Does quantum mechanics, in particular the phenomena of superposition and entanglement, provide a case for the revision of classical logic?

passing through one slit), nor can we say it is going through one or the other: due to wave-particle duality, the light we are describing as a single photon will also behave as a wave such that interference occurs, exactly like the interference between two water waves. If we take away our detection equipment and simply observe the light distribution on the screen we shall see a pattern that would be predicted by a wave theory of light. It seems that we are committed to simply take the mathematical formalism (which in actuality uses complex numbers, not simple fractions) at face value and say that the physical reality is one of uncertainty.Entanglement DescribedEntanglement is just as counter-intuitive. Two systems are considered entangled when it is impossible to formulate them independently of each other. An example of this is that of a pair of electrons emitted from a source. The electrons have correlated ‘spin’ properties in such a way that they form a single unified system, even though they may be spatially separated by significant distance. By measuring the spin properties of one electron, I gain knowledge of the other. The electrons are considered to be entangled because it is impossible, when faced with the mathematical statement of the system, to divide up the explanation into separate parts, each part corresponding to an electron. Rather, the entanglement of systems can only be considered as a whole. If one electron is measured then the entire system is affected – instantly. It is not that by measuring one part of the system my measurement causes a series of physically definable knock-on effects which change the system. Rather, it is that the moment I perform an act of measurement I am measuring the entire system, even if I am only measuring a single particle which is by all classical accounts physically separate from another entangled particle. Ontology of Uncertainty - Bell's TheoremThe phenomena of entanglement was once used by Einstein to attempt to show that Quantum physics was incomplete – an idea which later came to be known as involving “hidden variables” (Norton, 2010). Under a “hidden variables” theory a quantum system may be considered to be determinate, and uncertainty is the result of scientists’ ignorance of the hidden variables that determine the system. Unfortunately for Einstein, the hidden variables theory has been all but proven false. It is not a mere matter of interpretation: the statement of hidden variables has experimentally testable consequences. These were first discovered and tested by Bell in 1964 and the results have been replicated by numerous scientists (Shimony, 2009). The tests repeatedly show that when one assumes a hidden variables theory one calculates incorrect predictions (the case being tested is often the case of the correlated electrons as described above), whereas the standard quantum theory generates accurate predictions (* - is it just a matter of which theory has better predictions? Or is there a theoretical issue involved too? Something to do with locality?). We are therefore forced to take the uncertainty of quantum mechanics seriously: we cannot pass it off as an issue which has yet to be resolved. So far as our current best theories show, uncertainty is fundamental and intrinsic to the world. In short, quantum physics cuts to the core of metaphysics.Ontological Uncertainty and Classical LogicIt should be clear how these considerations throw doubt upon classical logic. The law of excluded middle – one of the most fundamental cornerstones of classical logic, and philosophical reasoning – has been undermined by superposition. The law of excluded middle states that it is impossible for something to both be the case and not be the case. Superposition, on the other hand, claims that this clear-cut dichotomy is simply not how the world works. Classical logic says that the photon should either be at A or not at A. Superposition means that this cannot be said. Further, the very notion of identity itself is attacked by entanglement. Two entangled particles are still two particles, and yet they form a single unified system. This suggests that an entangled electron does not possess reflexivity – it is not identical to itself, because it is not possible to formulate 'itself' independently. Identity is also undermined by wave-particle duality, since we cannot even say what it is that a photon or light wave is. It is not a particle, nor is it a wave. But it is not a third thing either: it is sometimes particle-like, and sometimes wave-like, depending on what aspect you highlight by your method of measurement. There appears to be no ‘thing in itself’ which light is: rather, it changes depending on the experimental context. This attacks the transitivity of identity: in one experimental context light is a particle, in another context it is a wave, but it is not the case that particles are waves.As in the case of geometry, it appears to be the case that an empirical theory has disproved an a priori claim about the nature of the world. If we can even point to a single case where it simply is not true that the law of excluded middle holds, then it is not the case that the law of excluded middle is a priori true (it is both not true, and not a priori). The same holds for identity. Since classical logic is clearly inadequate for describing the quantum world, Putnum argues that we need to revise classical logic into a quantum logic, just as we revised Euclidean geometry to non-Euclidean geometry (1975c, p.184).Against