1. Assumption A (logic)

2. Assumption B (locality)

You design an experiment which violates the theorem. ( Hint: it's a quantum system)

What can you conclude?

Either A is false, or B is false, or both.

But we know that A is false on account that Bell used a mathematical framework for classical system. In classical physics, we need to know the position and momentum in order to know everything about the particle, and these are points in phase space. So set theory, points in set theory to represent position and momentum, and Boolean algebra is the right mathematical framework for classical physics. Call that mathematical framework classical logic. So from that, we can say that Bell's theorem tells us that what we have is a classical system.

But for a quantum system, we need to represent the state of a particle by a vector in Hilbert space, and observables by operators acting on those states, not points from set theory. This is a totally different framework than a classical system. Call that quantum logic. So Bell's theorem applied to a quantum system is not going to work. Violations of Bell's theorem does not prove non-locality, or what was called spooky action at a distance.

Going back to our two assumptions:if A is false, we cannot conclude that B is false. Therefore we don't have any conclusive proof of non-locality.

**Thought Experiment**

You have to understand that Bell's theorem applies to classical system.

Now, you devise an experiment. You look at your results and they don't fit with that theorem.

You wonder why. You study that theorem carefully and you find that it is based on two assumptions. You don't know which one is false. Is it A, is it B, is it both?

Then some smart physicist called Susskind comes to you, and say, listen, the first assumption is wrong.

You ask why. He demonstrates. The logic applies to classical physics, and then points out that your experiment is about a quantum system.

Would you deduct from this that assumption B is true or false? I think not.

What I'm saying is if you want to investigate whether or not non-locality is a fundamental feature of the universe, you need to forget about Bell's theorem. You are going to need another yardstick from which you can design an experiment that will allow you to investigate that issue.

**One More Argument**

In Bell's theorem:

We make two assumptions in the proof. These are:

A. Logic is a valid way to reason.

B. Parameters exist whether they are measured or not. For example, when we collected the terms Number(A, not B, not C) + Number(A, B, not C) to get Number(A, not C), we assumed that either not B or B is true for every member.

Consider any measurements A, B and C.

Classical system:

You use Boolean algebra, based on set theory, you get (Bell's inequality):

(1) Number(A, not B) + Number(B, not C) is greater than or equal to Number(A, not C)

Quantum system:

You use a different mathematical framework, in which states are vectors in a Hilbert Space, and you get (violations of Bell's inequality):

(2) Number(A, not B) + Number(B, not C) is not greater than or equal to Number(A, not C)

What you need to realize is that CLASSICAL logic was used to derive Bell's inequality. A quantum system violating Bell's inequality can only mean that a classical system is different than a quantum system, which is based on a different kind of logic, quantum logic.