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. — Dusty of Sky
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. — Dusty of Sky
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, On Interpretation, §9
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. — Dusty of Sky
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. — Andrew M
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. — Dfpolis
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. — Dfpolis
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. — Dfpolis
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
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
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. — Dusty of Sky
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. — Dusty of Sky
I think it is possible for a being capable of ideation and understanding to perform this deduction regardless of his particular experiences. — Dusty of Sky
Oddly, one can say the same about i - the root of negative one. Despite this, we make use of them. — Banno
It's not real in the same way as other numbers are. — Dusty of Sky
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. — Dusty of Sky
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). — Quantum logic is alive ∧ (it is true ∨ it is false) - Michael Dickson
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. — Banno
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
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. — Andrew M
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. — Andrew M
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. — Andrew M
Hence the distributive law is wrong. — Quantum logic is alive ∧ (it is true ∨ it is false) - Michael Dickson
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. — Dusty of Sky
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. — Dusty of Sky
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. — Dusty of Sky
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. — Dusty of Sky
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). — Andrew M
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. — Andrew M
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. — Dusty of Sky
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. — Dusty of Sky
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? — Dusty of Sky
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. — Dusty of Sky
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. — Dusty of Sky
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. — Wigner's friend - Wikipedia
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? — Dusty of Sky
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. — Dusty of Sky
I think we must posit that the indefinite parts are grounded in the definite parts. — Dusty of Sky
That seems right to me. Or, in alternative language, that the potential is grounded in the actual. — Andrew M
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. — Andrew M
Then I think we may have reached a satisfactory point of agreement. — Dusty of Sky
it's been a good discussion. — Andrew M
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