Let's analyze the interplay between classical logic and probability theory.
In (a model of) classical propositional logic, we have n-ary predicates ##P(x_1,\ldots,x_n)## that (essentially, I don't want to introduce ##\models##) map the elements of a set ##X## into the truth values ##\top##, ##\bot##. These predicates (applied to elements) constitute statements. We can build new statements by using logical connectives like ##\neg##, ##\wedge##, ##\vee## in the usual way and these statements will also attain truth values.
Now, we can represent this algebra of statements by subsets of ##X## in the following way: To each unary predicate ##P(x)##, we associate a subset ##U_P = \{x : P(x)\}##. Using some basic set theory, we can check that ##U_{P\wedge Q} = U_P \cap U_Q##, ##U_{P\vee Q} = U_P \cup U_Q## and ##U_{\neg P} = X \setminus U_P##.
Let's say, we are interested in a list of unary predicates about the set ##X##. We can form the algebra of predicates, obtained from applying arbitrary logical connectives to that list. We can represent this algebra of predicates by a
set algebra ##\Sigma## of subsets of ##X##. This algebra can equivalently be obtained by taking arbitrary (finite) unions and intersections of the subsets ##U_P## associated to the predicates we're interested in. In order to make contact with probability theory, we need to allow for countably many unions and intersections and we need ##X\in\Sigma##.
Associating probabilities to statements like ##P(x)## means specifying a
probability measure on ##\Sigma##, i.e. a function ##p:\Sigma\rightarrow [0,1]## that satisfies some basic axioms like ##p(X)=1##. The triple ##(X,\Sigma,p)## forms a probability space.
We can define random variables on ##X## as (measurable) functions ##A:X\rightarrow\mathbb R##. We will later be interested in the random variables
$$\chi_P(x) = \begin{cases}1 & P(x) \\ 0 & \neg P(x)\end{cases} \text{.}$$
Using these random variables, the statement ##P(x)## is equivalent to ##\chi_P(x) = 1##. We can also check that ##\chi_P(x)^2 = \chi_P(x)##.
On the set ##\mathcal R## of random variables, we can introduce the so called valuation maps ##\nu_x : \mathcal R \rightarrow \mathbb R##, which just evaluate a random variable at ##x##: ##\nu_x(A)=A(x)##. If ##X## is not empty, then obviously at least one such ##\nu_x## exists. We can easily check that ##\nu_x(AB)=\nu_x(A)\nu_x(B)## and ##\nu_x(A+B)=\nu_x(A)+\nu_x(B)##. If we have ##\nu_x(A^2)=\nu_x(A)##, then it follows that ##A(x)=1## or ##A(x)=0##, i.e. ##A(x)## is a function of the type ##\chi_P(x)##.
We now want to know, whether we can represent all propositions about quantum systems using classical logic. So let's first define what we mean by a proposition about a quantum system. Given a Hilbert space ##\mathcal H##, a quantum proposition ##\hat P\in\mathcal P## is just an orthogonal projector ##\hat P:\mathcal H\rightarrow \mathcal H## (i.e. ##\hat P^\dagger = \hat P## and ##\hat P^2=\hat P##).
If quantum propositions ##\hat P## could be represented using classical logic, then there would be a map ##r:\mathcal P\rightarrow \mathcal R## of the quantum propositions into the random variables on some probability space ##(X,\Sigma,p)##. On a classical probability space, there exists at least one valuation map ##\nu_x##. If a such a map ##r## did exist, then we could define a valuation map on the set of quantum propositions by ##\nu=\nu_x\circ r##. Since commuting quantum propositions behave like propositions in classical logic, we would require ##\nu## to behave like a classical valuation map at least for commuting observables, i.e. we would require ##\nu(\hat P+\hat Q)=\nu(\hat P)+\nu(\hat Q)## and ##\nu(\hat P\hat Q)=\nu(\hat P)\nu(\hat Q)##. From that, we can also be sure that quantum propositions would be mapped to classical propositions, because ##\nu(\hat P^2)=\nu(\hat P)## implies that ##r(\hat P)=\chi_P## for some ##P##.
Now here comes the surprise: A well-known no-go theorem (Kochen-Specker) says that for ##dim(\mathcal H)>2##, no such valuation map ##\nu## with these properties exists! Not all propositions about quantum systems can be embedded into classical logic! If quantum mechanics is correct, we cannot reason about properties of quantum systems with only the laws of classical logic. This is independent of the interpretation and thus also applies to Bohmian mechanics.
Demystifier said:
Such a probability cannot be assigned in standard QM, but it does not imply that it cannot be assigned in any theory, perhaps some more fundamental theory (yet unknown) for which QM is only an approximation.
The above considerations show that we cannot hope for this unless quantum mechanics is incorrect.
There should exist general logical rules which can be applied to any theory of nature, not only to a particular theory (such as QM) which, after all, may not be final theory of everything. In such a general logical framework, the statement ##S_x=+1\wedge S_y=-1## must not be forbidden.
Such a theory cannot make the same predictions as quantum mechanics.
What can such a theory look like? Well, the minimal Bohmian mechanics is not such a theory, because minimal BM cannot associate a probability with ##S_x=+1\wedge S_y=-1##. This is because spin is not ontological in minimal BM. However, there is a non-minimal version of BM (see the book by Holland) in which spin is ontological and ##S_x=+1\wedge S_y=-1## makes perfect sense. Nevertheless, this theory makes the same measurable predictions as standard QM.
This must necessarily use the ##d=2## loophole and can't be extended to all quantum systems.
I am not saying that this non-minimal version of BM is how nature really works. Personally, I think it isn't. What I am saying is that there is a logical possibility that nature might work that way. Therefore it is not very wise to restrict logical rules to a form which forbids you to even think about such alternative theories.
You can of course leave this possibility open, but then you must propose a theory that disagrees with quantum mechanical predictions.
So my answer will be rather short.
