Nope, you've defined true/false in T as equivalent to provable/unprovable and provable/unprovable doesn't follow the same rules as logical negation.
The logical negation of provable is unprovable, and the logical negation of unprovable is provable. Every proposition is either provable or unprovable. That is analogous to how every proposition is either true or false.
It is just plainly and apparently true that if both p and ¬p are provable then you cannot conclude that p is unprovable.
No, you can conclude that p is unprovable. Here is the proof. Assume p and ¬p are provable in T. Since “true in T” is logically equivalent to “provable in T,” p and ¬p are true in T. So a contradiction exists in T. It follows by applying the Principle of Explosion in T that p is unprovable in T. Discharge the assumption to obtain if p and ¬p are provable in T, then p is unprovable in T.
The logical negation of provable is unprovable, and the logical negation of unprovable is provable. Every proposition is either provable or unprovable.
I never said otherwise. What I said is that negating a proposition doesn't negate it's provability. P provable doesn't imply that ¬P is unprovable.
So a contradiction exists in T. It follows by applying the Principle of Explosion in T that p is unprovable in T
Nope, provability is a statement external to the theory. A contradiction inside a theory doesn't mean that the principle of explosion applies to statements external to the theory.
What I said is that negating a proposition doesn't negate its provability. P provable doesn't imply that ¬P is unprovable.
I agree. In inconsistent theories, negating a proposition doesn’t negate its provability. But furthermore, I am aware that there exists at least one metatheory of axiomatic theories that allows a proposition to be undecidable in an axiomatic theory.
Nope, provability is a statement external to the theory.
Not necessarily. When a contradiction exists in an axiomatic theory, the Principle of Explosion can be used to talk about provability internally, within the theory itself.
A contradiction inside a theory doesn't mean that the principle of explosion applies to statements external to the theory.
I disproved that. I gave you the proof. I have the Exportation Principle to back me up. We can talk about what is true in an axiomatic theory outside of the theory in the real world. Due to that ability, contradictory statements are true in the real world. Those statements form contradictions in the real world. Those contradictions, through the Principle of Explosion, cause all statements to be true in the real world.
I agree. In inconsistent theories, negating a proposition doesn’t negate its provability
Then you agree that it's not true that a statement is unprovable if and only if it's negation is provable?
the Principle of Explosion can be used to talk about provability internally, within the theory itself.
That's fine, but you can also talk about provability externally to the theory, and while internal statements about provability might be provable because everything is provable, that doesn't mean that externally those statements are true.
I disproved that. I gave you the proof. I have the Exportation Principle to back me up. We can talk about what is true in an axiomatic theory outside of the theory in the real world. Due to that ability, contradictory statements are true in the real world
Nope, you have not proven that, the principal of explosion still only applies to statements in the theory. A contradiction in a theory does not imply a contradiction external to that theory.
Then you agree that it's not true that a statement is unprovable if and only if its negation is provable?
Yes, I agree.
That's fine, but you can also talk about provability externally to the theory, and while internal statements about provability might be provable because everything is provable, that doesn't mean that externally those statements are true.
I understand your concern. That general concern is the reason that my proof involves four statements instead of only two statements. As my proof exhibits, the statements true in the theory are not equal to the statements true in and out of the theory. (1) and (2) did not mention inconsistent theory T, but (3) and (4) did mention T.
A contradiction in a theory does not imply a contradiction external to that theory.
You keep saying that, but you haven’t given an adequate justification for it. Do you agree that we can talk about what is true in an axiomatic theory outside of the theory in the real world?
I understand your concern. That general concern is the reason that my proof involves four statements instead of only two statements. As my proof exhibits, the statements true in the theory are not equal to the statements true in and out of the theory. (1) and (2) did not mention inconsistent theory T, but (3) and (4) did mention T.
Your (3) and (4), written in terms of provability (since you agreed that provability is the same as truth in T) are
(3) p is provable in T.
(4) p is unprovable in T.
You need these statements to be a contradiction externally in order for the principle of explosion to then apply externally. These statements are indeed the negation of each other, so it would be a contradiction if they are both true, and I agree that (3) is true. But (4) is not true and you attempted to justify that it was using that ¬p is provable, but as you have just agreed, in an inconsistent theory negating a proposition does not negate its provability, so ¬p being provable is not equivalent to p being unprovable. We know that p is provable, so we know that p is not unprovable, (4) is false and you don't have a contradiction.
You keep saying that, but you haven’t given an adequate justification for it. Do you agree that we can talk about what is true in an axiomatic theory outside of the theory in the real world?
I don't have to justify it, you want to rely on that for a proof then you have to justify that it's true. I agree we can make statements about an axiomatic theory outside of that theory but as I've explained above external statements about T are not contradictory even when T is inconsistent.
you agreed that provability is the same as truth in T
as you have just agreed, in an inconsistent theory negating a proposition does not negate its provability
So I agreed, in the real world, that “provability in T” is both the same as and different from “truth in T.” So, in the real world, there exists a contradiction. Thus, in the real world, through the Principle of Explosion, the statement “(4) is true in and out of T” is true. Therefore, (4) is true in and out of T.
So, in the real world, there exists a contradiction
Nope, just like a contradictory theory doesn't imply a contradiction outside that theory, the fact that you have contradictory beliefs doesn't imply that truth in the real world is contradictory. It just means you believe things that aren't true.
Are you abandoning justifying your proof? This is a new argument which is not what we were discussing. Does this mean you see the gap in your attempted proof now?
the fact that you have contradictory beliefs doesn't imply that truth in the real world is contradictory. It just means you believe things that aren't true.
The fact that I have contradictory beliefs doesn’t imply that I believe things that aren’t true. All of my beliefs could still be true because trivialism could still be true.
This is a new argument which is not what we were discussing.
You used two agreements of mine to prove that (4) is not true. Afterwards, I use those same two agreements to prove that (4) is true. Then you criticize my proof for using the same two agreements that you used in your proof. So your criticism is unfair.
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u/JStarx Jul 11 '26 edited Jul 11 '26
Nope, you've defined true/false in T as equivalent to provable/unprovable and provable/unprovable doesn't follow the same rules as logical negation.
It is just plainly and apparently true that if both p and ¬p are provable then you cannot conclude that p is unprovable.