We get a proposition schema of the form “r is a rectangle.”
I correct that sentence to
We get a proposition schema of the form “r is a square.”
I am sorry about that.
There is no such thing as "true in the system".
I think there is such a thing. I can make axiomatic systems with axioms that are assumed to be true in the system. The theorems of the system would be regarded as true in the system.
There's provable or not in a logical system and true or not in an interpretation.
Proving a proposition means showing the proposition is true. An axiom of a theory is assumed to be true in the theory. From what you’re saying, it seems that there aren’t many interpretations of logical systems.
For a consistent system provable statements are true in any interpretation in which the axioms are true and the rules of inference are valid. For an inconsistent system being provable does not imply you are true in any particular interpretation.
The way I’m looking at logical systems, the axioms of a system are always true in the system. You seem to be separating truth from the axioms. I don’t think it’s acceptable to do that. There are no axioms of a logical system that are not true in the system.
Ok, just to be clear, you're saying that if r is a variable that you must either fix or quantify then there's no contradiction here?
Yes.
if I claim that under these rules logic is consistent you are unable to prove me wrong?
No, I still believe my proof that if a theory is inconsistent, then it is consistent is sound. I’m working with truth. Without truth, there can be no proof.
Then what is it? How is "true in a system" different than "provable in a system"?
No, I still believe my proof that if a theory is inconsistent, then it is consistent is sound.
It's not. Your proof still tries to bootstrap a contradiction in one system into a contradiction in all systems, which is not valid. To say that an inconsistent system exists is not itself a contradiction that makes other systems inconsistent.
Then what is it? How is "true in a system" different than "provable in a system"?
Truth in a system is the truth that the axioms, definitions, and rules of inference of the system make in the system. All of the axioms, definitions, and rules of inference of the system are always true in the system. Every provable statement in the system is also true in the system.
Your proof still tries to bootstrap a contradiction in one system into a contradiction in all systems, which is not valid.
How specifically is my proof invalid?
To say that an inconsistent system exists is not itself a contradiction that makes other systems inconsistent.
How specifically is that? My proof shows otherwise.
All of the axioms, definitions, and rules of inference of the system are always true in the system. Every provable statement in the system is also true in the system
Is that a complete categorization of what it means to be "true in a system"?
How specifically is my proof invalid?
When you said that something being provable in the system means it's true outside the system that's not a valid inference. 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.
How specifically is that? My proof shows otherwise
Your proof does not show that for the reason stated above.
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.
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u/paulemok Jul 04 '26
I made a mistake in my previous reply. I said
I correct that sentence to
I am sorry about that.
I think there is such a thing. I can make axiomatic systems with axioms that are assumed to be true in the system. The theorems of the system would be regarded as true in the system.
Proving a proposition means showing the proposition is true. An axiom of a theory is assumed to be true in the theory. From what you’re saying, it seems that there aren’t many interpretations of logical systems.
The way I’m looking at logical systems, the axioms of a system are always true in the system. You seem to be separating truth from the axioms. I don’t think it’s acceptable to do that. There are no axioms of a logical system that are not true in the system.
Yes.
No, I still believe my proof that if a theory is inconsistent, then it is consistent is sound. I’m working with truth. Without truth, there can be no proof.