is to apply to what is.

I think that you are correct if we understand a posteriori to mean learned through experience and a priori to mean known without having been learned. A child is not born with a developed understanding of the laws of logic. We must hone our intellects in order to comprehend the truth of a proposition like "a proposition cannot be both true and false". However, if a posteriori means contingent upon experience and a priori means knowable as true or false regardless of particular experiences, then I think you are incorrect. For as long as we possess an intellect capable of abstracting and experiences capable of being abstracted from, we should be able to deduce the same laws of logic. Even if I had no experiences to abstract from but the consciousness of my own existence, I should be able to deduce that I exist, therefore I don't not exist, and since not not existing is the same as existing, my only options are to exist or not exist. Moreover, I should see that these principles must be the same for all existing things due to the nature of existence. Given the weakness of the human mind and its dependence on sensation, a human solely conscious of his own existence would probably not being to perform this sort of deduction. But in principle, I think it is possible for a being capable of ideation and understanding to perform this deduction regardless of his particular experiences. As you say, the laws of logic are implicit in the nature of being, so even if the human intellect, due to its weakness, depends on many particular experiences to become conscious of the laws, the laws are contained implicitly within every possible experience. Perhaps you have reasons for objecting to the word a priori, but I think that if anything is a priori, it is logic.
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There's no law preventing us from thinking the words square circle, but we can't form a concept corresponding to these words.

Oddly, one can say the same about i - the root of negative one. Despite this, we make use of them.
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There's no law preventing us from thinking the words square circle, but we can't form a concept corresponding to these words.

We can't form an image of a square circle, but we can add the modifying concept <square> (not just the word "square") to the concept <circle>.

However, if a posteriori means contingent upon experience and a priori means knowable as true or false regardless of particular experiences, then I think you are incorrect.

I mean a posteriori with respect to the experiences required to learn, and a priori in is subsequent application, So, I think we agree.

Even if I had no experiences to abstract from but the consciousness of my own existence, I should be able to deduce that I exist, therefore I don't not exist, and since not not existing is the same as existing, my only options are to exist or not exist.

I think we only learn about ourselves as knowing subjects from reflecting on our acts of knowing. So, I don't think that what you envision is possible.

I think it is possible for a being capable of ideation and understanding to perform this deduction regardless of his particular experiences.

I agree if you mean any specific kind of experience, but not if you mean with absolutely no experience of knowing, as then we would have no means of grasping that we can know, This is because we can only understand what is actual/operational, not what is merely potential.

My objection to a priori, is that see no reason to believe that we know anything prior to all experience.
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Oddly, one can say the same about i - the root of negative one. Despite this, we make use of them.

True, but there's reason we say i is imaginary. It's not real in the same way as other numbers are. that being said, I'd love to hear a mathematician's insight into why i is so useful and why it can contribute to the generation of fractal patterns like the Mandelbrot set.
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It's not real in the same way as other numbers are.

I'm not sure that can be filled out...

How are numbers real? Apart, of course, from real numbers...
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Although the law of excluded middle may not be universal in as obvious a sense as the law of non-contradiction, I still think we can truthfully call it universal. Reality consists of things which exist, and as long as excluded middle applies to all things which exist, it applies to all things in reality. Therefore, it is universal. Since no definite position of the photon exists, the law of excluded middle does not apply to it. But the photon itself exists, so the law applies to the photon. For instance, it is either true or false that the photon has a definite position.

Indefiniteness can apply to existence as well. An electron could be in a superposition of an excited state and ground state (having emitted a photon). In this case the number of photons (0 or 1) is in superposition. That is, what exists or not can be in superposition.

So the issue seems to boil down to definiteness and indefiniteness. In everyday experience, an unobserved flipped coin is definitely heads or tails and we don't have any trouble visualizing that scenario. We can describe the situation formally using the law of excluded middle (with observed anomalies, such as the coin landing on its edge, having straightforward explanations).

QM upends that intuitive picture. You can have a quantum coin which, when flipped twice and then observed, will always be found in the same state it started in (e.g., always heads). There is no straight-forward way to visualize that process without giving up counterfactual-definiteness. And so the LEM is no longer applicable in the obvious way.

Further, since a classical coin flip scenario depends on quantum decoherence, that suggests the LEM is contingent. That is, there is no logical requirement for quantum systems to decohere - a quantum system could conceivably remain coherent indefinitely (i.e., isolated and not interacting with other systems). The universe, taken as a whole, may be an example of such a system.

