[bvssvni]

> Something being random and/or undetermined is not sufficient for it to be like a qubit. You need the linear algebra aspect for the name to be appropriate, IMO.

Naming things is hard. Given how constrained Propositional Language is as a language, I do not think there is much risk of misinterpreting it. I needed something to associate with "superposition" but also fit with "quality". Both "qubit" and "quality" starts with "qu", so I liked the name.

It does not bother me if people find another better name for it.

> I have looked through that section, and afaict, nowhere in it do you define an alternative notion of “provability”?

I do not want to create a controversy around Provability Logic by making too strong claims for some people's taste. What I meant is that this section is explaining HOOO EP and my interest is in communicating what it is on its own sake, without needing to compare it to Provability Logic all the time. However, since HOOO EP is so similar to Provability Logic, it requires some clarification. I hope you found the section useful even though you were not able to see how it is an alternative notion of provability from its definition.

> Why do you expect to be able to prove `□false => false` ? I.e. why do you expect to be able to prove `not □false`, i.e. prove the consistency of the system you are working in.

I think this is trying to think about logic as a peculiar way. HOOO EP was not developed to reason about consistency. It has its own motivation that makes sense. However, once you have HOOO EP, you can start discussing how it relates to consistency of theories.

It makes sense, in a sense of consistency, from the perspective where an inconsistent theory is absurd. Absurd theories can prove anything, so there is no distinction between true and false statements. Now, if you interpret `□false` as an assumption that one can prove false, then of course, one can prove false. `□false => false` is the same as `!□false`. Does this mean that it proves its own consistency? No, because you made the assumption `□false`. You have only talked about what you can prove in the context of `□false`. From this perspective, `□false => false` is trivially true.

Provability Logic does not allow you to think of `□false` as meaning "I can prove `false`". Instead, it is interpreted as "this theory is inconsistent" but without assigning this statement a particular meaning. This means, there is a gap between "I can prove `false`" and "this theory is inconsistent". Now, if you ignore the gap, then you are just making an error of interpretation. You have to respect the formal sense, where Provability Logic can have two different notions of absurdity while naturally you would think of them as one. However, if you want to have one instead of two, then you need HOOO EP.

> Also, if you want to do reasoning from within an inconsistent theory, then I’d hope it is at least paraconsistent, as otherwise you aren’t going to get much of value?

It sounds like you are assuming HOOO EP is inconsistent? Why are you speculating about my motivations in a such context?

[drdeca]

> Provability Logic does not allow you to think of `□false` as meaning "I can prove `false`". Instead, it is interpreted as "this theory is inconsistent" but without assigning this statement a particular meaning.

“It is provable in system X that False” and “system X is inconsistent” are the same statement. That’s what it means for a system to be inconsistent: there is a proof in the system of the statement False.

So, no, you’re wrong: in probability logic, []False can be interpreted as “it is provable (in this system) that False”.

> It sounds like you are assuming HOOO EP is inconsistent?

While I do suspect it to be, that isn’t why I said that. I was saying that because you were seemingly saying that probability logic has a bias favoring reasoning from within a consistent system. And, I was saying, “why would you want to be reasoning within an inconsistent system? Shouldn’t something be suited for reasoning within a consistent system, seeing as reasoning in an inconsistent system gives you nonsense?”

> I hope you found the section useful even though you were not able to see how it is an alternative notion of provability from its definition.

It didn’t provide me with the answer I was seeking, and so I will instead ask you directly again: what alternative notion of probability do you have in mind?

[bvssvni]

Logic by default does not have a bias toward consistency. The bias is added by people who design and use mathematical languages using logic. It does not mean that the theory you are using is inconsistent.

Asking "why do you want to be reasoning within an inconsistent system?" is like facing a dead end, because you are supposing a bias that was never there in the first place. As if, logic cares about what you want. You only get out what you put in. Bias in => bias out.

I am speculating about the following: If we don't bias ourselves in favor of consistency at the meta-level, then the correct notion of provability is HOOO EP. If we are biased, then the correct notion is Provability Logic.

In order to see HOOO EP as a provability notion, you have to interpret the axioms as a theory about provability. This requires mathematical intuition, for example, that you are able to distinguish a formal theory from its interpretation. Now, I can only suggest a formal theory, but the interpretation is up to users of that theory.