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

• I
• Happiness
In summary: The sign always says "the cat is alive" even if the cat is found dead. However, the physical state of the box is in a superposition of death and alive once the measurement of the decay is taken and the mechanism triggered accordingly.
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?

Abhishek11235 and PeroK
It can. Put a sign inside the box with the cat that says the cat is alive.

Abhishek11235, hilbert2, atyy and 1 other person
Happiness said:
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 Schrodinger’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?

Schrodinger 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.

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Abhishek11235, etotheipi, martinbn and 1 other person
Happiness said:
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.

Abhishek11235 and phinds
PeroK said:
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.

"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".

Halc said:
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?

Nugatory said:
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 Schrodinger’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.

Happiness said:
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.

## 1. What does it mean for a statement to be in a superposition of being true and false?

In quantum mechanics, a superposition refers to a state in which a particle or system exists in multiple states simultaneously. In the context of a statement, it means that the statement is both true and false at the same time.

## 2. How is a statement in a superposition of being true and false different from a statement that is simply unknown?

A statement in a superposition of being true and false is different from an unknown statement because it exists in a state of uncertainty, whereas an unknown statement is simply lacking information or evidence to determine its truth value.

## 3. Can a statement in a superposition of being true and false ever be resolved to a single truth value?

Yes, according to the principles of quantum mechanics, a measurement or observation of the statement can collapse its superposition and determine a single truth value. This is known as the collapse of the wave function.

## 4. Are there any real-world applications or implications of a statement being in a superposition of being true and false?

While the concept of superposition is primarily studied in quantum mechanics, it has also been applied in fields such as computer science and decision-making. Some theories suggest that human decision-making may involve superposition-like processes.

## 5. How does the concept of a statement in a superposition of being true and false relate to the concept of Schrödinger's cat?

Schrödinger's cat is a thought experiment that illustrates the concept of superposition in quantum mechanics. In the experiment, a cat is placed in a box with a radioactive substance that may or may not have decayed, leading to the cat being both alive and dead at the same time. Similarly, a statement in a superposition of being true and false exists in a state of both truth and falsehood until it is observed or measured.

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