# Does the principle of explosion mean we can prove objects must “fall up” because QM is inconsistent with GR?

(Not sure if this is more appropriate to the Maths.SE)

The Principle of Explosion is a law of classical logic which says that if we accept inconsistency, then everything becomes possible. I am wondering how this works in the physical world, since we know QM is inconsistent with GR. The Wikipedia article goes into some details of how the proof works, so here's an attempt:

1. Objects move on a single well-defined path known as a geodesic (true in GR)
2. Objects follow all paths (Feynman's path integral in QM)
3. Therefore the statement "objects move on a single well-defined path known as a geodesic OR objects fall upwards" is true
4. However given that the above statement is true, since we know that "objects follow all paths" (i.e. objects don't follow a single well-defined path) is true, the first half of the statement is false
5. Since the statement is true overall, it must follow that the latter statement, "objects fall upwards", is true
6. Therefore we have proven that objects fall upwards

This result is so nonsensical I'm sure something is wrong. What? The only thing that makes sense to me is that the Principle of Explosion doesn't work in physics, in which case the next question is: why not?

I thought about using QM only for this proof, invoking Schrodinger's cat and using the two statements "the cat is alive" and "the cat is dead", but that doesn't work because neither statement is necessarily true.

1 is not true in GR. Objects only follow geodesics under a certain set of very restrictive conditions.

2 is not a rigorously well-defined statement.

Anyway, of course you can start from an inconsistency and prove all propositions. Why do you want to prove something as trivial as the proposition that objects fall up? Why not prove both that $$2+2=5$$, and also that $$2+2\ne 5$$?

This result is so nonsensical I'm sure something is wrong. What? The only thing that makes sense to me is that the Principle of Explosion doesn't work in physics, in which case the next question is: why not?

Problems with this:

• The propositions you list are not all rigorously well-defined.
• The propositions you list are not all correct, or even good approximations, even within their domain of applicability.
• We don't expect physical theories to be true in any absolute sense of formal logic. We expect them to be good approximations under certain conditions.

Sorry, but this simply doesn't make any sense. The way GR and QM are set up is different, but this is not a logical contradiction because both are merely models that are useful in two different domains.

Your argument is essentially the same as the following:

1. Milk costs \$1 in supermarket A. 2. Milk costs \$2 in supermarket B.
3. Therefore, $$1 = 2$$.
4. But we also know that $$1 \neq 2$$, so milk prices prove that logic is inconsistent.

If you understand why this argument doesn't work, you'll also see why yours doesn't. The main difference is that you have a few extra steps and fancier words.

In general, notions from logic and set theory have absolutely no relevance to physics. The idea that they do is a classic trap that, sadly, many otherwise able minds have fallen into.

• I will go to supermarket A, though. – Exocytosis May 12 at 7:32