So there are at least two broad options available. One option is that there is a more fundamental logic (say, quantum logic) that applies universally with classical logic emerging as a special (or approximate) case that applies in decohered environments. A second option is that classical logic is universal, with indefiniteness being just a placeholder for what has not yet been satisfactorily explained in definite terms.

It doesn't seem that either option can be definitively ruled out at present. Michael Dickson discusses the first option in his paper:

The Fundamental Claim of QL. QL claims that quantum logic is the ‘true’ logic. It plays the role traditionally played by logic, the normative role of determining right-reasoning. Hence the distributive law is wrong. It is not wrong ‘for quantum systems’ or ‘in the context of physical theories’ or anything of the sort. It is just wrong, in the same way that ‘(p or q) implies p’ is wrong. It is a logical mistake, and any argument that relies on distributivity is not logically valid (unless, of course, distributivity has been established on other grounds).
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There's no law preventing us from thinking the words square circle, but we can't form a concept corresponding to these words.
— Dusty of Sky

Oddly, one can say the same about i - the root of negative one. Despite this, we make use of them.

i can be conceptualized as a quarter rotation on the complex plane. So, 1 * i * i = -1. As Gauss said:

That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question. — Carl Friedrich Gauss
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Indefiniteness can apply to existence as well. An electron could be in a superposition of an excited state and ground state (having emitted a photon). In this case the number of photons (0 or 1) is in superposition. That is, what exists or not can be in superposition.

Couldn't we say that the electron exists but no definite state of the electron exists and no definite number of photons exists? Saying no definite state exists just means that the state is indefinite. I see why it seems problematic that we can neither affirm nor deny that a photon exists. Could we perhaps resolve this problem by thinking of the photon's indefinite existence as a property of the electron. Existence is only existence as such when it is definite. If something exists indefinitely, it only exists as a property of something which exists definitely. For instance, the position of a particle in superposition exists indefinitely as a property of the particle, which exists definitely.

QM upends that intuitive picture. You can have a quantum coin which, when flipped twice and then observed, will always be found in the same state it started in (e.g., always heads). There is no straight-forward way to visualize that process without giving up counterfactual-definiteness. And so the LEM is no longer applicable in the obvious way.

I'm not sure I understand this argument. Are you saying that giving up counterfactual definiteness also forces us to give up LEM? I've argued that this isn't the case because you can't meaningfully predicate something of a non-existent subject. LEM even applies to indefinite states of affairs because all states of affairs are either definite or indefinite. It just doesn't apply to particular determinations of indefinite states of affairs because no such determinations exist.

If the unobserved universe exists in an indefinite state at both the micro and the macro level, and the only parts of the universe that are definite are the parts we observe, I see how this could be problematic. If indefinite existence is a property of definite existence, and the unobserved universe exists indefinitely, then the unobserved universe is a property of the observed universe. This would imply that the existence of the universe is dependent upon our observation of it. I think it would be better to avoid this conclusion if possible, but it's still preferable to allowing things which don't definitely exist to have properties. You said, "what exists or not can be in a superposition". This strikes me as not only counterintuitive but inconceivable. If something does not exist, then it is nothing, and it can have no properties. Therefore, a thing can only have properties insofar as it exists. If it is indefinite whether a thing exists or not, then it is indefinite whether it has properties.

So there are at least two broad options available. One option is that there is a more fundamental logic (say, quantum logic) that applies universally with classical logic emerging as a special (or approximate) case that applies in decohered environments. A second option is that classical logic is universal, with indefiniteness being just a placeholder for what has not yet been satisfactorily explained in definite terms.

I won't speak to the second option, but if the first option entails that things are fundamentally indefinite, then I think we should reject it. It seems to me that, although we can allow some things to be indefinite, our description of the world must bottom out in something definite. For instance, the particle can only have an indefinite position if it definitely exists. If the particle exists indefinitely, then I argue that we should think of its indefinite existence not as an independent entity but as a property of something definite. For only things which exist can have properties, so if a thing doesn't definitely exist, it doesn't definitely have any properties, so nothing we can say about it is definitely true.

Hence the distributive law is wrong.

