# Quantum mechanics and the principle of bivalence

Does quantum mechanics, due to the phenomenon of superposition (Schrodinger's cat is both dead and alive), give reasons to alter the laws of logic, specifically the principle of bivalance (something is either true or untrue). What would be the consequences of such a step?

• I 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 Apr 2 at 21:30
• 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 Apr 2 at 22:41

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.

• Your RCP just seems to be the bivalence. Is there a difference? And LEM/LNC can be applicable and hold even if the bivalence fails, as in quantum logic, for example. Although Heisenberg phrased himself in terms of LEM when talking about his "potentiae", it is clear from context that he means restricting bivalence, not LEM. It certainly seems too restrictive to impose bivalence as a precondition for applying "laws of thought", even mundane reasoning often proceeds without such assumption. You yourself talk about "rules of dialectic" with the RCP "unnecessary" in the dialectical logic. – Conifold Apr 3 at 20:06
• @Conifold - The PB and LEM are quite different, as Wiki notes. One is a rule for meaningful statements, the other a rule for the operation of the dialectic. They may often overlap. If we abandon Aristotle's rules then we cannot expect sensible results from the dialectic method. Most philosophers abandon do them, with predictable results. They tend to forget the RCP and so turn metaphysical problems into logical paradoxes. Heisenberg specifically states that it is the tertium non datur rule that requires modification. . – PeterJ Apr 4 at 12:10
• Can you define LEM and LNC in your answer? – Giancarlo Apr 9 at 6:45
• @Giancarlo - Done. – PeterJ Apr 9 at 8:12

No.

A common misinterpretation of quantum superposition (something being in two states that appear to be exclusive for perception, like dead and alive) is bivalence, where states are effectively exclusive. The term perception is a key factor, because a phenomenon of superposition does not mean something being in a state A OR B, and neither A AND B, but that quantum state A has multiple perceptible values, which are complementary with B. Quantum probabilities are therefore able to be expressed with imaginary numbers. Such mechanics are logic at the quantum scale of existence, but cannot be grasped by the mechanics of our macroscopic perception.

Bivalence is not at the same level: probabilities in the macroscopic world are a simple percentage. Bivalent states are also exclusive (XOR function: either A is true or false, not both, not none).

Therefore, only by altering the principles of perception it would be possible to grasp the principles of logic from the quantum domain (note that it's not the same as "altering the principles of logic": logic is just the same, except that our perception is not able to approach the logic of the quantum realm; a consequence of such problem is that quantum behavior cannot be described but by using math).

• I agree with the first paragraph up to "Quantum probabilities are therefore able to be expressed with imaginary numbers". Quantum probabilities, like any other probabilities, are real numbers between 0 and 1, they are squared absolute values of complex amplitudes. Superposition of amplitudes is, therefore, not expressible through classical logical operations. But one can choose to grasp that as physics and keep the principles of classical logic. – Conifold Apr 9 at 9:04