[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.

[drdeca]

> 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.