Comment by drdeca
2 years ago
> 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?
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.
> In order to see HOOO EP as a provability notion, you have to interpret the axioms as a theory about provability.
A notion can generally be prior to a particular formalization. If you have an alternative notion of probability in mind, you should be able to express it.
> Now, I can only suggest a formal theory, but the interpretation is up to users of that theory.
Ok, well, it has no users other than yourself, so if you want to communicate how it could be useful, I recommend you find a way of expressing/communicating an interpretation of it.
—- Also, I think your idea of a “bias towards consistency” is unclearly described at best.