Yes, he said:
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“Excluded middle does not apply” is not the same thing as “excluded middle is violated.” What he’s saying (and I agree) is that “dead” turns out not to mean the same thing as “not alive.”
Everything is either alive or not alive–S’s cat is not a violation of this, nor is anything in quantum theory. For excluded middle to be violated with regard to the characteristic “alive,” something would have to be both alive and not alive. This cat ain’t that.
I think you’re giving mathematicians a poor rap and giving common sense way too much credit. I’ve heard that joke before, and the punchline is that while the physicist considers himself far more strict than the philosopher by pointing out that he can only make statements about observed facts, the mathematician points out that the physicist has made a leap of faith of his own by assuming that all sheep are unicolor. Physicists strive to adhere to strict mathematics as much as any mathematician. At the same time, there’s plenty of math that goes against ‘common sense’. Statisticians love examples like the Birthday Problem or the Monty Hall paradox that go against all manner of common sense.
Quaternions are another fun mathematical feature that I’ve had the pleasure to work with in the past, basically vectors in a four dimensional vector space that can be used to describe direction in three dimensional space regardless of a fixed plane, all using complex number (mathematicians, please correct any mistakes in the previous sentence.) I mean, try and wrap common sense around that?
Screw common sense, is that I’m basically saying.
First of all, quantum mechanics does not violate classical logic. This is straightforwardly demonstrated through the possibility of so-called ‘hidden variable’-interpretations, like Bohmian mechanics. They do have unusual properties (non-locality, contextuality, etc.), and they essentially refer to ‘surplus structure’, i.e. unobservable stuff that the OP (rightly) has a problem with, but it’s always at least possible, if perhaps inconvenient. In such interpretations, one can always reason classically about every observable, and weirdnesses of quantum theory are indeed only apparent, and due to our ignorance of the hidden variables.
Nevertheless, you can also choose to look at quantum theory using a logic different from classical logic; however, in this case, it’s not the principle of the excluded middle that goes, but rather, distributivity. This is known as quantum logic (and while it has been proposed that it should replace classical logic as the ‘proper’ logic, because of the above argument, that doesn’t actually follow).
The reason for this is so-called complementarity. Two propositions are complementary roughly if both can’t simultaneously be known exactly. This is the basis of the uncertainty principle, which thus provides the classic example of why quantum logic does not distribute. First of all, the principle itself says that the accuracy of both position and momentum can’t exceed a specific threshold – the formalization is typically: ΔxΔp > h, where Δx and Δp are the uncertainties in position and momentum, respectively. One can translate this into the following proposition:
z: The particle’s momentum is in the interval Δp, and its position lies in the interval Δx.
Now we can think about the following three propositions:
p: The particle’s momentum is in Δp,
q: The particle’s position is in Δx[sub]1[/sub]
r: The particle’s position is in Δx[sub]2[/sub]
Where Δx[sub]1[/sub] and Δx[sub]2[/sub] are just the, say, left and right halves of the interval Δx. Then the following proposition: p ˄ (q ˅ r) (where ˄ denotes the logical and, and ˅ denotes the logical or) is clearly true, because it is equivalent to z, and thus, just a restatement of the uncertainty principle. However, the proposition: (p ˄ q) ˅ (q ˄ r), which is equivalent in classical logic, fails to be true, as both p ˄ q and q ˄ r are false; each asserts a proposition incompatible with the uncertainty principle, e.g. ‘the particle’s momentum is in Δp and the particle’s position is in Δx[sub]1[/sub]’, which would mean Δx[sub]1[/sub]Δp < h.
Now it is in this complementarity that the key to superposition lies. Let’s say there’s a property, which can obtain either of two values (typically, this is a particle’s spin along some axis), which is complementary to a similar property (spin along another axis, for example). If the particle thus has a definite spin along one axis, its spin along the other axis must be fully indeterminate – which means, it must be in a superposition of both possible values. Now Schrödinger’s cat just consists in coupling the welfare of a macroscopic being to such a superposition – in principle, it’s a straightforward consequence of the theory.
The superposition is then not of the logical form ‘A and not-A’, i.e. it’s not a violation of the principle of the excluded middle. Rather, if ‘A’ is ‘alive’, it’s just ‘not-A’ – the cat’s two-valued property doesn’t have a definite value along the alive-dead axis (so it’s certainly not alive, but also not dead – one way of viewing this is simply that alive and dead don’t exhaust the possibilities in the quantum world, but one can get rather tangled up thinking about it this way).
