Q&A: A Question in Logic
A Question in Logic
Question
I heard that the uncertainty principle contradicts the distributive law in logic (I don’t know whether I’m formulating this correctly, but):
In classical logic, the statement:
“A implies (B or C)” is equivalent to the statement:
“A implies [(A and B) or (A and C)]”
But in quantum theory, because of uncertainty, this does not hold.
If the momentum is such-and-such, then I know that the position is between here and here. I can divide the position and say that it is between one end and the middle, or between the middle and the other end. So now I have a logical statement: if the momentum is such-and-such, the position is (from here to here or from here to here). My calculation came out that the position is between two endpoints, and I divide it into two halves. According to ordinary logic, this should be equivalent to the statement: the momentum is A and the position is in the first half, or the momentum is A and the position is in the second half. The first, inclusive, statement was true—if the momentum is such-and-such, the position is (either between here and here or between here and here)—but the parallel statement, which according to ordinary logic should also have been correct, does not hold here.
That is the gist of it.
Is that correct? It seems very strange to me that logic would not be valid in some world..
Answer
I didn’t follow the entire description. There is an article about this by Daniel Weil in issue 1 of Higayon; see there.
As for the matter itself, obviously logic is not measured in a laboratory. Measurements presuppose logic, and we have no way whatsoever to depart from logic. Quantum theory itself is examined within the framework of classical logic. When people speak about changing logic, they mean this metaphorically: parts of the theory under discussion can be described within a framework that looks like a different logic. These are relations between variables, not principles of thought. Thought is always classical.
And this is where the proponents of the quantum-logic approach are mistaken: they think that speaking about a different logic provides an explanation for the oddities of quantum theory. I already commented on this in my article “What Is Chalut?” in Tzohar:
Thanks (though it’s still not clear to me how what you wrote solves the problem. You stated that logic can’t be contradicted, etc. But you still need to answer this proof).
I thought about it a bit and came to the conclusion that there is a mistake in the argument I brought above (which I’ve now also found on the English Wikipedia page on quantum logic), but that would take us too far afield.