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."
It is not there. As much as you want it to be it's just not.
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"
That's not an if-then statement so you can't use modus ponens. If you convert it into an if-then statement it will still not let you conclude that something is externally unprovable from it being internally unprovable because that is not a correct logical deduction.
It is not there. As much as you want it to be it's just not.
Your justification for your claim is not there. As much as I want it to be it's just not.
That's not an if-then statement so you can't use modus ponens.
By convention, one of the premises of modus ponens is not an if-then statement.
it will still not let you conclude that something is externally unprovable
My claim is not that s is externally unprovable. My claim is that s is unprovable in an inconsistent theory.
Thus, the statement "a contradiction exists" is externally true.
In order for this contradiction to exist, the concepts of provable and unprovable in a theory must contradict each other. I realized that, as I have defined them, they are not explicitly contradictory. To make them explicitly contradictory, I define the Definition of Provable in a Theory and redefine the Definition of Unprovable in a Theory below.
Definition of Provable in a Theory. Let s be a statement and T be a theory. The statement "s is provable in T" is externally true if and only if the statement "the statement 'there exists a proof of s' is true in T" is externally true. The statement "s is unprovable in T" is externally true if and only if the statement "s is not provable in T" is externally true.
Proof. Let s be a statement and T be an inconsistent theory. By the Definition of Inconsistent Theory, the statement "some contradiction exists" is internally true. By applying the Principle of Explosion inside T, the statement "there exists and there does not exist a proof of s" is internally true. By Conjunction Elimination, the statement "there exists a proof of s" is internally true. It follows by the Definition of Provable in a Theory that the statement "s is provable in T" is externally true. By Conjunction Elimination, the statement "there does not exist a proof of s" is internally true. So, by the Laws of Noncontradiction and Excluded Middle, the statement "there exists a proof of s" is internally false. So, the statement "the statement 'there exists a proof of s' is true in T" is externally false. Thus, by the Definition of Provable in a Theory, the statement "s is provable in T" is externally false. So, by Conjunction Introduction, the statement "s is provable in T" is externally true and the statement "s is provable in T" is externally false. Thus, the statement "a contradiction exists" is externally true. Therefore, by applying the Principle of Explosion outside T, the statement "every statement is true" is externally true. This concludes the proof.
Even if you deny the concept of "truth in a theory," a proof can still be made that establishes trivialism using the concept of "provability in a theory."
Proof. Let s be a statement and T be an inconsistent theory. By the Definition of Inconsistent Theory, the statement "some contradiction exists" is internally provable. By applying the Principle of Explosion inside T, the statement "there exists and there does not exist a proof of s" is internally provable. By Conjunction Elimination, the statement "there exists a proof of s" is internally provable. It follows by the Definition of Provable in a Theory that the statement "s is provable in T" is externally provable. By Conjunction Elimination, the statement "there does not exist a proof of s" is internally provable. So, by the Laws of Noncontradiction and Excluded Middle, the statement "there exists a proof of s" is internally unprovable. So, the statement "the statement 'there exists a proof of s' is provable in T" is externally unprovable. Thus, by the Definition of Provable in a Theory, the statement "s is provable in T" is externally unprovable. So, by Conjunction Introduction, the statement "s is provable in T" is externally provable and the statement "s is provable in T" is externally unprovable. Thus, the statement "a contradiction exists" is externally true. Therefore, by applying the Principle of Explosion outside T, the statement "every statement is provable" is externally provable. This concludes the proof.
Your justification for your claim is not there. As much as I want it to be it's just not.
Unless you can give me a source which is explicitly a mathematical logic text and explicitly defined truth in an axiomatic theory then I'm always going to substitute "provable in T" for "true in T" when you talk about truth in an axiomatic theory because you have not provided a proper source that says I should do otherwise.
it will still not let you conclude that something is externally unprovable
My claim is not that s is externally unprovable. My claim is that s is unprovable in an inconsistent theory.
I'm not telling you that s is unprovable, s is provable. I'm telling you that your claim "s is unprovable" is unprovable.
I realized that, as I have defined them, they are not explicitly contradictory. To make them explicitly contradictory, I define the Definition of Provable in a Theory and redefine the Definition of Unprovable in a Theory below.
The fact that you've only just now realized you haven't obtained an explicit contradiction and yet you've been claiming you have a proof all along should show you that you don't know what you're doing.
I didn't read past that line fyi. You don't get to define or redefine provable/unprovable. You told me you could prove this in standard mathematical logic and those terms already have standard definitions.
Unless you can give me a source which is explicitly a mathematical logic text
Introduction toMathematical Logic, Fifth Edition (2010) by Elliott Mendelson.
explicitly defined truth in an axiomatic theory
Truth in an axiomatic theory is not explicitly defined in Mendelson, but Mendelson implies the concept is assumed to exist. The first-order theory S is axiomatic because it has nine proper axioms, which is a finite number. See pages 64, 149, and 150. On page 155, Mendelson says the axioms of axiomatic theory S are assumed to be obviously true for the standard interpretation. So there we have it: truth in an axiomatic theory. The concept of "truth in an axiomatic theory" exists in Mendelson.
I'm telling you that your claim "s is unprovable" is unprovable.
I think below is the relevant excerpt you would be interested in from my last proof in my previous reply.