Revision - "Subject Matter"There are, however, two disanalogies between the geometrical case and the issue of classical logic. The first is a problem of 'subject matter'. The subject matter of geometry with regards to the world is clear: it is about space. Logic, on the other hand, is not so clearly about anything. Classical logic may be used to model language, as in linguistics, or as a foundation for mathematics, as Russell attempted, or indeed to formulate physical laws such as the Laplace-Poisson theory of the gravitational field, as shown by Hartry Field (Leng, 2010, p.57). I wish to say, therefore, that classical logic is not analogous to the case of geometry because it is not something that we should claim to be a priori true (that is to say, true of the world but not true by the world). Rather, it is simply a formal system that we may apply to certain phenomena. If it is the case that quantum physics cannot be modelled by classical logic then we simply say that this is not an appropriate area of application for classical logic. We do not say that the whole of classical logic is false – but nor do we say that it is true.Mathematicians may object to this argument. Euclidean geometry, they will say, is also a formal system with various applications which may prove fecund or ill-advised. What is so different between geometry and logic that justifies this different treatment? To this I would make two replies. Firstly, there is a sense in which Euclidean geometry, when it is used to describe actual space, may be right or wrong independent of the usefulness of the application. For example, we may use Euclidean geometry in various works of engineering because it is a close enough approximation to the truth. However, it is still the case that, despite being a useful approximation, it is incorrect – the correct geometry is non-Euclidean (* is this true? Is applying Euclidean geometry to real space an approximation, or is it something like a manifold, where the space is locally Euclidean, within a larger non-Euclidean world?). On the other hand, in the case of applying logic, it is not the case that if I say “The cat is either on the mat or not on the mat” that a formulation of this statement in classical logic would be an approximation of the quantum logical formulation. Indeed, if I were to apply the quantum logical formulation to this statement I would be committing an error that resulted in nonsense (this is part of the problem of Schrodinger's cat). Quantum effects do not occur on the macroscopic level. Secondly, I would say that geometry, while a formal system, is not quite so abstract as logic. While we may find other applications for it, it is also the case that it lends itself firstly to a description of space. With respect to the space of the universe, it is false (or so our current best theories say). I do not say that Euclidean geometry is logically false as a formal system – it is simply not true of the world. There are, therefore, two standards of truth for Euclidean geometry: one formal, and one empirical. It meets the formal standard (which we might call 'validity', rather than truth) but not the empirical. I wish to propose that the only standard of truth suitable for classical logic is formal. Against Revision - "Applicability"The second disanalogy is that we can think of situations where it is in fact perfectly acceptable to apply classical logic to a quantum system – though within certain limits. So long as variables do not refer to those things which are limited by the uncertainty principle, classical logic appears to avoid most of its troubles. For example, while we cannot claim that the law of excluded middle applies to the position or momentum of a photon, we can apply it to two different wavefunctions. The photon either has one wavefunction or another. This is a result of our becoming realist about uncertainty: the basic elements of the system are not the particles but rather the mathematical expressions which describe their probable behaviour. This extends the argument based on applicability: classical logic is neither true nor false, but it does have some applications which are appropriate and some which are not.ConclusionI have described various quantum phenomena, and how they may be considered to undermine classical logic. I have presented two problems for this view, both of which related to applicability. Given these considerations, it does not appear that classical logic is something which can be revised in the face of empirical evidence. Rather, we say that the empirical evidence guides our selection of both which logic may be the most appropriate to apply to a set of phenomena, and of which elements of a phenomenon we should highlight with our logical model. Quantum physics, therefore, does not provide a case for the revision of classical logic. However, the various quantum phenomena mentioned do undermine a certain view of the role of classical logic, which is that of delimiting the set of possibilities (and therefore also the set of actualities). As we have seen, it can be the case that certain empirical discoveries can feed back upon our view of how a given logic applies to the world. The role of logic, then, is that of a useful formal system, not that of a metaphysical arbiter of what can and cannot be.BibliographyHeisenberg, W (1989), “Physics and Philosophy”, Penguin: London.Leng, M (2010), “Mathematics and Reality”. Oxford: Oxford University Press.Norton, J., (2010), Lecture: “Einstein on the Completeness of Quantum Theory” from the series Einstein for Everyone, University of Pittsburgh, Spring 2010. 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