There are infallible documents

[godelian]

Then why did you go through the mental gymnastics of "therefore..." and pretend it's some kind of theorem e.g a conclusion which necessarily follows from axioms.

First-order logic is itself an axiomatic system. The proof is valid in first-order logic, assuming the LEM.

[godelian]

https://en.wikipedia.org/wiki/Rice%27s_theorem

all non-trivial semantic properties of programs are undecidable.

...such as truth-value.

The truth value of axioms is not the result of a runtime/semantic computation. The program must at compile time already assume the truth of the axioms in order to be able to computationally prove a theorem at runtime.

[Skepdick]

First-order logic is itself an axiomatic system. The proof is valid in first-order logic, assuming the LEM.

Why are you assuming LEM, and not proving it?

The truth value of axioms is not the result of a runtime/semantic computation. The program must at compile time already assume the truth of the axioms in order to be able to computationally prove a theorem at runtime.

Precisely. But you aren't peddling "there are infallible documsnts" as axiomatic.

You are peddling it as a theorem - something you've proven at runtime.

So what are the axioms you are using to compute this theorem? Over and above LEM.

[godelian]

Why are you assuming LEM, and not proving it?

The problem with "Every proposition is false", is that it means that this proposition is itself false and therefore that there must exist true propositions. The language is first-order logic because of the use of the universal quantifier.

ChatGPT: When is every proposition -- no exception -- in an axiomatic system necessarily false?

For every proposition to be false, the system must be inconsistent (allow contradictions) or have no models (be unsatisfiable). In a well-formed, consistent system with a model, at least some propositions must be true.

I am still looking into actually proving the LEM in this context. That is harder than it at first glance looks like. For the time being, I just assume the LEM, assuming that it is necessary to assume it.

Precisely. But you aren't peddling "there are infallible documsnts" as axiomatic.

I prove that the opposite is contradictory.

So what are the axioms you are computing with?

Good question. If you formalize it, it certainly uses the universal quantifier in first-order logic. So, there is a requirement of language.

But then again, the formalization may actually also make use of the truth predicate. In that case, we do indeed need to distinguish between metatheory and object theory, since the truth predicate for the object theory is defined only in the metatheory. It is easy to write all of that down in natural language, but if you formalize it, it turns out to be more complex.

[Skepdick]

The problem with "Every proposition is false", is that it means that this proposition is itself false

Fine.

and therefore that there must exist true propositions.

Non-sequitur.

The falsehood of "All propositions are false" doesn't entail Some propositions are true.

Which inference rule/axiom are you using to arrive here?

I am still looking into actually proving the LEM in this context. That is harder than it at first glance looks like. For the time being, I just assume the LEM, assuming that it is necessary to assume it.

That's just called "motivated reasoning"

Precisely. But you aren't peddling "there are infallible documsnts" as axiomatic.

I prove that the opposite is contradictory.

Negation isn't "oppsite". It's just negation. Treating negation as opposite/complementary is LEM.

Negation by itself just says "not this" is contradictory - it doesn't automatically give you "therefore this is not contradictory."

It is easy to write all of that down in natural language, but if you formalize it, it turns out to be more complex.

Probably because English isn't a 1st order language...

[godelian]

So what are the axioms you are using to compute this theorem? Over and above LEM.

ChatGPT: ∀T∀P ( T ⊨ ¬P)
The statement:

∀T∀P ( T ⊨ ¬P)

means that for every theory T and every proposition P, the theory T semantically entails ¬P.
Interpretation:

Universal quantification over T and P:
This means that no matter what theory T you choose and no matter what proposition P you choose, the statement T⊨¬P must hold.
Semantic entailment (⊨):
T⊨¬P means that in every model where T is true, ¬P must also be true.
This implies that P must be false in all models of T.

Consequence:

If this statement were true, then for every theory T, every proposition P is false in all models of T.
This would mean that no proposition is ever true in any model of any theory.
The only way this could be the case is if no models exist at all, meaning that every theory is inconsistent.

Conclusion:

The statement:

∀T∀P(T⊨¬P)

is false in general, because there exist consistent theories (e.g., Peano Arithmetic, ZFC set theory, etc.) that have models in which some propositions are true. For this statement to be true, every theory T would have to be inconsistent, which is clearly not the case in normal mathematical logic.

There is no need to use a truth predicate to prove this statement.

[Skepdick]

So what are the axioms you are using to compute this theorem? Over and above LEM.

ChatGPT: ∀T∀P ( T ⊨ ¬P)
The statement:

∀T∀P ( T ⊨ ¬P)

means that for every theory T and every proposition P, the theory T semantically entails ¬P.
Interpretation:

Universal quantification over T and P:
This means that no matter what theory T you choose and no matter what proposition P you choose, the statement T⊨¬P must hold.
Semantic entailment (⊨):
T⊨¬P means that in every model where T is true, ¬P must also be true.
This implies that P must be false in all models of T.

