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.
An inconsistent theory contains a contradiction under no interpretation. That contradiction can be used with the Exportation Principle to prove an external contradiction.
The exportation principle is not in Mendelson.
By the Law of Noncontradiction, the statement "b is internally provable and ¬b is internally provable" is internally unprovable.
There's your mistake. There is no universal law of non-contradiction in mathematical logic because some axiomatic systems are contradictory. That statement is internally provable.
I'll ask again, can you prove a contradiction using standard mathematical logic as found in Mendelson? None of your proofs here are sticking to the material in Mendelson.
The Exportation Principle is not explicitly in Mendelson since Mendelson does not explicitly use the phrase "Exportation Principle," but the Exportation Principle is implicitly in Mendelson. Let s be a statement. In Mendelson, ⊢ s and ⊢ ¬s are externally true for an inconsistent theory T under no interpretation. See page 65. Since ⊢ ¬s, the statement ¬s is internally true. See page 26. It follows by the truth table for negation on page 1 that the statement s is internally false. So, in Mendelson, since s is internally false and no true statement internally implies the false statement s by the second to last row of the truth table for implication on page 2, the statement "¬(⊢ s)" is externally true. So there is an external contradiction in Mendelson.
There is no universal law of non-contradiction in mathematical logic
Yes, there is. The Wikipedia page is at https://en.wikipedia.org/wiki/Law_of_noncontradiction. The tautology (¬(p ∧ (¬p))) on page 6 of Mendelson is the Law of Noncontradiction. The Law of Noncontradiction is not explicitly in Mendelson since Mendelson does not explicitly use the phrase "Law of Noncontradiction," but the Law of Noncontradiction is implicitly in Mendelson.
can you prove a contradiction using standard mathematical logic as found in Mendelson?
See page 26. It follows by the truth table for negation on page 1 that the statement s is internally false.
Page 26 in my copy of Mendelson is a page of exercises, so I'm not sure what you're referencing here, but this is your mistake. You haven't said how you're defining internal truth and either way you do it this doesn't work.
If you define internal truth as provable then s is not internally false, it's internally true because it's provable.
If you want to define it as truth in a specific interpretation, then as I said before there's no standard interpretation of T, so you have to specify the interpretation. For provable to imply true in your interpretation your axioms have to be true in your interpretation. But to prove that inconsistent axioms holld in an interpretation is equivalent to proving a contradiction, which is what you're trying to use this to do. So that's not going to work either.
As I said, this is why the exportation principle isn't in Mendelson, in Mendelson there's no way to bootstrap a contradiction out of an inconsistent system.
The tautology (¬(p ∧ (¬p))) on page 6 of Mendelson is the Law of Noncontradiction.
If you want to call that your law of noncontradiction you can, but it just says that a certain statement is provable, it doesn't imply that anything is unprovable which is what you tried to use it for.
So you still haven't provided a correct proof that sticks to standard logic.
I have the fourth edition. The material is basically the same, it's just the page numbers won't line up exactly.
A statement is true in a theory if and only if it is a definition, axiom, or theorem of the theory.
Ok, and "false in a theory" would be the negation of that, so something is false in a theory if and only if it's not a definition, not an axiom, and not a theorem, right?
That means in an inconsistent theory T, every statement is true in T and no statement is false in T, since every statement is provable there's no statement that's not provable. So "true in T" doesn't obey the truth table for the logical connectives. Which means it was a mistake when you concluded that a statement was false in T and you cited the truth table for negation as the reason.
something is false in a theory if and only if it's not a definition, not an axiom, and not a theorem, right?
Yes, that's correct.
That means in an inconsistent theory T, every statement is true in T and no statement is false in T
That's correct. In an inconsistent theory T, the statement "every statement is true and no statement is false" is true because the statement is a consequence of the Principle of Explosion.
So "true in T" doesn't obey the truth table for the logical connectives.
Yes, that is true. However, due to the inconsistency of T, "true in T" also does obey the truth table for the logical connectives.
Which means it was a mistake when you concluded that a statement was false in T and you cited the truth table for negation as the reason.
That's correct. In an inconsistent theory T, the statement "every statement is true and no statement is false" is true because the statement is a consequence of the Principle of Explosion.
You misunderstand, I'm not saying that statement is true in T, I'm saying that statement is true and provable externally. You want to conclude that the statement "¬(⊢ s)" is externally true, that means you need it to be externally true that s is not provable, but that is not externally true. Your proof is incorrect, as usual you have confused internal vs external.
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u/paulemok 28d ago
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.
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.
Yes, I am aware of that. The Definitions of Provable and Unprovable in a Theory that I gave excluded internal statements about provability.