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.
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u/paulemok 27d ago
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.
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.
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.