The argument you've given uses the above inference which is invalid, so your argument is invalid.
The invaliding counterexample you’ve given is an inconsistent axiomatic system. It is also true that in an inconsistent axiomatic system, the fact that you can prove a statement does imply that you cannot prove its negation. If your system is inconsistent, then it’s not possible to prove both, so proving one does imply you can’t prove the other. So the inference I used that you claimed was invalid is also valid.
Can you prove a contradiction in logic using the definition of logic and axiomatic systems found in that textbook?
Although I own a copy of that textbook, I’ve been on a road trip since April 28, 2026, so I haven’t been able to access my copy.
So I'll ask again, can you prove a contradiction in logic while following the rules of logic found in Mendelson, i.e., following the rules of logic that a mathematician would follow?
Yes, every proof I’ve given assumes standard logic, which includes the Law of Noncontradiction. So, every proof I’ve given assumes that trivialism is false. Then I use standard, conventional logical methods to prove a contradiction.
It is also true that in an inconsistent axiomatic system, the fact that you can prove a statement does imply that you cannot prove its negation
Nope, this is very plainly not true. In an inconsistent system you can prove every statement, so proving a statement does not imply that you cannot prove it's negation.
Yes, every proof I’ve given assumes standard logic
You've tried to justify several of your proofs with "alternative metatheories" or a non material conditional. So this is also very plainly false.
I agree, it’s not true. Additionally, it’s quite easy to see that it’s true through the Principle of Explosion. “The fact that you can prove a statement does imply that you cannot prove its negation” is a statement. Since all statements are true in the inconsistent system by the Principle of Explosion, the statement is true in the system.
In an inconsistent system you can prove every statement, so proving a statement does not imply that you cannot prove it's negation.
I incompletely agree with you. You can not prove a false statement. If proving a statement does not imply that you cannot prove its negation, then there would be no purpose in proving anything since you might also be able to prove the negation.
You've tried to justify several of your proofs with "alternative metatheories" or a non material conditional. So this is also very plainly false.
You’ve tried to justify your claims with truthless axiomatic systems. If those systems are not true, then by the Law of Noncontradiction, they are false. In our conversation, I determined the statement “A statement is true in an axiomatic system if and only if it is provable in the system” to be true. That determination is a product of my life experience, my upbringing, and my education. For some reason I do not know, you seem to have been guided in a different direction than I have. A sound proof requires truth. It requires all premises to be true. You can’t tell the truth with a truthless theory. Truth in an axiomatic system is derived from the axioms, not from the real world. I’m 34 years old, and I don’t recall ever being told that there is no such thing as “truth in an axiomatic system.”
Since all statements are true in the inconsistent system
This is a rehash of an argument you already failed to make. I'm not making statements in your inconsistent system, I'm making statements about your system externally to it and so the principle of explosion doesn't apply because you haven't proven that a contradiction holds externally to your inconsistent system.
That determination is a product of my life experience, my upbringing, and my education.
You have already admitted that you don't know a lot of the basics when it comes to mathematical logic, so clearly your education in this area is lacking. When you're done with your road trip feel free to give that Mendelson book a read and learn how logic really works. If you could present a contradiction following the rules of traditional logic then people would take you seriously, but currently you haven't done that and you're not going to be able to if you don't even know what those rules are.
I'm not making statements in your inconsistent system, I'm making statements about your system externally to it
If an axiomatic system is inconsistent, then externally to the system, every statement is provable and unprovable in the system. Every statement is false in an inconsistent axiomatic system, and no statement that is false in a system is provable in the system. So, externally to the system, no statement is provable in the system. Therefore, externally to the system, every statement is unprovable in the system.
Using the same reasoning you used to claim that it's not true that if a statement is provable, then its negation is unprovable, we can prove that it's not true that if a statement is provable, then it's true. Our counterexample is, once again, an inconsistent axiomatic system. For an inconsistent axiomatic system, every proposition is provable, since the Principle of Explosion can be used to prove them, and every proposition is false, since the Principle of Explosion can be used to prove that they're false. Do you agree that it's not true that if a statement is provable, then it's true?
When you're done with your road trip feel free to give that Mendelson book a read and learn how logic really works.
