In order to accommodate this property with a logical form of reasoning, you have to deny the exclusion of the middle, which says that a property is either possesed or it isn't. In this cse the fact that the particle is here would contradict the possibility that it is over there. So excluded middle has to go.
Now mathematicians are familiar with logical systems like that, some nondistributive lattices are like boolean algebra (a distributive lattice) without the law of the excluded middle. So these structures have been studied as "Quantum Logic" (qg). A lot of work has been done, but I don't think anything earth-shattering has been discovered.
The law of the excluded middle is a statement about statements:
LEM: For any statement X, X or not X
Suppose LEM is false, it therefore follows that there is at least one statement X, such that not (X or not X). Let S denote one such statement, hence
not (S or not S)
Using truth tables the previous statement is logically equivalent to the following statement:
S and not S
At this point we must conclude that the LEM is true, and cannot be abandoned.
Next I would like to say that I do not see why one would have to abandon the LEM, in order to reach the conclusion that one particle can be in two places simultaneously.
Consider the following argument.
Suppose that anybody is composed of an integral number of particles. It would follow that the simplest body of all is a two particle body. Consider the creation of a body. You would have to have two particles headed for the same place at the same time. Neither could ever get there. Eventually they would be as close as they are ever going to get. The question is this, when they are one state change away from being at the same place at the same time, where will they be at the very next moment in time, since neither of them can be at the collision point. Use conservation of energy and momentum.
Either they reflect or they do not reflect. If they must reflect then bodies don't exist, contrary to fact, hence there are collisions in which two particles "stay together". A proper analysis of this will reveal that energy and momentum are conserved if at the very next moment in time neither particle is in the plane of collision, but rather both particles are now in orbit around a common center of mass, forming a new body. AND THIS BRINGS ME TO THE POINT. Suppose you accept that bodies are created by point particles colliding, and then spinning around a center of mass, why would they spin clockwise as opposed to counterclockwise?
Now, the mathematical logic of this will lead you to the conclusion that either they spin clockwise or they spin counterclockwise. But the 'OR' here is inclusive OR, not exclusive OR, hence there is no contradiction in saying that right after collision, the particles spin clockwise AND counterclockwise. But if that is true, then each particle is now at two locations on the circumference of a circle, and so it would be correct to say that it is possible for one particle to be at two places at the same time. This is the most intriguing concept of physics I have ever come across.
I have done the analysis for a collision angle of 90 degress, and the right conclusion is that each particle is now at two different vertices of a regular hexagon, in an orbital plane.
Now here is an interesting point. If particles are literally point masses, the probability of any two just happening to be headed for the exact same place at the exact same moment in time is zero, and so one would expect that there shouldn't be too many large bodies of matter in the universe, contrary to fact.
So let me really go into detail here. Suppose we have two particles just moving, they will not be headed for the same place at the same time by luck alone. But if there MUST be a force of attraction between them, then they will pull each other towards a collision of some kind. This kind of reasoning really leads to new physics. Can any particle be truly free from all force? If not, then every particle must be accelerating in an inertial reference frame, and hence there are no truly free particles. But supposing that there must be a force of attraction, what mediates the force?