It’s interesting that almost all of quantum theory follows from the existence of complementary propositions – the way you can ‘build’ ordinary probability theory on classical logic, you can build quantum mechanics on quantum logic, too. I’ve written up a sketch of how this works (it will contain a lot of redundancy if you’re familiar with classical logic, set theory, and probability, so you can probably skim those parts), and also of how thinking about quantum mechanics in terms of such a novel theory of probability illustrates the origin (and meaning) of interference effects, such as in the classical double slit experiment. And for the complete three-fer, I discuss superpositions in the beginning of this bit on quantum entanglement in the light of what’s known as ‘Zeilinger’s principle’ (who, in a word, considers the origin of complementarity to be the finiteness of information in any given system – once something about it is known perfectly, there simply isn’t enough information to decide its other properties). In the interest of openness, however, I should mention that my perspectives probably can’t be considered entirely mainstream.
So much for shameless self-promotion… 
We’ve talked about this before, of course. Here I’ll just note that as far as I can tell (still) z is not equivalent to p ˄ (q ˅ r) unless the particle has a definite position, which, according to mainstream quantum mechanics, it does not.
More specifically, I don’t think there’s an equivalence between:
P: The particle’s position is in delta-x
and
Q: The particle’s position is in delta-x[sub]1[/sub] ˅ The particle’s position is in delta-x[sub]2[/sub].
For P’s being true does not require that the position have any property of being in any smaller interval than x. The position is spread out, in a sense, over the whole of x.
Similarly, the statement “Line segment X is in interval [a, c]” does not require that it be true that “either line segment X is in interval [a, b] or line segment X is in interval [b, c]” (where, of course, b is in between a and c).
As far as I can tell, the sense of ‘the particle’s position’ which makes your statement z true in QM is better modeled by line segments than points.
I think it’s somewhat misleading to use the ‘either… or’ construction here, as the logical or is inclusive; so ‘line segment is in interval [a, b] or [b, c]’ can well be viewed as ‘true’ if (a part of) the line segment is in [a, b] and (a part of) the line segment is in [b, c], i.e. if it’s false that the line segment is not in [a, b] and also false that the line segment is not in [b, c]. Of course, it’s not really a good way of talking about quantum mechanics to say things like ‘the particle is partially in [a, b]’, but it’s certainly right that it is unambiguously false to say that the particle is not in [a, b].
Nevertheless, the failure of the distributive law in quantum logic is, as far as I can tell, the accepted view (that’s at least how it seems to be regarded both by wiki and the Stanford Encyclopedia of Philosophy), so I think it’s the appropriate position for GQ.
Also, particle positions are not really well-modeled by lines, since in measurement, one can measure them arbitrarily well (at the expense of arbitrarily bad knowledge about momentum), which would be impossible for extended objects. The bottom line is that there’s really no unambiguous structure ‘behind’ the quantum formalism, not one of points, and neither one of lines – there’s a lattice of propositions about these properties, which happens to be non-distributive.
To be clear, by “in” I meant “contained entirely within.”
As for the mainstream status of the view, note the article you linked to does not claim that QM violates distributivity. It says some philosophers have thought so, while (it seems to me) being careful to neither reject nor endorse this view. And it says that there are non-classical logics which do not validate a theorem we might recognize as encapsulating a “distributive property,” and which model QM well. But none of that means QM violates distributivity.
That’s fine, but it doesn’t mean QM “violates distributivity.” In particular, the three propositions you offered, if interpreted consistently, validate Propositional Logic’s distributive property rather than violating it.
Put yet another way (sorry): The fact that subtraction isn’t commutative doesn’t mean that the physical phenomena we model through subtraction “violate commutativity.” It doesn’t mean that somehow the commutative properties of addition and multiplication are false or invalid.
Similarly, the fact that quantum logic isn’t distributive doesn’t mean that the phenomena modeled by quantum logic “violate distributivity.” It doesn’t mean that somehow the distributive properties of the “and” and “or” of propositional logic are false or invalid.
THANK YOU for finally actually answering the question. Or at least starting an answer. It’s infuriating watching so many people say that quantum theory somehow disproves classical logic.
What no one seems to be addressing is that we are dealing not with quantum mechanics, but with the Copenhagen interpretation of said mechanics. That particles exist in a superposition that collapses upon measurement is not something that has been scientifically proven, and the OP is right to reject that it has been.
The interpretation of quantum mechanics is the realm of philosophy, not science. There are many interprettions of what is going on. In particular, whether a superposition is a mere mathematical entity or an actual value is hotly debated. Or to put it in the terms of the paradox: whether the cat alivedead, or is its value merely undetermined is actually an open question on the quantum level.
And let us not forget that there is no theory of everything, so even asserting that quantum mechanics is true is not without peril. In fact, many physicists refuse to do anything but say that quantum theory works, and even refuse to interpret the results.
