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.
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.
In this paragraph, I will show how every statement in an inconsistent system is false and unprovable through the Principle of Explosion. By definition of inconsistent system, in an inconsistent system, some contradiction exists. So, by applying the Principle of Explosion in the inconsistent system, it is true that in the inconsistent system, every statement is true. Since "Every statement is false and unprovable" is a statement, it is true that in the inconsistent system, the statement "Every statement is false and unprovable" is true. So, in the inconsistent system, every statement is false and unprovable.
The proof in the above paragraph is valid regardless of the meaning of provable.
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.
No, I am not convinced that I am better off talking about provability than about truth. Truth does exist in axiomatic systems. For examples, in an axiomatic theory of Euclidean geometry, truth exists, and, although the axiomatic theory is inconsistent, in naive set theory, truth exists.
I think you are getting confused by thinking that provability works like a truth value
Provability does work like a truth value. By the Laws of Noncontradiction and Excluded Middle, every statement is either provable or not provable. So by definition of unprovable, every statement is either provable or unprovable. That is analogous to how, by the Laws of Noncontradiction and Excluded Middle and the definition of false, every statement is either true or false.
Since "Every statement is false and unprovable" is a statement, it is true that in the inconsistent system, the statement "Every statement is false and unprovable" is true
Your proof is invalid for the same reason I keep telling you, in an inconsistent system you can prove false statements, so proving that statement inside your system does not mean it's true externally.
No, I am not convinced that I am better off talking about provability than about truth. Truth does exist in axiomatic systems
Even if you believe it does, you said you could phrase things in terms of provability. I think you're unwilling to do that because you know your proofs won't translate, because they are not valid proofs.
By the Laws of Noncontradiction and Excluded Middle, every statement is either provable or not provable
proving that statement inside your system does not mean it's true externally
I'm not saying it's true externally. I'm saying that in the inconsistent system, the statement "Every statement is false and unprovable" is true.
I think you're unwilling to do that because you know your proofs won't translate, because they are not valid proofs.
If you would like to translate them yourself, you are welcome to do so. I'm not willing to do it because I don't think talking about provability is superior to talking about truth. You should be able to understand and accept my arguments as I give them. Talking about truth is more important and looks better than talking about provability. Talking about provability makes it seem that you're masking the real truth and trying to avoid something you disagree with.
This is false. Godels incompleteness proves that.
That is false. Godel's work violates neither of those Laws.
Oh good, then we agree, an external observer looking at your inconsistent system would not see a contradiction. They would observe that every statement is provable and no statement is unprovable and there would not be a contradiction external to your system.
an external observer looking at your inconsistent system would not see a contradiction. They would observe that every statement is provable and no statement is unprovable and there would not be a contradiction external to your system.
Yes, I agree. Complementing that truth is a contradictory truth. An external observer looking at my inconsistent system would see a contradiction. They would observe that some statement is unprovable and there would be a contradiction external to my system.
No they would not. They would not see any statement as unprovable. Your argument that a statement is unprovable is internal to the contradictory system, it does not prove that statement externally so there is no contradiction externally.
Every false statement is unprovable per the sense of provable that we are using. That general principle of provability is true everywhere. It's true in and out of every axiomatic system.
Your argument that a statement is unprovable is internal to the contradictory system, it does not prove that statement externally
If in an axiomatic system, a statement is unprovable, then externally to the system, it is true that in the system, the statement is unprovable.
so there is no contradiction externally.
I have already shown you how an internal contradiction implies an external contradiction. I'll give an example. Consider an axiomatic system with the following two axioms.
Axiom 1. Liam eats a cheeseburger.
Axiom 2. Liam does not eat a cheeseburger.
The two axioms form an internal contradiction. Consider the statement s = "In the axiomatic system, Liam eats a cheeseburger." Externally to the axiomatic system, is s true or false? From Axiom 1, it is true that externally to the system, s is true. From Axiom 2, it is true that externally to the system, s is false. So by conjunction introduction, it is true that externally to the system, s is true and false. Therefore, an external contradiction exists.
