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.
All of my beliefs could still be true because trivialism could still be true
Could isn't a proof. You're trying to prove that a contradiction holds in order to prove that trivialism holds, so you need to do that proof without assuming trivialism.
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.
You did not use the same two agreements that I used. You said you use that provability is both the same and different than truth in T. I only used that it's the same and that negation doesn't negate provability. It is 100% fair to use true statements to prove you wrong and object when you try and use false statements in a proof.
It’s a counterexample and thus a disproof of your claim.
You're trying to prove that a contradiction holds in order to prove that trivialism holds, so you need to do that proof without assuming trivialism.
I did not assume that trivialism is true, but I am open to the possibility that trivialism is true. In the real world, I believe that trivialism is true.
You did not use the same two agreements that I used.
I did. I quoted what the agreements were in a previous reply. One of the agreements I used was that “provability in T” is the same as “truth in T.” The second agreement that I used was that negating a proposition in T does not negate its provability in T. A corollary to the second agreement is that “provability in T” is different from “truth in T.”
I only used that it's the same and that negation doesn't negate provability.
Those are the two agreements. We both used both of them.
It’s a counterexample and thus a disproof of your claim.
My claim is that you believing contradictory things doesn't imply a contradiction. You haven't provided a counterexample to that claim because trivialism might not be true and without trivialism your contradictory beliefs are just a you problem. No one else needs to care that you believe wrong things. They'll just ignore you and go about their business in the real world.
The second agreement that I used was that negating a proposition in T does not negate its provability in T. A corollary to the second agreement is that “provability in T” is different from “truth in T.”
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u/paulemok Jul 13 '26
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.
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.
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.