It's an excellent question.

Heisenberg thought that QM forced us to modify the *tertium non datur* rule. So do many scientists. They are wrong, and here's why.

The principle of bivalence is not the issue here since it is unnecessary in dialectical logic that all statements are true or false, only that the statements we subject to our logical processes are. Aristotle builds this principle into his logic with his rule for contradictory pairs (RCP).

'Of every contradictory pair one member must be true and the other false.'

Notice that in order to apply the LEM or LNC we must know, before we begin, that one member is true and one false. How often do we forget this?

In QM this is not our situation. For instance, when we say an electron is a wave and also a particle there is no contradiction. This is because we know that an electron is not exclusively one or the other and must actually be neither but something capable of being either. The RCP is not satisfied so the LEM and LNC do not apply.

If you work through the various seemingly contradictory phenomena of QM you'll find that they can all be dealt with in this way. They may seem bafflingly contradictory but they are not actually so in formal logic. Or, at least, nobody has shown them to be so.

Logic allows us to combine false or partially true statements as we wish. Only when the RCP is satisfied do the 'laws of thought' come into play.

In QM and in metaphysics most of the dilemmas, (wave/particle, freewill/determinism, mind/matter and so forth) do not take a form that satisfies the RCP so they are not formal contradictions. For each of them a third option is possible. Heisenberg was wrong. What is needed for QM and metaphysics is not a modification to the LEM but a close examination of the rules for the dialectic.

In my opinion this simple point, once grokked, unlocks the secrets of metaphysics.

EDIT: The law of non-contradiction (LNC) states that for any A it is impossible for both A and ~A to be true. That is to say, if the assertion ‘x is square’ is true, then the assertion ‘x is-not square’ cannot also be true. The law of the excluded middle (LEM) states that it is necessary for one of A and ~A to be true and the other to be false. Either x is square or it is not, there is no third alternative. Where there is a third alternative then A and ~A are not a legitimate dialectical pair.

2I think you have to set bounds on what can and cannot be subject to logic. Qubits don't prevent the rules of boolean logic from operating for example. And really it's not correct to suggest that a qubit can be both true and false, it's more correct to think of them as being undecided.uninstantiated. – Richard – 2019-04-02T21:30:44.983

2

Keep in mind that quantum mechanics does not determine logic and dynamics separately: we can introduce quantum logic and simplify dynamics, or we can keep the classical logic and make the dynamics more involved. The path through quantum logic has been tried and did not take, but it does exhibit a logical peculiarity: the bivalence is false (a vector may not belong neither to a subspace nor to its orthogonal complement), but the excluded middle is, nonetheless, true (the sum of a subspace and its orthogonal complement is the whole space).

– Conifold – 2019-04-02T22:41:28.297