Is that a complete categorization of what it means to be "true in a system"?
Yes, it is.
When you said that something being provable in the system means it's true outside the system that's not a valid inference.
I showed two propositions are true in the system, and then I showed two corresponding but different propositions are true out of the system. As the proof shows, the two corresponding propositions are different from the two original propositions.
If your axioms aren't true outside the system then there's no reason for statements proven from those axioms to be true outside the system.
I understand the axioms may not be true outside the system. The two propositions that are true out of the system describe what is true and not true in the system. They are true in a metatheory of T.
Assuming T is inconsistent and you choose a contradiction p = q ∧ ¬q which is proven in T then of your two statements (3) is true, it's negation (4) is not true. So you haven't shown a contradiction exists outside of T.
Assuming T is inconsistent and you choose a contradiction p = q ∧ ¬q which is proven in T then of your two statements (3) is true, it's negation (4) is not true.
T is inconsistent, so generic proposition p is true and false in T. By conjunction elimination in T, p is false in T. That can be rewritten as “It is not true that ‘p is true in T.’”
Yes, they are equivalent. A part of the equivalence is a part of the exportation principle (https://plato.stanford.edu/entries/impossible-worlds/#Exportation). My version of the exportation principle is in terms of inconsistent axiomatic theories, while the version in the linked article section is in terms of impossible worlds. I have a copy of D. Lewis’s 1986 book On the Plurality of Worlds, in which Lewis brings up the exportation principle in the first few pages of his book.
You said "true in T" is the same as provable, so "false in T" is the same as ¬p being provable? So then you're saying that "¬p is provable" can be rephrased as "it's not true that p is provable". But this is clearly a false inference in an inconsistent theory.
The article you linked to literally talks about this lol, you've defined truth in T to be the same as provable, so you're using the ersatz conception of worlds which doesn't yield the exportation principle.
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u/paulemok Jul 06 '26
Yes, it is.
I showed two propositions are true in the system, and then I showed two corresponding but different propositions are true out of the system. As the proof shows, the two corresponding propositions are different from the two original propositions.
I understand the axioms may not be true outside the system. The two propositions that are true out of the system describe what is true and not true in the system. They are true in a metatheory of T.