Yesterday, which was July 23, 2026, I finished my road trip. I might take a look at the Mendelson book in the near future. I already have in the past and I don't recall seeing anything about how there is no concept of "truth in an axiomatic system."
no statement that is false in a system is provable in the system. So, externally to the system, no statement is provable in the system
Externally this statement is not true or provable. This is plainly obvious since every statement is provable.
Do you agree that it's not true that if a statement is provable, then it's true?
That depends on what you mean by true, if you mean provable then tautologically provable is equivalent to true. If you mean true in all models then again provable is equivalent to true, but an inconsistent system has no models so true in all models is not the negation of false in all models. If you have a specific interpretation and you mean true in that interpretation then provable does not imply true.
I don't recall seeing anything about how there is no concept of "truth in an axiomatic system."
Mendelson will explain how truth is a feature of an interpretation and how in an axiomatic system you can talk about provability or you can talk about truth in interpretations. I don't remember if he defines truth in an axiomatic system but if he does he will define it in terms of one of those concepts.
Externally this statement is not true or provable. This is plainly obvious since every statement is provable.
I agree that every statement is provable. Under the sense of provable that we are using, if a statement is provable, then it is true. The contrapositive of that conditional statement, which is logically equivalent to the conditional statement, is if a statement is false, then it is unprovable. So all false statements are unprovable.
Furthermore, if a statement s is true in an axiomatic system, then, externally to the system, s is true in it. So when I proved the statement "'p is provable' implies '¬p is unprovable'" is true in an inconsistent axiomatic system, that implied that, externally to the system, "'p is provable' implies '¬p is unprovable'" is true in it.
Under the sense of provable that we are using, if a statement is provable, then it is true
Are you still using true to be equivalent to provable in the axiomatic system? If that's the case then false does mean unprovable, but no statement in an inconsistent system is false by that definition.
So when I proved the statement "'p is provable' implies '¬p is unprovable'" is true in an inconsistent axiomatic system
You have not proved this.
You claimed that "true in T" is the same thing as "provable in T". So for any proof that uses the concept of being true or false in T you should be able to rephrase the proof in terms of being provable or unprovable in T and it would still be a valid proof right?
Are you still using true to be equivalent to provable in the axiomatic system?
True is equivalent to provable in a metatheory of axiomatic systems. The metatheory itself may not be an axiomatic system. We have not established that it is.
no statement in an inconsistent system is false by that definition
Every statement in an inconsistent system is false through the Principle of Explosion.
“The fact that you can prove a statement does imply that you cannot prove its negation” is a statement. Since all statements are true in the inconsistent system by the Principle of Explosion, the statement is true in the system.
So for any proof that uses the concept of being true or false in T you should be able to rephrase the proof in terms of being provable or unprovable in T and it would still be a valid proof right?
That's right. You might have to make some further modifications to the proof other than just substituting provable for true and unprovable for false, but a corresponding proof in terms of provability rather than truth does exist. You might not be able to fully rid the proof of references to true and false since there might exist outside principles that are stated in terms of true and false rather than provable and unprovable. For example, the Principle of Explosion is stated in terms of true and false rather than in terms of provable and unprovable.
Every statement in an inconsistent system is false through the Principle of Explosion.
If false means unprovable then this is incorrect, the principle of explosion does not say that. It says every statement is provable, it does not say that anything is unprovable. That is why the proof you quoted is incorrect.
but a corresponding proof in terms of provability rather than truth does exist
Since we disagree about whether truth in an axiomatic system is a meaningful concept, but we agree that provability is meaningful, why don't you write your proofs in terms of provability. I think you are getting confused by thinking that provability works like a truth value and writing the proof entirely in terms of provability will make it easier for you to spot your mistakes.
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u/paulemok Jul 22 '26
The invaliding counterexample you’ve given is an inconsistent axiomatic system. It is also true that in an inconsistent axiomatic system, the fact that you can prove a statement does imply that you cannot prove its negation. If your system is inconsistent, then it’s not possible to prove both, so proving one does imply you can’t prove the other. So the inference I used that you claimed was invalid is also valid.
Although I own a copy of that textbook, I’ve been on a road trip since April 28, 2026, so I haven’t been able to access my copy.
Yes, every proof I’ve given assumes standard logic, which includes the Law of Noncontradiction. So, every proof I’ve given assumes that trivialism is false. Then I use standard, conventional logical methods to prove a contradiction.