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In quantum mechanics it is thought to be possible that something could be at two places at the same time. But if that is really the case then perhaps law of non-contradiction is no longer valid.

So to what level are the fundamentals of logic really in danger?

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"In quantum mechanics it is thought to be possible that something could be at two places at the same time." What does this mean exactly? Does it mean that "Some people who don't understand quantum mechanics believe that something could be at two places at the same time?". That might be true. Or does it mean that "Quantum mechanics allows the possibility that something could be at two places at the same time?", which is of course entirely false. If the latter, why should we worry about the implications of something so thoroughly at variance with the truth? – WillO Mar 10 at 3:22
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it is thought to be possible that something could be at two places at the same time - no one who has basic background in physics thinks like that. – user36790 Mar 10 at 6:10
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This idea is popularised by the so-called laymen books by Kaku and co. This is a nonsense. We deal with probability amplitudes- there can be probability amplitude that something exists at more than one place prior to measurement. But it doesn't, in any way means something exists at two places. Possibility is not synonymous with Actuality. Better discard such lame thinkings. – user36790 Mar 10 at 6:14
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@WillO: You can't do much with that. Unless one studies QM, he will have such nonsense ideas boosted by several laymen sites in internet followed by those pop-sci books. – user36790 Mar 10 at 6:16
    
@user36790 We all know what she means, and it is not nonsense. The wave function pervades space and things that are not currently interacting in an observable way are potentially anywhere. You don't have to be a condescending pedant about inexact vocabulary. – jobermark Mar 10 at 16:58

The idea that quantum mechanics fundamentally challenges the rules of logic was popular for a while, but has fallen out of favor in recent years.

While intuitively it might seem that quantum superposition (i.e something being in more than one base state at the same time) is what challenges the rules of logic, by invalidating the law of non-contradiction, this is not the case. An electron in a superposition of spin |+> and spin |-> might seem like a contradiction, but it can simply be treated as being in a distinct third state of being "either |+> or |->".

The real challenge to classical logic from Quantum Mechanics comes from the uncertainty principle, which leads to situations where:

[(p and x) or (p and y)] is different from [p and (x or y)].

Birkhoff and Von Neuman proposed in the 1930s that the paradoxes of Quantum Mechanics can be explained if we abandoned classical logic and used some form of Quantum logic instead (Birkhoff, Garrett; von Neumann, John. "The Logic of Quantum Mechanics". Ann. Math. 37 (4): 823–843.). Such a Quantum logic would change or abandon all together some of the rules of classical logic, and would be a perfect case of logical axioms arrived at by observation.

Hilary Putnam discussed this in depth in his paper "Is Logic Empirical?", later republished as "The Logic of Quantum Mechanics." ("The Logic of Quantum Mechanics" in Mathematics, Matter and Method (1975), pp. 174-197). In it, he argued that, just as empirical physical results - relativity - forced us to abandon Euclidean geometry, so it is possible that the results of quantum mechanics will force us to abandon classical logic.

Von Neumann, Birkhoff and Putnam all seemed to have moved away from this position in later years. Quantum logic didn't really solve any physics problems or provide any new insights into the epistemic challenges posed by Quantum Mechanics.

Although Quantum logic is still an active field of study up to the present day, it does not get much attention from most philosophers and had been abandoned completely by physicists. The only people who are paying attention to it are pure mathematicians who study different types of logic as mathematical structures (Quantum Logic's relation to Orthomodular Lattices and its relation to Fuzzy Sets), without paying any attention to the semantic or epistemic value of such non-classical logics. See for example "Quantum Logic, M.L. Dalla Chiara, R. Giuntini, arXiv:quant-ph/0101028".

You will occasionally come across the term "Quantum Logic" used in the quantum computing literature, but by that they do not mean the QL of Von Neumann and Birkhoff. Instead, what is meant by that is classical Boolean logic applied to quantum states and quantum bits.

