I think we understand each other! Thank you for clarifying.
The way I translate this: some logical statements are true (to you) but not provable (to you), because you are not living in a world of mathematical logic, you are living in a messy, probabilistic world.
It is nevertheless true, by the rule of necessitation in provability logic, that if a logical statement is true within the system, then it is also provable within the system. P → □P. Because the fact that the system is making the statement P is the proof. Within a logical system, there is an underlying assumption that the system only makes true statements. (ok, this is potentially misleading and not strictly correct)
This is fascinating! So my takeaway is something like: our reasoning about logical statements and systems is not necessarily “logical” itself, but is often probabilistic and messy. Which is how it has to be, given… our bounded computational power, perhaps? This very much seems to be a logical uncertainty thing.
I think we understand each other! Thank you for clarifying.
The way I translate this: some logical statements are true (to you) but not provable (to you), because you are not living in a world of mathematical logic, you are living in a messy, probabilistic world.
It is nevertheless true, by the rule of necessitation in provability logic, that if a logical statement is true within the system, then it is also provable within the system. P → □P. Because the fact that the system is making the statement P is the proof.
Withina logical system, there isan underlying assumptionthat the system only makes true statements.(ok, this is potentially misleading and not strictly correct)This is fascinating! So my takeaway is something like: our reasoning about logical statements and systems is not necessarily “logical” itself, but is often probabilistic and messy. Which is how it has to be, given… our bounded computational power, perhaps? This very much seems to be a logical uncertainty thing.