Well, you’ll see that in the beginning of my post, I explicitly stated that quantum mechanics does not violate classical logic; nevertheless, it can be described using a logic different from classical logic, in which distributivity does not hold. That’s not controversial; the algebra of Hilbert space subspaces, which can be viewed analogously to the algebra of sets in the classical case, simply is not distributive (more accurately, the projection operators – sort of the ‘membership functions’ – aren’t). The structure you get from this has the same properties as what one calls a ‘logic’ (sans distributivity), so that’s what people do.
Again, think about the example in terms of ‘empirical semantics’, where by ‘the particle is in [a, c]’ one just means ‘experiment will find the particle within [a, c]’. From this point of view, it’s clearly true that ‘experiment will find the particle within [a, c]’, and ‘experiment will find the particle within [a, b] or within [b, c]’ are equivalent – one can just check one’s measurement records afterwards, and it will always be the case that the particle was found within [a, c], and it will always be the case that the particle was found in [a, b] or [a, c], since it was always found at some point in [a, c], and the union of [a, b] and [b, c] is just [a, c] – thus it was always found at a point in [a, b] or [b, c].
Or, consider another case, a computer build using a three-valued logic. This does not violate two-valued logic, since one can build a computer using two-valued logic that emulated it; but nevertheless, three-valued logic is an obviously valid and equivalent description (you could just as well use the computer based on three-valued logic to emulate a two-valued logic computer – neither is any more fundamental).
Again, to be perfectly clear, I don’t mean to say that quantum mechanics ‘violates’ distributivity in that distributivity is now somehow wrong – fundamentally, distributivity is just a way to regroup symbols on a page, and you can do that according to whatever rules you may want to use. But for quantum mechanics, a set of rules in which distributivity doesn’t hold appears to be particularly apt, because it models complementarity well. You can use classical logic for the task – see the first paragraph of my first post in this thread --, but one can argue against this on different grounds (Occam’s razor is sometimes put to this task); you can also use three-valued logic (as Reichenbach did), or fuzzy logic, or nonlinear logic, or other kinds of rules one might come up with. There is just no one true right system handed down from above, its rules set in stone – we have to make do with what we can come up with (and that’s a good thing, because if there were one true set of rules, where’d they be supposed to come from?).
I did see that, but I guess I was confused about your position. Later on you said
which appeared to me to be making a point that classical distributivity is violated by quantum phenomena. But is that not what you’re saying here?
Well, you could translate what I wrote into classical logic, using for example a hidden variable theory, where particles simultaneously do have a well-defined position and momentum, and we’re just prohibited from finding them out. Thus things only appear to be non-distributive due to conspirative dynamics that keep us from accessing the basement layer, which is classical and distributive – this would essentially amount to emulating quantum logic within classical logic. (Something like this was done, for example, by Edward Moore in his somewhat famous paper ‘Gedanken-Experiments on Sequential Machines’, where he used automata nowadays called ‘Moore machines’ to construct an analogue of the uncertainty principle.)
But you can just as well not do that, and just use quantum logic, and only the properties and relations between them quantum mechanics dictates; for those, it is indeed the case that distributivity does not hold, meaning that classical logic, without ‘embellishments’, just isn’t the right tool to do the job of modelling the behaviour of quantum systems (which does not mean that it’s wrong anymore than the insufficiency of real numbers to model quantum dynamics means that the real numbers are ‘wrong’; this kind of question is just not an empirically decidable one).
After reading through the posts again I don’t really see anyone stating that quantum theory disproves classical logic. I think everyone was trying to inform the OP that standard reasoning (and physics) don’t always apply in the quantum world.
I thought that was pretty well understood since the OP was specifically asking about Schrodinger’s cat.
The term “scientifically proven” is marketing jargon and has nothing to do with the work of real scientists, especially in the realm of physics. This kind of statement is what leads people to saying things like “evolution is only a theory.”
There is a plethora of experimentally verified phenomena in this field. Spend five minutes researching entanglement and you’ll understand that while physicists disagree about things like what a grand unified theory will look like, there are still widely accepted theories in the community.
People are far too concerned with this alive-dead cat. This thought experiment was NOT intended to be a way to explain the Copenhagen interpretation to people. It was actually an illustration of the differentiation between the quantum world and our everyday interactions.
I do not believe that there is anything about the Schrodinger’s cat concept that is tied uniquely to the Copenhagen interpretation. Certainly Many-Worlds interpretation still has the alive/dead cat, and just adds to it by making the observer a lookingatalivecat/lookingatdeadcat entity once the box is opened.