If the disjunction symbol means what it means in classical logic, the distributive principle is correct. If it means what it means in quantum logic, it is incorrect. The disjunction symbol can mean whatever we want it to mean, so I don't think either application is fundamentally right or wrong.
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Couldn't we say that the electron exists but no definite state of the electron exists and no definite number of photons exists? Saying no definite state exists just means that the state is indefinite. I see why it seems problematic that we can neither affirm nor deny that a photon exists. Could we perhaps resolve this problem by thinking of the photon's indefinite existence as a property of the electron. Existence is only existence as such when it is definite. If something exists indefinitely, it only exists as a property of something which exists definitely. For instance, the position of a particle in superposition exists indefinitely as a property of the particle, which exists definitely.

The superposition can extend to the electron's existence as well. Consider Schrodinger's Cat where there can conceivably be lengthy alternative histories in superposition (and exhibiting interference). This also plays out in the Wigner's Friend thought experiment which describes a scenario where a system's state is definite for one observer (the friend) but indefinite for another observer (Wigner, for whom the friend is in superposition).

I'm not sure I understand this argument. Are you saying that giving up counterfactual definiteness also forces us to give up LEM? I've argued that this isn't the case because you can't meaningfully predicate something of a non-existent subject. LEM even applies to indefinite states of affairs because all states of affairs are either definite or indefinite. It just doesn't apply to particular determinations of indefinite states of affairs because no such determinations exist.

I just meant that we can't assume that if the (unobserved) state of the quantum coin is not heads then it must be tails. A superposition of heads and tails is also a possible state. (And, in this single quantum coin flip example, is the state.)

You said, "what exists or not can be in a superposition". This strikes me as not only counterintuitive but inconceivable. If something does not exist, then it is nothing, and it can have no properties. Therefore, a thing can only have properties insofar as it exists. If it is indefinite whether a thing exists or not, then it is indefinite whether it has properties.

OK. But be careful not to read "what exists or not can be in a superposition" as more than a mathematical description (e.g., a line somewhere on a plane that represents the probability of measuring the definite alternatives). It's just that quantum mechanics allows such a state, however that be conceived (which is supplied by the various quantum interpretations).

Hence the distributive law is wrong.
— Quantum logic is alive ∧ (it is true ∨ it is false) - Michael Dickson

If the disjunction symbol means what it means in classical logic, the distributive principle is correct. If it means what it means in quantum logic, it is incorrect. The disjunction symbol can mean whatever we want it to mean, so I don't think either application is fundamentally right or wrong.

Dickson would agree that you can define logical connectives either way. But he argues that only (non-distributive) quantum logic has empirical significance, and thus relates to correct reasoning, since it derives from quantum theory. He discusses this further under "The Motivation" on p3.
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The superposition can extend to the electron's existence as well. Consider Schrodinger's Cat where there can conceivably be lengthy alternative histories in superposition (and exhibiting interference). This also plays out in the Wigner's Friend thought experiment which describes a scenario where a system's state is definite for one observer (the friend) but indefinite for another observer (Wigner, for whom the friend is in superposition).

This is the first I've heard of Wigner's friend, but I just read that the purpose of the thought experiment is to support the theory that consciousness causes collapse i.e. that everything is in a superposition until a conscious being observes it. But maybe I misunderstood what I read. Theoretically, it is possible that nothing exists definitely until it is observed. In that case, consciousness is necessary for existence. I'll call this theory quantum idealism. If something could be definite for one observer but indefinite for another, then perhaps a many worlds interpretation of quantum idealism would follow. Each consciousness exists in its own world. Where two conscious beings observe the same thing, their worlds converge and where they observe contrary things (e.g. I observe the cat is alive and you observe it's dead) their worlds diverge. Since Wigner's friend observes something definitely which remains indefinite for Wigner, the consciousness of Wigner's friend has diverged into two separate worlds, each with its own version of Wigner's friend. Until Wigner contacts his friend, it is undetermined which of the two divergent worlds Wigner's world will converge with.

I don't necessarily endorse either a many worlds or a single world version of quantum idealism, but I think this is one possible way to account for lengthy alternative histories existing in superposition without contradicting LEM. As long as reality bottoms out in definite facts, such as being x observes y, LEM remains in tact. Whatever is indefinite exists only relation to what is definite. The cat is only indefinitely alive or dead in relation to the definite fact that it is in the box.

Dickson would agree that you can define logical connectives either way. But he argues that only (non-distributive) quantum logic has empirical significance, and thus relates to correct reasoning, since it derives from quantum theory. He discusses this further under "The Motivation" on p3.

