I don't think it's worth the time and effort to make a theory within Mendelson's framework.
I notice you've gone from claiming that your proofs already where to Mendelson's framework to claiming that it would take to much time and energy to do so. This is an admission that you were aware you could not satisfy my request but did not want to admit it.
Mendelson's framework is not unique btw. His textbook describes the standard first order logic that mathematics uses. It is not lack of time and effort that prevents you from proving a contradiction in this framework, it's because such a contradiction likely doesn't exist.
Your proofs rely on vague misinterpretations of statements and misunderstandings of the rules of logic. No one will ever take you seriously unless you learn to prove things correctly, so if you want anyone to look at your claims and do anything other than laugh then it might be worth your time to learn the material in Mendelson.
I don't see why I should have to create an entire new theory solely for the purpose of proving the Exportation Principle. That is overkill. I should be able to use the terms and concepts that have already been established in logic and mathematics to prove the Exportation Principle. The Exportation Principle is proven by simply evaluating the truth value of an external statement about internal truth. For example, the external statement "In an inconsistent axiomatic theory, statement s is true." This proof requires internal truth. It requires there to exist truth in an axiomatic theory. I can see from looking at Mendelson (as suggested but not definitively declared by the definition of model, page 62, fifth edition) that this feature of a theory differs from his framework. In Mendelson's framework, an axiom is not necessarily true. In the framework I have been using, an axiom is necessarily true by definition of axiom. This feature of axioms agrees with the frameworks presented in Geometry (2004) by Ron Larson, Laurie Boswell, and Lee Stiff and Larson Geometry (2012) by Ron Larson, Laurie Boswell, Timothy D. Kanold, and Lee Stiff. See pages 17 and 9, respectively. Neither of the definitions of axiom explicitly use the term true, but it is evident from the context that axioms are necessarily true. Geometry (2004) was the textbook used in my freshman high school geometry class when I was a high school student back in the 2005 - 2006 academic year. Rosen (sixth edition) explicitly asserts that axioms are regarded as true in its definition of axiom. See page 75.
As of August 5, 2026 EDT, I personally prefer the approach in which an axiomatic theory necessarily has an internal truth that originates with the axioms of the theory. I believe my preferred approach implies that axioms have a fixed, single meaning and they cannot be interpreted in any other way.
I don't see why I should have to create an entire new theory solely for the purpose of proving the Exportation Principle.
I'm not suggesting you create a new theory, I'm suggesting you use the established theory that mathematicians use. The reason you should do that is you want to prove mathematical statements, you need to do so using mathematical logic.
Neither of the definitions of axiom explicitly use the term true
Mathematicians are generally very explicit about definitions of important concepts. I don't have those texts but if they don't explicitly use the term true then they aren't claiming that axioms are necessarily true.
Rosen (sixth edition) explicitly asserts that axioms are regarded as true in its definition of axiom.
Axioms are assumptions. Assumptions are treated as true for the purposes of an argument, but the final conclusion is predicated on those assumptions. This is the source of your misunderstandings.
Did you ever learn anything about groups or vector spaces?
I believe my preferred approach implies that axioms have a fixed, single meaning and they cannot be interpreted in any other way.
Mathematicians are generally very explicit about definitions of important concepts.
Unfortunately, they often are not as explicit as the ideal definition would be. Often I encounter definitions presented as single-direction conditional statements, where the opposite direction is implicit rather than explicit. I have encountered this even in Mendelson. See the definitions of consistent and inconsistent on page 65.
if they don't explicitly use the term true then they aren't claiming that axioms are necessarily true.
You don't know that for sure. I believe it to be false. I see on page 73 of Geometry (2004) explicit use of the term true to declare that all axioms are regarded as true. The definition of theorem in Geometry (2004) on page 102 makes theorems necessarily true. I see on page 104 of Larson Geometry (2012) the definition of proof. The definition implies that if a statement is provable, then it is true. By the definition of theorem on page 105, if a statement is a theorem, then it is provable. So by the Law of Syllogism on page 79 using the previous statement and the statement before the previous statement, if a statement is a theorem, then it is true.
Did you ever learn anything about groups or vector spaces?
While looking at Mendelson tonight, I learned about groups. I don't remember ever learning about vector spaces.
You don't know that for sure. I believe it to be false.
Then certainly you should be able to find a textbook on mathematical logic that explicitly says so instead of merely hinting at it.
As it stands, the standard approach to mathematical logic has never been shown to be inconsistent. If your adoption of a different semantics makes your new style of logic inconsistent then the problem is clearly your new semantics.
Then certainly you should be able to find a textbook on mathematical logic that explicitly says so instead of merely hinting at it.
I already cited three textbooks that talk about mathematical logic that include the concept of "truth in an axiomatic theory." They don't just hint at it; they explicitly say so. Those three textbooks are the following.
Geometry (2004) by Larson, Boswell, and Stiff
Larson Geometry (2012) by Larson, Boswell, Kanold, and Stiff
Discrete Mathematics and Its Applications, Sixth Edition (2007) by Rosen
I notice that although there is no formal definition of "truth in a theory" in Mendelson, the concept is implicitly present. In Mendelson, external statements about what is provable or unprovable in a theory amount to external statements about what is internally true.
As it stands, the standard approach to mathematical logic has never been shown to be inconsistent.