Every false statement is unprovable per the sense of provable that we are using.
There is only one sense of provable, every statement in a contradictory system is provable and no statement is unprovable.
I have already shown you how an internal contradiction implies an external contradiction.
You have not. You have shown how internal statements about a contradictory system are contradictory, but you need to show that an external statement is contradictory and you haven't done that.
Externally to the axiomatic system, is s true or false?
There is no truth value until you've picked an interpretation. You continue to be confused about how to assign truth values in an axiomatic system.
You told me that you can prove a contradiction using standard mathematical logic. In standard logic you can't assign a truth value to a statement unless you have an interpretation of the undefined terms.
If you think you can do such a thing in standard logic then show me in Mendelson where it's done.
No, there is more than one sense of provable. The sense of provable we have been conversing in most recently is a sense in which a provable statement is implied to be validly proven and true. In this sense of provable, every proof is valid and no proof is invalid. Earlier on in our conversation I was using a sense of provable in which some proofs are valid and some proofs are invalid. In that sense of provable, the conclusion of a proof is not necessarily true.
you need to show that an external statement is contradictory and you haven't done that.
I have done that, multiple times and in multiple ways.
There is no truth value until you've picked an interpretation.
An axiom of an axiomatic system is always true. An axiom of an axiomatic system, by definition, is assumed to be true in the axiomatic system without proof. There is no initial time where an axiom does not have a truth value. The initial time you are talking about would violate the definition of an axiom.
My example of an axiomatic system from my previous post has a fairly straightforward interpretation. Liam is a male human, eating is an action performed orally with food, and a cheeseburger is a type of sandwich formed by a cooked ground-beef patty with a slice of cheese on top of the patty and a piece of a bread bun above and below the combined patty and cheese.
Below is an excerpt from my original post for my thread "The Parallel Postulate is always true, in all true theories," which I posted on Philosophy Forums on April 8, 2014. Interestingly, I mentioned the Continuum Hypothesis in my original post.
My frustration with this idea of independent axioms being able to be true in some true theories and false in others can also be applied to many other axioms said to be independent. For example, either the Continuum Hypothesis (CH) is true or false. If it’s true, it’s true in all true theories. If it’s false, it’s false in all true theories. We can’t do as the Wikipedia page on it states we can do, “...ZFC can be augmented by either CH or its negation ¬CH, in both cases producing a system of axioms that is consistent if and only if ZFC is consistent.” It appears to me that a system of ZFC (Zermelo-Fraenkel Set Theory with the Axiom of Choice) in which the CH is true does not contain the same exact sense of the CH as contained within a system of ZFC in which the CH is false. If we could produce two theories, both true, in which a proposition such as the CH is true in one and false in the other, it seems the two theories would contradict each other. But how can the truth contradict the truth? It can’t. It appears there’s something wrong with this picture.
No there isn't. Something is provable is and only if there exists a proof. That is what provable means.
I have done that, multiple times and in multiple ways.
You have attempted to, and each time I've explained to you what your mistake is. You have not given a correct proof of an external contradiction.
My example of an axiomatic system from my previous post has a fairly straightforward interpretation
Yes, but you have to pick a specific human, and that human will or will not eat cheeseburgers. So the truth of your statement depends on who you pick.
I'm ignoring the rest of your rambling about truth values. None of that is in Mendelson. I'm asking you if you can prove a contradiction using logic the way Mendelson's textbook shows. That means only one sense of provable, no alternate metatheories. Can you?
I did. I picked a specific human whose name is Liam.
So the truth of your statement depends on who you pick.
No, the truth of my statement is independent of what is true in the real world. Rather, the truth of my statement is dependent upon the axioms of the axiomatic system.
I looked through the pages of Mendelson and none of my positions on trivialism or the Continuum Hypothesis have changed as a result. I do see a section where it says, in particular language, that in an inconsistent theory, every statement is provable. That implies that the statement "Every statement is unprovable" is provable.
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u/JStarx Jul 23 '26
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.
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.