Hidden variable interpretations get rid of the superposition, but replace it with quantities that are subject to the same criticism that the OP makes about the superposition itself – that we’re introducing these fundamentally non-observable things. (Plus, due to Bell’s theorem, they need to be non-local, which buggers up relativity, too).
But it’s not that bad. A simple solution has been staring physicists in the face for decades, has been mentioned by luminaries like Wheeler and Feynmann, and avoids the Bell’s Paradox completely!
I’m untrained but I hope any response to me here is a serious refutation, not a frivolous comparison with “purple cows.”
In the Bell’s paradox, twin photons are created on Monday, then read Tuesday and Wednesday on distant asteroids. Note that the Tuesday is not preferentially “earlier” than the Wednesday reading, since each is outside the other’s cause-effect cone. Yet in the Paradox, the reading on one asteroid seems somehow telepathized to the other asteroid, faster than light, or even Wednesday’s filter setting can be seen to affect Tuesday’s reading.
The paradox disappears if you consider that the paired photon is a single photon, arriving from some future point beyond the Wednesday asteroid, travelling backward to the actual pair creation event, then forward in time to Tuesday’s asteroid. Cause-effect behaves trivially: Wednesday’s filter setting is imparted to the photon, and the polarization is “later” detected by the Tuesday receiver.
Is there a flaw in this argument? Wheeler-Feynmann argue, IIRC, that reverse-causality photons of the type described imply “advanced” radiation waves whereas only “retarded” radiation waves are observed. On the other hand, wouldn’t advanced waves be quite hard to detect? (I am not quite espousing the traditional Transactional model: it has an unnecessarily complex notion of causation, though I won’t try to synopsize mine in this post.)
Just as rising entropy is a consequence of the happenstance that we inhabit a world of low entropy, couldn’t the prevalence of retarded waves be an artifact of the same thermodynamic state? Probably not, or geniuses like Feynmann would have seized on the elegance of this solution. Still, it would be nice to understand where the model fails.
Some notes:
(1) I think you are talking about the EPR paradox.
(2) A description that attempts to prevent a causality problem by positing that the information is routinely transmitted backwards in time is somewhat defeating the purpose. If we posit that we can sent information backwards in time, why not just send it between the measurements directly.
(3) The Feynman-Wheeler description of “particles going backwards in time” is a description of antimatter. I.e. that an X with its time-direction reversed is an anti-X. That means your description is fine for entangled photons, or particle-antiparticle pairs, but entanglement is not limited to those kinds of particles. There is nothing in principle stopping an electron from being entangled with another electron, or a proton, or something else, which can’t be described the way you have it.
Yes, I think all these “paradoxes” are closely related; indeed are all manifestations of quantum physics’ “mysterious” implication.
The information doesn’t magically travel back in time; it just follows a normal cause-effect relation, but with time-sense reversed. Counterintuitive? Of course. Problematic? Perhaps; I’d like to understand the specific objections. But the beauty of the scheme, if it is at all workable, is that quantum mystery disappears.
Au contraire, haven’t Feynmann et al themselves modeled electrons as positrons going backward in time?
septimus, perhaps the discussion of your idea would be something for a separate thread? Personally, I don’t see where you differ from Cramer’s transactional interpretation, and as with that interpretation, I’m not sure I really see the need. As leahcim notes, interpretational problems of EPR experiments mainly focus on how to save causality in the presence of apparently faster than light ‘spooky actions’ (which I consider neither spooky nor ‘actions’ in most senses of the word). To just throw out the notion of causality does seem like cheating somehow. And of course, the problems you get are those you get with any retrocausal proposal, such as paradoxical influences on the past, causal loops (i.e. A caused B retrocaused A), and of course, there’s also the problem of how our experience of a unidirectional arrow of time comes about; the usual argument of ‘entropy was lower in the past’ doesn’t work, since for every point in time, if causal influences propagate in both time directions symmetrically, one should expect that entropy is higher both towards the past and towards the future, as the dynamics that generate entropy increase work in both directions equally well.
I think I see where whc.03grady is misunderstanding something…
I can see where you get this impression, if you’re starting from the Schrodinger’s Cat idea. However, we do make observations that tell us about this uncertainty at the heart of the quantum world. Before we observe the particle at the wall behind the two slits, there was uncertainty and it seemed to pass through both slits simultaneously (as a wave). Our observations show us that this happened. Flash memory works because there’s uncertainty about the position of an electron, and sometimes they just appear inside an insulator. But we can observe that flash memory works.
Our observations lead us to the idea that these wave/particle things exist in a superposition of states. It’s not, like you seem to be saying, that we base these ideas on simple ignorance of what may be really happening.
Further, the Geiger counter that triggered (or not) the breaking of the vial of poison, is itself an observer.