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Quantum logic validates the law of non-contradiction because intersection of a subspace with its orthogonal complement is always zero. So the aspect of quantum theory that puts pessure on that law, if any, is unrelated to quantum logic. – Conifold Mar 9 at 21:38
    
The issue wasn't just the with the law of non-contradiction. The uncertainty principle and the non-commutativity of quantum operators forced logicians to reexamine the distributive law of classical logic. Also negation itself gets confusion when up move from simple space of proposition to a hilbert space of operators. – Alexander S King Mar 9 at 21:42
    
Everything "weird" about quantum logic is already expressed in traditional accounts of the double slit or EPR experiments, which are in perfect consistency with classical logic and non-contradiction (except in certain popular books). Quantum logic is just a technical rephrasing of that, where we get to talk about "objects" in a more classically sounding way at the expense of using less classically sounding logic. So whatever quantum mechanics has to say about "fundamentals of logic" substantively is completely independent of the status of quantum logic. – Conifold Mar 10 at 3:23
    
@Conifold see edit. – Alexander S King Mar 10 at 17:59
    
This is a great answer. Would it be possible for you to expand also covering the implications of Bell's theorem? – Alpha Mar 12 at 2:34

Let me clarify a confusion first. Logic applies to sentences, not to objects, so object's ability to be in two places at once is not a contradiction, unless definition of "object" rules out such a possibility. It certainly does in classical mechanics, but classical mechanics does not apply to quantum objects that can be "two places at once". And quantum mechanics explicitly allows objects to be "everywhere at once" to the extent that this metaphorical language makes sense there. More precisely, it makes no sense to talk about where in space objects "are" unless they are in an eigenstate of the position operator, and in that case they can not be "two places at once".

In fact, classical logic does apply to "properly phrased" sentences of quantum mechanics. This however means that some sentences like "electron will be detected in such and such area" have no truth value most of the time. If one wants to make them have it one has to redefine what "truth value" means, and use logic that is non-classical, that is what is called "quantum logic". The difference with the standard description however is purely technical, we now allow some sentences that previously made no sense to have it, by changing the meaning of what "sense" is. In quantum logic "truth values" are subspaces of a Hilbert space, "negation" is their orthogonal complement, and "conjunction" is their intersection. So the value of "electron will be detected at such and such area" will be the subspace defined by projection to that area. This rephrasing is indeed not very popular lately, but it has nothing interesting to say about the non-contradiction law. This law still holds in quantum logic because a subspace intersects its orthogonal complement only trivially at 0.

This being said, the law of non-contradiction only applies to situations where one can ignore effects of change, such as mathematics, formal theories, and perhaps some very stable aspects of reality. It is obviously violated by objects that change over time, bullet is at rest now, and flies later, rabbit is alive today, and dead tomorrow, etc. One can try to "save" it by slapping temporal labels on sentences and adding something like "at the same time", this is similar to the idea of quantum logic, it is called temporal logic, there is a great variety of them, and they are also non-classical. For example, they typically do not assign truth values to the happening of uncertain future events.

But it is questionable that the law of non-contradiction holds even with "at the same time". Heraclitus famously said "we do and do not step into the same river, we are and we are not", this is what Plato termed "becoming". According to him, "what becomes and never is" is not subject to "logos", only "aesthêsis alogos", irrational sense. If the becoming and sense perception are taken seriously the law of non-contradiction fails already for sentences about classical objects, without any need for quantum mechanics. This point of view is taken in yet another kind of logic, dialetheic logic.

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Right, the Church has recorded cases of bilocation for centuries, and even in its pickiest periods, it has not questioned Aristotle's syllogisms. – jobermark Mar 10 at 17:46

Quantum mechanics does not threaten the rules of logic - that's the positive message.

During a short period in the interpretation of quantum mechanics it was discussed, whether a different calculus of logic had to be introduced to interpret quantum mechanics (quantum logic). Today this approach is no longer in the focus.

That a particle can be at two different positions at the same time, seems sharpening the probability interpretation of quantum mechanics. How is the precise statement and its context to which you refer?

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Logic is an abstract concept which does not necessarily directly tie to reality. We like to argue that there is at least a very solid indirect relationship, in that logic does a good job of describing reality and we must implement said logic in reality, but it is an abstract concept.

What QM does is shake up the linguistic choices we use when using logic to describe reality around us. For example, it forces us to completely rethink the concept of an object being "at a position in space." QM does not say that an object can be in two places at the same time in the way you or I might think of it. What it does say is that the fundamental concept of the position of a particle is best described as a probability distribution, not a single value. Moreso, if you do interesting things like create entangled pairs, you can create probability distributions with very unintuitive shapes. Some of those shapes lead us to simplify the actual QM and simply claim "a particle can be in two places at the same time."

