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Given the anti-intuitive results of Quantum Mechanics, it is not surprising that Physicists would look for a deeper reason in the structure of the theory to explain what was then (and still is) startling phenomena.

Von Neumann & Birkhoff pointed out that propositions about a classical physical system was encoded in its phase space. Each proposition determined a subset of this space. And the entire family of propositions describes a boolean algebra. That is we have classical propositional logic.

Although Quantum Mechanics differs dramatically from classical mechanics there are broad features that correspond from Classical to Quantum - as there should be - since Quantum Mechanics should converge towards Classical Mechanics as one takes plancks constant towards zero. This is the case for phase space - there is a quantum phase space.

The question then is what kind of logic does this quantum phase space support. Von Neumann & Birkhodd showed that this is non-distributive lattice, and by interpreting the join as or, and the meet as and they had a logic which they christened Quantum Logic. Its main drawbacks is that its difficult to interpret as a logic because of the failure of distributivity:

  • no satisfactory implication operator has been found;

  • it remains only a propositional language:no generalisation of a predicate form of this logic has been found.

  • physically, because its not simply an epistemological issue that we do not know whether a proposition p holds or not, we cannot say that p or ~p=1 holds. That is the law of the excluded middle is suspect.

These are substantial drawbacks despite the attractiveness of the idea. Its also been argued that pure states do not assign truth in the usual binary mode (true, false); so truth itself becomes obscure and has to be rethought. Recently there has been a couple of attempts to ground the theory in intuitionistic logic for which the law of the excluded middle and choice doesn't hold.

Now, I've only recently understood that the dual of intuitionistic logic is paraconsistent. It doesn't seem a priori plausible to me that we must interpret the logic of these new constructions intuitionistically, why not paraconsistently?

Indeed, its because of the formalisation of paraconsistent logics that a physicist like Doering can say he's lost his fear of inconsistency.

Now if a physical system is modelled intuitionistically, and suppose every physical event is modelled intuitionistically; then dually it must be also modelled paraconsistently.

The question is: are we biased in our pre-emptive choice of intuitionistic logic? Are there any conditions we can demand such that a paraconsistent interpretation of a physical event is chosen over an intuitionistic one?

For example could one suppose a super-position of truth values? In the early days of quantum theory it was startling to discover that a particle had both particulate & wave properties, as these seemed mutually exclusive phenomena. It was finally understood that a particle was a field; and the questions one asked determined whether one saw a particle or wave. Is there a similar possibility for truth?

  • N.B. It is straightforward to interpret Birkhoff and von Neumann's quantum logic propositionally. Considering the straightforward model of the lattice as the lattice of subspace containments in Hilbert space, each subspace may be regarded as a proposition which holds of a vector. For a subspace A⊆ℋ and a pure state-vector |ψ⟩, we may regard |ψ⟩∈A to be the proposition that A holds of |ψ⟩. The objects are thus maximally precise (non-vacuous) meets of propositions. In this model, the join is the direct sum of spaces, and logical negation is orthocomplementation; then |ψ⟩∈(A v ¬A) is a tautology. – Niel de Beaudrap Jul 2 '13 at 10:17
  • They are dual in that LEM becomes the principle of contradiction, but the principle of contradiction also exists in intuitionism, and what would it become in a paraconsistent logic? So Inutitionism must be stronger than the dual of a normal paraconsistent setup, no? It has one more rather powerful aspect, the possibility of proof by contradiction, as long as it is constructive proof. You would have to somehow inject a limited form of LEM that applied only under conditions of wave-collapse or something. – jobermark Feb 4 at 21:22
  • A contradiction in ordinary logic requires that one of A/not-A is false and the other true. Where both or neither is true there is no contradiction for it is not a case of A/not-A. These logical conundrums you mention are usually caused by an incorrect application of Aristotle's rules for contradictory pairs. If we apply them properly then paraconsistent logic usually becomes unnecessary. I would suggest that it is this simple misunderstanding of logic that causes Graham Priest to so badly misunderstand Buddhism and physicists to think that the world contradicts ordinary logic.. – PeterJ Feb 5 at 16:17
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Essentially, yes, though the details are still being worked out.

For example, da Costa and Ronde give a paraconsistent axiomatisation in `The Paraconsistent Logic of Quantum Superpositions'. Some additional discussion can be found here.

As an aside, I do think we are biased towards intuitionism. But that might not be such a bad thing; full blown dialethism is a bitter pill to swallow! But that's another story, the relationship between intuitionism, dialethism, and paraconsistent logic has a vast and ever expanding literature.

  • It's actually sensible to simply say that dialethism is nonsensical. It is often brought up in vain attempts to circumvent the incompleteness theorems, but actually dialethism makes things much worse! Moreover, the generalized incompleteness theorems are true for any foundational system, regardless of how crazy its logic is. The incompleteness theorems can be evaded only if there is no real-world model of PA (see here). – user21820 Dec 13 '18 at 9:16
  • PA stands for Peano Axioms or Peano Arithmetic (I'm not sure which, but I don't think that matters). – elliot svensson Feb 4 at 21:27

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