All the math in this stuff goes over my head, especially given the limited amount of time I have to peruse it. It seems to me like the main difference between quantum and classical logic is that "a or b" in quantum logic means "a or b or it is indefinite whether a or b". So is the reason he sees quantum logic as more empirically significant that physical states of affairs can be indefinite? I see how this would make quantum disjunctions more useful to apply in certain contexts, but I don't think it makes the classical disjunction false. And if the classical disjunction is not inherently false, then neither is the principle of distributivity. And if I am right that reality bottoms out in something definite, then the classical disjunction applies to what is most fundamental.
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This is the first I've heard of Wigner's friend, but I just read that the purpose of the thought experiment is to support the theory that consciousness causes collapse i.e. that everything is in a superposition until a conscious being observes it. But maybe I misunderstood what I read. Theoretically, it is possible that nothing exists definitely until it is observed. In that case, consciousness is necessary for existence. I'll call this theory quantum idealism. If something could be definite for one observer but indefinite for another, then perhaps a many worlds interpretation of quantum idealism would follow. Each consciousness exists in its own world. Where two conscious beings observe the same thing, their worlds converge and where they observe contrary things (e.g. I observe the cat is alive and you observe it's dead) their worlds diverge. Since Wigner's friend observes something definitely which remains indefinite for Wigner, the consciousness of Wigner's friend has diverged into two separate worlds, each with its own version of Wigner's friend. Until Wigner contacts his friend, it is undetermined which of the two divergent worlds Wigner's world will converge with.

While Wigner held that "consciousness causes collapse", the thought experiment is independent of that interpretation. The key issue is that there can be two apparently incompatible reports - one by the friend who reports having measured a definite value (i.e., collapse has occurred) and another by Wigner who reports an indefinite value (i.e., collapse has not occurred). Those reports could be generated by unattended measuring devices, perhaps time stamped and read by someone at a later time.

That aside, your interpretation is a possible one, something like a many-minds interpretation.

I don't necessarily endorse either a many worlds or a single world version of quantum idealism, but I think this is one possible way to account for lengthy alternative histories existing in superposition without contradicting LEM. As long as reality bottoms out in definite facts, such as being x observes y, LEM remains in tact. Whatever is indefinite exists only relation to what is definite. The cat is only indefinitely alive or dead in relation to the definite fact that it is in the box.

OK. But note that per the Wigner's Friend thought experiment above, that box can itself be in an indefinite state with respect to some further observer or measurement device. In this sense, such definite facts assume a measurement or decohered context.

It seems to me like the main difference between quantum and classical logic is that "a or b" in quantum logic means "a or b or it is indefinite whether a or b". So is the reason he sees quantum logic as more empirically significant that physical states of affairs can be indefinite?

Yes. More precisely: per quantum theory, the world can be represented as a non-distributive lattice. So a logic that seamlessly captures that structure will also be non-distributive.

There are two separate claims there. The first is about the geometry of the world (analogous to whether spacetime is fundamentally curved or flat). The second is about the propositional logic that naturally applies to that geometry.

I see how this would make quantum disjunctions more useful to apply in certain contexts, but I don't think it makes the classical disjunction false. And if the classical disjunction is not inherently false, then neither is the principle of distributivity. And if I am right that reality bottoms out in something definite, then the classical disjunction applies to what is most fundamental.

That is true for quantum logic as well. Any claim about the universe as a whole is definite since it's a claim about the whole geometrical space. Any claim in a measurement or decohered context is also definite (at least relative to that context) since, on measurement, an indefinite state collapses to a definite state.

The difference between a classical disjunction and a quantum disjunction only emerges in those cases where two orthogonal subspaces in a higher-level space (such as two orthogonal lines on a plane) don't exhaust all the possibilities. Which is to say, those situations characterized by indefinite states such as the dual-slit experiment or the quantum coin.
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If we accept "consciousness causes collapse" but reject the many minds interpretation, then wouldn't we conclude that whatever is definitely the case for Wigner's friend is also definitely the case for Wigner? Once Wigner's friend performs a measurement and collapse occurs, the result he observes becomes definite for all potential observers even if they do not observe it themselves. Wigner doesn't know what his friend observed, but once his friend observes it, it is no longer in a superposition. We could also reject both the many-worlds interpretation and "consciousness causes collapse" and hold that unattended measuring devices can also cause collapses which become definite for all potential observers, conscious and mechanical alike. Does that not work?

