Quantum mechanics and logical statements I am a math student and currently working on my bachelor thesis with a philosophy professor. The subject is paraconsistency and thus also dialetheism which is the believe that a statement can be true and false at the same time.
I had a general/introductory course to QM and one experiment I recall is the Stern-Gerlach experiment where you measure spin with a magnetic field.
Consider the statement (I choose $\varphi$, $J$ to specify a atom and measurement, is that a good idea?):

The next specific silver atom $\varphi$ I send trough the Stern-Gerlach experiment has spin up during that specific measurement $J$.

As I understood the validity of this statement can only be determined after executing the experiment. Because of our understanding of QM and the current accepted theory this is inherently probabilistic and not predictable?
What do you think about the formulation of the statement? Do you think I can say that the above statement is neither true or false (or arguably both) before actually sending the atom through the experiment? Or do i have some misunderstanding about QM and what statements you can make about it?
 A: Fools rush in where angels fear to tread.
Surely your statement (like the statement, "It will rain here tomorrow.") is a prediction. Predictions are true or false according to what (later) turns out to be the case. They are not simultaneously true and false (or neither true nor false) just because we haven't yet been able to check whether they are true or false.
What have I missed?
A: Your question is not purely about physics, but I don't think that your statement depends on anything quantum mechanical for the purposes of your exercise.  I think you are mixing general issues of probability and decision-making under uncertainty with the specific way those concepts arise within the framework of quantum mechanics.  Some of QM is definitely "new" relative to classical physics, in particular what's asserted about the source of the uncertainty and how much you can reduce it.  But some things that appear to be "quantum" features at first glance are really just generic features of reasoning under uncertainty, regardless of the source of that uncertainty.
Getting to your specific example:
Your statement as regards a quantum system is neither true nor false until after you make the measurement, as you said.  I would not say that it's both true and false simultaneously though.  Prior to the measurement, the best you could say is that it has some defined probability of being true or being false.  That seems different than saying that it is definitely BOTH true and false at the same time.
For the purposes of your exercise, I think this is no different than saying the next time I flip my coin it will be heads.  Well that might also be true or false, and you don't know now.  It's not simultaneously true and false though.  The source of the uncertainty in this classical case is different, and the degree to which you could drive it down with better information about the initial conditions is different, but the fundamental problem with evaluating the truth of the statement in context of your project, I think, only depends on the existence of the uncertainty not on its source.
Inspired by a comment by @lalala, consider this as a "counterexample" to your situation.  Let's say you know that the particle has just gone through a SG apparatus that is oriented in the same direction.  Then from QM, you know with 100% certainty that it is in one orientation or the other, let's say spin-up to be concrete.  But now imagine that I don't know that it has gone through an SG apparatus immediately before, and you ask me to make a prediction.  I don't know the initial conditions, but I do know from QM that there are only two possible outcomes, so my best estimate is probably to say the chances are 50-50 that it comes out spin-up.  We both came up with different estimates that were as good as we could do with the information available to us at the time, but those estimates are pretty far apart.  From your perspective, the proposition does not yield in any meaningful sense to the idea that your statement is BOTH true and false. (You know that it is true with probability 1!)  From my perspective, the situation is as ambiguous as it could possibly be (though it's for you to philosophize about whether that amounts to being BOTH true and false).  In neither case, do I think there's a meaningful sense that the statement is BOTH true and false at the same time.  This example does use a quantum system, but nothing about the underlying issues really depended on that. You could easily make similar scenarios that did not. (See, e.g. the "Monty Hall problem", which is just about choice and not particularly about any physical system.)
A: There's a difference between paraconsistent and quantum logics (there's more than one of each, and there is some overlap). In paraconsistent logics, the explosion theorem $(p\land\neg p)\to q$ fails in general. In quantum logics, the distributive laws $p\land(q\lor r)=(p\land q)\lor(p\land r)$ and $p\lor(q\land r)=(p\lor q)\land(p\lor r)$ fail. Both are weaker than classical logic, and classical logic and the strongest quantum logic need to be somewhat weakened to obtain something paraconsistent.
Critics of quantum logic have argued classical logic is adequate to describe quantum phenomena. The easiest way to do this is to phrasing everything in terms of eigenstates rather than the values of classical variables. We thereby consider an observable's value undefined in states which aren't eigenstates of its associated operator on the Hilbert space.
A: Let us say that the setup is such that you can get up or down with some probability. Then the truth of your statement depends on who do you ask, that is, on the interpretation of QM that that person has. If he is a local non-realist, the statement, before the measurement, is false because the particle has no determined value of J, the value does not exist until you measure. For someone with a non-local hidden variables interpretation, the value of S does exist before the measurement, and the statement would be true. I dont know if there are other interpretations of QM in which the statement could be both false and true, or any other combination you like.