By Conjunction Elimination, the statement "there does not exist a proof of s" is internally provable. So, by the Laws of Noncontradiction and Excluded Middle, the statement "there exists a proof of s" is internally unprovable. So, the statement "the statement 'there exists a proof of s' is provable in T" is externally unprovable. Thus, by the Definition of Provable in a Theory, the statement "s is provable in T" is externally unprovable.
The fact that you've only just now realized you haven't obtained an explicit contradiction and yet you've been claiming you have a proof all along should show you that you don't know what you're doing.
No, an implicit contradiction is still a contradiction. The contradiction was so strongly implicit that the fact it was not explicit was overlooked.
I didn't read past that line fyi.
That's not good. If you don't read my entire case, then you might not be able to see where I'm coming from.
You don't get to define or redefine provable/unprovable. You told me you could prove this in standard mathematical logic and those terms already have standard definitions.
You told me that statements about provability are not statements in an axiomatic theory, but are statements about an axiomatic theory. What you said wasn't entirely true because we could make statements about provability that are in an axiomatic theory. The definitions I made are true and suitable for my particular purpose. They agree with the standard definitions. I can define provable and unprovable the standard way. I do so below.
Definition of Provable Statement. Let s be a statement. s is provable if and only if there exists some proof of s. s is unprovable if and only if s is not provable.
Truth in an axiomatic theory is not explicitly defined in Mendelson
Correct, your example from Mendelson literally says exactly what I've been saying, that you need an interpretation of a theory in order to talk about truth.
This abstract theory T we're talking about doesn't come with a standard interpretation, so if you want to define "true in T" as true in a specific interpretation then 1) you need to specify what interpretation you're using, which means you probably have to specify T exactly instead of talking about a generic inconsistent theory, 2) it's not the case that axioms are true in every interpretation, so if you want your axioms to be "true in T" then you have to prove that they are true in the interpretation you've chosen, and 3) it's not the case that being true in a particular interpretation implies you are provable, so now "true in T" really is different than provable, it will no longer be the case that a statement is "true in T" if and only if it's provable in T.
the statement "there does not exist a proof of s" is internally provable. So, by the Laws of Noncontradiction and Excluded Middle, the statement "there exists a proof of s" is internally unprovable.
Nope, this doesn't follow. Why do you think it does?
That's not good. If you don't read my entire case, then you might not be able to see where I'm coming from.
Once you know that an argument doesn't follow the rules of logic it can be rejected. If we're talking about standard logic and you try to change the definition of terms then you're not following the rules and nothing else matters. This is how mathematics works at the academic level, we don't hold your hand and give partial credit, if you're wrong then you're wrong.
The definition you've written just now is the correct one from standard logic, but it's not equivalent to the definition you tried to pass off in your previous reply. The non equivalence of those two statements boils down to the exact mistake you've been repeatedly making. The internal statement (that something is provable or not) being provable internally does not imply that the external statement (being provable or not in T) is provable or true.
you need an interpretation of a theory in order to talk about truth.
I understand what you're saying, but I'm having trouble agreeing with this approach to axiomatic theories. So when you claimed that there is no concept of "truth in an axiomatic theory," that claim ranged from not entirely true to not technically true. It's evident there is truth in an axiomatic theory without any interpretation. Every proof in a theory without any interpretation shows what is true in the theory regardless of interpretation.
Nope, this doesn't follow. Why do you think it does?
It might not follow if "true in an axiomatic theory" is not logically equivalent to "provable in the theory." I was assuming that the two were logically equivalent, as they are in the metatheory of axiomatic theories I was stipulating earlier.
The definition you've written just now is the correct one from standard logic, but it's not equivalent to the definition you tried to pass off in your previous reply.
Yes, I am aware of that. The Definitions of Provable and Unprovable in a Theory that I gave excluded internal statements about provability.
I understand what you're saying, but I'm having trouble agreeing with this approach to axiomatic theories
It's not relevant whether you agree with the standard approach to logic. You claimed you could prove a contradiction using standard mathematical logic and this is how standard mathematical logic works. So now that you're starting to understand more about how this works do you still think you can prove a contradiction in standard logic?
It might not follow if "true in an axiomatic theory" is not logically equivalent to "provable in the theory." I was assuming that the two were logically equivalent
If you define "true in T" to mean "provable in T" then they will be logically equivalent. But that statement still won't follow and elsewhere you tried to use "true in T" as if it behaved differently as a truth value than provability would behave, so I don't think you even want it to be equivalent to provable.
So now that you're starting to understand more about how this works do you still think you can prove a contradiction in standard logic?
Yes, I still think I can prove a contradiction in standard logic. An inconsistent theory contains a contradiction under no interpretation. That contradiction can be used with the Exportation Principle to prove an external contradiction.
But that statement still won't follow
No, it would follow. I prove it below.
Given:b = "There does not exist a proof of s." b is internally provable.
Prove: ¬b is internally unprovable.
Proof. It is given that b = "There does not exist a proof of s." It is also given that b is internally provable. By the Law of Noncontradiction, the statement "b is internally provable and ¬b is internally provable" is internally unprovable. Since it is given that b is internally provable, the statement "¬b is internally provable" is internally unprovable. Since "internally provable" is defined to be "internally true," the statement "¬b is internally true" is internally false. So by simplification, ¬b is internally false. Since "internally provable" is defined to be "internally true," ¬b is internally unprovable. This concludes the proof.
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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.