If it does work, then the physical world can be divided into observed (consciously or mechanically) parts, which are definite, and unobserved parts, which are indefinite. If it doesn't work, then perhaps we must adopt a many-worlds or many-minds interpretation. Either way, definite parts of the universe (whether it's only one observer's universe or the universe for all observers) exist definitely and indefinite parts exist indefinitely. I think we must posit that the indefinite parts are grounded in the definite parts. And I think we must do so regardless of what portion of the physical universe is indefinite. Here's an argument:

Firstly, it seems impossible for the definite parts to be grounded in the indefinite parts. For if what exists most fundamentally exists indefinitely, then it does not definitely exist. But we exist definitely, and our existence can't be grounded in things which don't definitely exist. Our existence might be grounded in things with some indefinite properties, but these grounding things must at least definitely exist and possess some properties definitely.

Secondly, it seems necessary that the indefinite is grounded in the definite. For if a thing exists indefinitely, it does not definitely exist, hence it does not definitely have any properties, hence nothing definitely true can be said about it. Perhaps we can only make probabilistic claims about the indefinite, but even these claims are definite insofar as the probabilities we predicate of something indefinite are definitely its probabilities. So unless we want to treat indefinite things like Kantian noumena about which nothing can be meaningfully asserted, we must posit that what exists indefinitely is grounded in what exists definitely, even if what exists definitely has no other definite properties than set of definite probabilities.
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If we accept "consciousness causes collapse" but reject the many minds interpretation, then wouldn't we conclude that whatever is definitely the case for Wigner's friend is also definitely the case for Wigner? Once Wigner's friend performs a measurement and collapse occurs, the result he observes becomes definite for all potential observers even if they do not observe it themselves. Wigner doesn't know what his friend observed, but once his friend observes it, it is no longer in a superposition.

Certainly that was Wigner's conclusion:

Wigner considers a superposition state for a human being to be absurd, as the friend could not have been in a state of "suspended animation"[1] before they answered the question. This view would need the quantum mechanical equations to be non-linear. It is Wigner's belief that the laws of physics must be modified when allowing conscious beings to be included.

We could also reject both the many-worlds interpretation and "consciousness causes collapse" and hold that unattended measuring devices can also cause collapses which become definite for all potential observers, conscious and mechanical alike. Does that not work?

As Wigner implies above, it doesn't work if you want to retain standard physics. Changing the physics leads to theories such as objective collapse and de Broglie-Bohm (which each introduce non-locality to make the standard predictions).

Note that Wigner still observes interference after the friend's measurement. So if the friend's result "becomes definite for all potential observers", then it is a hidden variable. That amounts to a rejection of locality, per Bell's Theorem.

Either way, definite parts of the universe (whether it's only one observer's universe or the universe for all observers) exist definitely and indefinite parts exist indefinitely.

The way I would put it is that what is indefinite has the potential to be definite (for an observer).

I think we must posit that the indefinite parts are grounded in the definite parts.

That seems right to me. Or, in alternative language, that the potential is grounded in the actual.

Note that I haven't argued that what exists depends on what does not exist (or exists indefinitely). I don't think that is true. I've only argued that part of the universe can be indefinite, or lack definite form, for a particular observer. It takes an interaction or measurement to actualize that potential, so to speak.
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That seems right to me. Or, in alternative language, that the potential is grounded in the actual.

I like that. It sounds very Aristotelian.

Note that I haven't argued that what exists depends on what does not exist (or exists indefinitely). I don't think that is true. I've only argued that part of the universe can be indefinite, or lack definite form, for a particular observer. It takes an interaction or measurement to actualize that potential, so to speak.

Then I think we may have reached a satisfactory point of agreement. I still think that certain logical principles must be regarded as necessarily and universally valid, but I've gained a greater appreciation for how QM forces us to modify how we apply and qualify these principles. A universe which contains indefinite portions is significantly different from the universe Aristotle conceived of, so it makes sense that we've had to develop new methods of logic to describe it. I just don't think this amounts to a revolution in logic and an overthrow of the old paradigm (and it doesn't sound like you do either). Methodologically, logic has had to adapt and innovate to keep pace with the natural sciences, but I think the deepest truths of logic have remained the same. The natural sciences have inspired us to develop a more nuanced understanding of the principles of logic and reject certain false conclusions which follow from a lack of nuance (e.g. that superpositions are impossible). But the underlying, conceptual structure of logic does not depend on how we conceive the natural world.
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Then I think we may have reached a satisfactory point of agreement.

:up:

And thanks for your comments throughout - it's been a good discussion.
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it's been a good discussion.

Agreed. And thank you as well. This has been educational for me, and I think my point of view on these topics has become more sophisticated as a result.
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