The only way this affects logic is in the predicates which attempt to describe the world. These must be updated to account for new discoveries. This is no different than when we realized that the "atom," the supposedly indivisible object was actually composed of subatomic particles (which themselves, are broken down further). Logic only breaks down in this case if you choose not to update your axioms to consistently represent reality. If anything, you can think of it similar to the explosion of provable statements which occur once you introduce a contradiction into your logic.

Our intuition updates our axioms slowly. That is why Einstein considered the behavior of entangled particles "spooky action at a distance."

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Entangled pairs are also commonly misunderstood. The popular image is "particles aren't normally entangled, and if we entangle them, they do interesting stuff". The reality seems to be more like "particles are usually entangled to brazillions of other particles, but in a special case where only two particles are entangled to each other, they do interesting stuff". The entanglement doesn't break down by being lost - it's just blurred a bit by the tons of other entanglements that include that particle. You get closer to classical physics when everything is entangled with everything else. – Luaan Mar 10 at 9:48

In quantum mechanics it is thought to be possible that something could be at two places at the same time. But if that is really the case then perhaps law of non-contradiction is no longer valid.

This is a bit of a misstatement. In quantum mechanics, an object exists in multiple versions, and different versions of the same object may be in different places. Those different versions of an object may sometimes interfere with one another in single particle interference experiments, see "The Fabric of Reality" by David Deutsch, Chapter 2. And there are other experiments that rule out explanations in which only a single version of a system exists.

So to what level are the fundamentals of logic really in danger?

"Logic" is used in different ways by different people. Quantum mechanics does not destroy our ability to do rational argument, or maths, or propositional logic or whatever. Applying propositional logic to reality will sometimes have to take into account that the relevant statements are about relations between multiple versions of the same object. I can't see any particular reason to think this is problematic.

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QM is in one sense a continuation of an antique idea - that of atomism - that what there is, is made of particles; the 'quantum leap' was to recognise that energy is too; in fact everything other than space and time - and it's an open question if they are too: the results of Loop Quantum Gravity posit that it is - and it's the departure point for Causal Set Theory.

Standard formal logic accepts that just one contradiction renders the entire theory trivial - in that every possible proposition is both at once true and false - this is unacceptable as then we have no criteria to discriminate between propositions; this is called the principle of explosion in that one contradiction 'explodes' the logic.

Paraconsistent logics adjust formal logic such that this 'explosion' of doesn't occur; that this is a live option is shown in that there is such a thing as intuitionistic logic - where the law of the excluded middle no longer holds, and has both an associated set theory and algebra - Heyting algebra.

The original Von Neumann & Birkhoff notion of Quantum Logic introduced as an explanatory factor for Quantum Logic itself has persisted into quantales, and more recently into linear type theory - this is where it makes contact with intuitionistic methods.

In the same way that Quantum Mechanics has invigorated the study of non-classical algebra, that is algebra in which the commutation rule doesn't hold; then one can say it has also invigorated the study of non-classical logics.

One noticeable effect is the notion of truth; whereas the classical notion is simply to draw on 'platonic' ideas of truth/falsity; the intuitionistic is to that truth, is only truth when it's justified - that is its proveable - a notion that goes back to Platos Theatatus; this makes its logic dynamic and kinetic.

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Einstein's theory of relativity shattered physicists' conception of the universe. It was no longer as simple as F=ma or any of the other further derivations made.

Yet, a hundred years later, mechanical engineers are still using Newtonian mechanics to build bridges and skyscrapers. Why would they do this when those formulas have been definitively proven to be false? They do because the usefulness of those formulas is still true for the scales for which they're applied. We're not modeling buildings at the quantum level, so the Newtonian equations are still enough to properly model the building for structural soundness.

So while it may be true that the invariants that make logic possible are broken at the quantum level, most logic is not done at the quantum level. Thus, we may continue to use that logic on the condition that we assume the invariants. We will find that the usefulness of logic is still there at the scale for which we apply it.

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I take it that "being in two different places simultanuously" refers to the phenomenon of entanglement.

Measurents, however, show a very short time elapse between the changement of state of the first and second photon. So, the "being in two places" disappeared.

The time elapse is so small that entangkement appears simultanuously to the unarmed eye. It only could be measured because of the advancement of equipment.

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"being in two different places simultanuously" refers to superposition of states, not entanglement. – Alexander S King Mar 14 at 18:38
    
Also, make clearer how you believe this answers the question. – virmaior Mar 15 at 0:56

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