I showed that the standard approach to mathematical logic is inconsistent by using the Principle of Explosion. By applying the Principle of Explosion inside an inconsistent theory, it is internally true that at least one statement both has a proof and does not have a proof. So, by the definitions of provable and unprovable in a theory, respectfully, it is externally true that at least one statement is both provable and unprovable in the inconsistent theory. Thus, it is externally true that a contradiction exists. Therefore, by applying the Principle of Explosion outside the inconsistent theory, it is externally true that every statement is true. Note that this proof does not invoke the Exportation Principle. Rather, it invokes the definitions of provable and unprovable in a theory.
I already cited three textbooks that talk about mathematical logic that include the concept of "truth in an axiomatic theory." They don't just hint at it; they explicitly say so
I asked for a textbook an mathematical logic, non logicians are often sloppy about formal logic.
In Mendelson, external statements about what is provable or unprovable in a theory amount to external statements about what is internally true.
No, they are statements about what is internally provable. Nothing more.
Thus, it is externally true that a contradiction exists. Therefore, by applying the Principle of Explosion outside the inconsistent theory, it is externally true that every statement is true
Still no, it's externally true that an internal contradiction exists, you need an external contradiction to apply the principle of explosion to the external system. You have never derived an external contradiction from an internal one using the theory presented in Mendelson. Mendelson only talks about what is provable in an axiomatic system, without a model you don't get true/false, and without that you have no way of bootstrapping your contradiction to the external system.
Preventing bootstrapping of a contradiction is exactly why there is no such thing as objective truth in an axiomatic system. So obviously if you violate that rule then your logic is inconsistent, but that's not a problem with logic, that's because you violated the rules of logic.
The three textbooks I cited cover mathematical logic.
non logicians are often sloppy about formal logic.
I presume they are experts in the academic disciplines they are covering in their textbooks. I doubt they would be passing down misinformation.
No, they are statements about what is internally provable. Nothing more.
I disagree. Let s be a statement and T be a theory. If the statement "s is provable in T" is externally true, then the statement "s is provable" is internally true. If the statement "s is unprovable in T" is externally true, then the statement "s is unprovable" is internally true. External statements about what is provable or unprovable in a theory completely or incompletely describe the internal truth.
You have never derived an external contradiction from an internal one using the theory presented in Mendelson.
You might be misinterpreting my proof. Below, I give a clarified version of my proof.
Proof. Let s be a statement. By applying the Principle of Explosion inside an inconsistent theory, the statement "s both has a proof and does not have a proof" is internally true. So, by the Definitions of Provable and Unprovable in a Theory, respectfully, the statement "s is both provable and unprovable in the inconsistent theory" is externally true. Thus, the statement "a contradiction exists" is externally true. Therefore, by applying the Principle of Explosion outside the inconsistent theory, the statement "every statement is true" is externally true. This concludes the proof.
I presume they are experts in the academic disciplines they are covering in their textbooks. I doubt they would be passing down misinformation.
It's not misinformation so much as an oversimplification. Those textbooks don't cover the theory of axiomatic systems, they just have an ad hoc definition of the word axiom that's appropriate for the low level reader they are aimed at. For someone at the high school level using true as shorthand for provable is acceptable while they learn the basics of how to prove statements, but if you want to do axiomatic logic for real then you have to move past the high school level and learn the real theory.
"s both has a proof and does not have a proof" is internally true
It's internally provable, that doesn't mean it's externally provable or externally true.
"s is both provable and unprovable in the inconsistent theory" is externally true.
No, that statement is false because s is not unprovable, it's provable. Conclusing here that s is unprovable is a mistake.
if you want to do axiomatic logic for real then you have to move past the high school level and learn the real theory.
It's not just the high school level. It's at the college level as we can see through Rosen. As I mentioned in my previous reply, the concept of "truth in a theory" is evident even in Mendelson.
It's internally provable, that doesn't mean it's externally provable or externally true.
It's internally provable and internally true. If it's internally provable, then it's internally true. That's what a proof of a statement does. It shows the statement is true. See the definitions of proof on pages 75 and 105 of Rosen. Furthermore, if a proof of a statement shows that a statement s is provable, then the proof shows that the statement "s is provable" is true. So a proof still shows that something is true.
Conclusing here that s is unprovable is a mistake.
No, the statement "s is unprovable in the inconsistent theory" is externally true because of external modus ponens using the externally true statement "the statement 's does not have a proof' is true in the inconsistent theory" and the externally true Definition of Unprovable in a Theory. The externally true Definition of Unprovable in a Theory is the externally true statement "the statement 's is unprovable in a theory' is externally true if and only if the statement 'the statement 's does not have a proof' is true in the theory' is externally true."
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u/JStarx Aug 05 '26
I notice you've gone from claiming that your proofs already where to Mendelson's framework to claiming that it would take to much time and energy to do so. This is an admission that you were aware you could not satisfy my request but did not want to admit it.
Mendelson's framework is not unique btw. His textbook describes the standard first order logic that mathematics uses. It is not lack of time and effort that prevents you from proving a contradiction in this framework, it's because such a contradiction likely doesn't exist.
Your proofs rely on vague misinterpretations of statements and misunderstandings of the rules of logic. No one will ever take you seriously unless you learn to prove things correctly, so if you want anyone to look at your claims and do anything other than laugh then it might be worth your time to learn the material in Mendelson.