Undergrad A statement in a superposition of being true and false?

[Happiness]

If a cat can be in a superposition of being dead and alive, why can't a statement be in a superposition of being true and false?

 

[Halc]

It can. Put a sign inside the box with the cat that says the cat is alive.

 

[Nugatory]

why can't a statement be in a superposition of being true and false?

“Be in superposition” is a sloppy way of saying “is a physical system whose wavefunction can be written as a linear combination of orthogonal basis vectors”. A statement isn’t a physical thing so it doesn’t have a wave function so cannot possibly qualify.

However, as @Halc points out above, we could have a sign upon which we have written some statement about the cat. That sign is a physical thing like the cat, so now we have the same problem as the original Schrödinger’s cat: instead of wondering whether the cat is dead or alive before we open the box, we wonder what’s written on the sign (we can imagine a robot in the box that monitors the cat’s health and writes the sign if it dies).

Now the resolution is the same as for the cat: macroscopic objects like cats and signs will never be in coherent superpositions of macroscopic observables. There’s no such thing as a cat that is in a superposition of dead and alive, just a 100% dead cat or a 100% alive cat and we don’t know which we have unless we look. For more explanation, you’ll want to read about quantum decoherence - and surely I’ve suggested Lindley’s book “Where does the weirdness go“ to you in some older thread?

Schrödinger knew this himself. When he wrote about the cat he was not saying that the cat would be in a superposition of dead and alive. He was saying that something was wrong with the then-current understanding of quantum mechanics because it couldn’t explain why the cat was never in such a superposition.

 

[PeroK]

If a cat can be in a superposition of being dead and alive, why can't a statement be in a superposition of being true and false?

There's the statement at the heart of Russell's paradox:

This statement is false.

That's in a superposition of true or false, in the sense that it's neither one nor the other.

 

[PeterDonis]

That's in a superposition of true or false, in the sense that it's neither one nor the other.

I don't think this is a valid analogy. A superposition in QM is a combination of states which are each valid states of the system on their own. But the whole point of Russell's paradox is that neither truth value is valid for the statement. Or, to put it another way, for a quantum system to be in a superposition means that it is in some valid state; but the statement in Russell's paradox can't be assigned any valid truth value.

 

[vanhees71]

"A superposition" is an empty phrase. One has to say "a superposition of...".

E.g., it makes sense to say, an eigenvector of the $\hat{s}_x$ is a superposition of eigenvectors of $\hat{s}_z$ but it doesn't say anything, if you say "an eigenvector of $\hat{s}_x$ is a superposition".

 

[Happiness]

It can. Put a sign inside the box with the cat that says the cat is alive.

Does the sign always say "the cat is alive", even if the cat is found dead when the box is opened? And why must the sign be inside the box? It seems the sign just has a certain probability of being true, say 60% true and 40% false. But is that the same as being in a superposition of being true and false?

 

[Halc]

However, as @Halc points out above, we could have a sign upon which we have written some statement about the cat. That sign is a physical thing like the cat, so now we have the same problem as the original Schrödinger’s cat: instead of wondering whether the cat is dead or alive before we open the box, we wonder what’s written on the sign (we can imagine a robot in the box that monitors the cat’s health and writes the sign if it dies)

No, my sign had unconditional "the cat is alive" on it, so we don't wonder what's written on it. We wonder about the truth of it before any physical measurement is taken, and that truth (my bold) is not a physical thing, so the point in the OP is arguable, I agree.

Does the sign always say "the cat is alive", even if the cat is found dead when the box is opened?

As I posted, the sign say "the cat is alive". We write that with a sharpie or whatever before closing the box. Later, once the measurement of the decay is taken and the mechanism triggered accordingly, the physical state of the box is in superposition of a live cat with a true sign and a dead cat with a false sign. That makes the sign definitely in superposition of being true and false.

And why must the sign be inside the box?

A reasonable point. We've already measured what it says, so putting it in the box doesn't change that. We've measured what it says, but until we've opened the box, the truth of the sign is arguably in superposition of being true or false. So I'm with you on that, but it isn't a physical relationship compared to the same sign being in there entangled with the cat, so that might disqualify it as being in such a state.

It seems the sign just has a certain probability of being true, say 60% true and 40% false. But is that the same as being in a superposition of being true and false?

Indeed, superpostion does not require 50/50 odds. It implies the various states, unmeasured, are capable of interfering with each other from a certain observer's perspective. I don't know how not to make it a relation with a potential measurement. I also don't know how one might measure interference from the cat being in superposition of dead and alive. They've put some macroscopic (visible without aid) objects into such states, but a cat? No. It's not a practical scenario except by discussing a cat on another planet that is isolated by virtue of being outside the past light cone of the observer.