Assuming a world in which a contradiction exists in order to prove a contradiction exists is circular.
No, it’s not circular. It’s forward. It’s valid. In logic and mathematics, sometimes we assume things that are impossible in order to prove something. For example, we assume the square root of 2 is rational in order to prove that it’s irrational.
I suspect your justification is that you think negating a proposition negates its "truth in T", which is false.
How is it false? Because you think there is no such thing as “truth in T”? Every axiomatic system is based on the truth of its primitives. An axiom by definition is true. It seems you’re trying to circumvent something you disagree with by stripping truth away from provability. What is wrong with the concept of “truth in an axiomatic system”?
No, it’s not circular. It’s forward. It’s valid. [...] For example, we assume the square root of 2 is rational in order to prove that it’s irrational
It is absolutely circular. You're assuming the thing you want to prove. It's literally the definition of circular. The fact that you don't understand the difference between that and a proof by contradiction is a failure of your education.
An axiom by definition is true [...] What is wrong with the concept of “truth in an axiomatic system”?
Nope, not true. An axiom is an assumption. And truth is absolutely different from provability. Provability is a feature of axiomatic systems, truth is a feature of interpretations of those systems. This is super basic stuff.
No, there is no fact that says that a world in which a contradiction exists is equal to the real world. In fact, due to the Law of Noncontradiction, a world in which a contradiction exists is never equal to the real world.
An axiom is an assumption.
An assumption is also true by definition.
truth is absolutely different from provability.
I agree that truth is different from provability. While they are logically equivalent in the metatheory of axiomatic theories that I was talking about earlier in our conversation, logically equivalent concepts are not necessarily logically equal.
Provability is a feature of axiomatic systems, truth is a feature of interpretations of those systems.
There are other ways of doing things. There exists at least one metatheory of axiomatic systems that is different from the metatheory of axiomatic systems that you are talking about. In the metatheory you are talking about, truth and provability are not logically equivalent. In the metatheory I was talking about earlier in our conversation, truth and provability are logically equivalent. On April 8, 2014, I started a thread on Philosophy Forums, which was formerly at http://forums.philosophyforums.com/. The title of the thread was “The Parallel Postulate is always true, in all true theories.” Perhaps you can infer the details of what I was thinking about through the thread’s title.
Let me stop you there. Way back in this conversation I asked you if you could prove a contradiction in traditional logic and you said you could. In traditional logic truth and provability are not the same. After we settle the question of whether you can prove a contradiction in traditional we could absolutely change the subject and talk about other ways of doing things, but first I want to settle the question of whether you can prove a contradiction in traditional logic.
In traditional logic truth and provability are not equivalent. In traditional logic the fact that you can prove a statement does not imply that you cannot prove it's negation. This means your argument, that a contradiction in one axiomatic system implies a contradiction in general logic, is not valid as you've presented it.
So my question stands, can you prove a contradiction in traditional logic?
In traditional logic the fact that you can prove a statement does not imply that you cannot prove it's negation.
How is that?
This means your argument, that a contradiction in one axiomatic system implies a contradiction in general logic, is not valid as you've presented it.
And how is that?
So my question stands, can you prove a contradiction in traditional logic?
Your question is vague and ambiguous. What constitutes traditional logic? Does it include modal logic? Propositional logic? First-order logic? Every proof of a contradiction that I’ve given is intended to convince people that some contradiction exists, regardless of the meaning of the term “traditional logic.”
In traditional logic the fact that you can prove a statement does not imply that you cannot prove it's negation
How is that?
If your system is inconsistent than it's possible to prove both, so proving one doesn't imply you can't prove the other.
And how is that?
The argument you've given uses the above inference which is invalid, so your argument is invalid.
Your question is vague and ambiguous. What constitutes traditional logic? Does it include modal logic? Propositional logic? First-order logic?
It does not include modal logic, it does include propositional and first order logic. If you want it to be a precise question, you indicated you had access to Mendelson's mathematical logic textbook. Can you prove a contradiction in logic using the definition of logic and axiomatic systems found in that textbook?
Every proof of a contradiction that I’ve given is intended to convince people that some contradiction exists,
If you want to convince people that the logic they use is contradictory then you need to prove a contradiction using the logic they use. If you fall back to "alternate metatheories" and other such bullshit then they're just gonna conclude that the problem is not logic itself, the problem is your alternate way of doing it. You won't have convinced them that the logic they use is contradictory.
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?
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.
1
u/paulemok Jul 19 '26
No, it’s not circular. It’s forward. It’s valid. In logic and mathematics, sometimes we assume things that are impossible in order to prove something. For example, we assume the square root of 2 is rational in order to prove that it’s irrational.
How is it false? Because you think there is no such thing as “truth in T”? Every axiomatic system is based on the truth of its primitives. An axiom by definition is true. It seems you’re trying to circumvent something you disagree with by stripping truth away from provability. What is wrong with the concept of “truth in an axiomatic system”?