Consequence:

If this statement were true, then for every theory T, every proposition P is false in all models of T.
This would mean that no proposition is ever true in any model of any theory.
The only way this could be the case is if no models exist at all, meaning that every theory is inconsistent.

Conclusion:

The statement:

∀T∀P(T⊨¬P)

is false in general, because there exist consistent theories (e.g., Peano Arithmetic, ZFC set theory, etc.) that have models in which some propositions are true. For this statement to be true, every theory T would have to be inconsistent, which is clearly not the case in normal mathematical logic.

There is no need to use a truth predicate to prove this statement.

Non-sequitur. A statement that is false in general may still be true in particular.

https://en.wikipedia.org/wiki/Rice%27s_theorem

A non-trivial property is one which is neither true for every program, nor false for every program.

[godelian]

The falsehood of "All propositions are false" doesn't entail Some propositions are true.

It does.

There exist witness theories for T as such that the following cannot be generally true:

∀T∀P ( T ⊨ ¬P)

Which inference rule/axiom are you using to arrive here?

All you have to do, is to produce one witness theory. ChatGPT mentions PA and ZFC, but you can concoct something even simpler and that would be the existential witness.

[Skepdick]

The falsehood of "All propositions are false" doesn't entail Some propositions are true.

It does.

There exist witness theories for T as such that the following cannot be generally true:

∀T∀P ( T ⊨ ¬P)

It doesn't need to be generally true.

For some T and some P T ⊨ ¬P
For some T and some P T⊭¬P

All you have to do, is to produce one witness theory.

You are proving my point.

The falsehood of "All propositions are false" doesn't entail Some propositions are true.

It WOULD entail "Some propositions are true." IF you produce a witness theory.

IF. Without the witness, you don't have the proof for "Some propositions are true"

[Dr Faustus]

Imagine the opposite:

There are no infallible documents.

Fine.

Now let's create a document that contains only the statement above. This document cannot be infallible. Therefore, the only statement that it contains must be false. Therefore, the claim that there are no infallible documents must be false.

Hence, by contradiction, we must conclude:

There are infallible documents.

While I certainly object to the infallibility of an organization such as a church, or of a person such as the Pope, I do not object to the infallibility of particular documents.

This was exactly Martin Luther's position during his trial in front of His Imperial Majesty, Charles V, in 1521:

Unless I am convinced by Scripture and plain reason - I do not accept the authority of the popes and councils, for they have contradicted each other - my conscience is captive to the Word of God.

The prosecutor of the Holy Apostolic Church answered that there are no infallible documents:

The Bible itself is the arsenal whence each heresiarch from the past has drawn his deceptive arguments.

Christianity does not believe in the existence of infallible documents but in the existence of infallible Popes. As demonstrated above, this particular belief is in violation of the very rules of logic. That is why Christian doctrine cannot possibly be closed under logical consequence.

To be completely rigorous, by contradiction we must conclut that there exist at least one document that is not infaillible.
But I agree with you there are many document that are infaillible.
A=A
B=B
C=C
D=D
...=....

[Skepdick]

To be completely rigorous, by contradiction we must conclut that there exist at least one document that is not infaillible.

That is not a valid inference.

To claim "at least one document that is not infallible" you must produce at least one such document.

Not all balls in the bag are blue doesn't mean any are green.

[godelian]

The falsehood of "All propositions are false" doesn't entail Some propositions are true.

It WOULD entail "Some propositions are true." IF you produce a witness theory.

IF.

ChatGPT: ∀T∀P ( T ⊨ ¬P). Find an example theory for which this statement is false.

The given logical statement:

∀T∀P (T⊨¬P)

asserts that for every theory T and every proposition P, T semantically entails ¬P. In other words, every possible theory refutes every possible proposition.

To find a counterexample, we need to provide a theory T and a proposition P such that T⊭¬P. This means T does not necessarily refute P, implying that there is at least one model of T where P holds.

Counterexample:

Consider the theory T={}, i.e., the empty theory, which contains no axioms. Take any proposition P, say P itself as a single atomic formula.

Since T is empty, it imposes no constraints on P. In particular, there exist models where P is true and models where P is false. Thus, T⊭¬P because there exists at least one model of T in which P holds.

Since we found a case where the universal quantification fails, the given logical statement is false.

[Skepdick]

The falsehood of "All propositions are false" doesn't entail Some propositions are true.

It WOULD entail "Some propositions are true." IF you produce a witness theory.

IF.

ChatGPT: ∀T∀P ( T ⊨ ¬P). Find an example theory for which this statement is false.

The given logical statement:

∀T∀P (T⊨¬P)

asserts that for every theory T and every proposition P, T semantically entails ¬P. In other words, every possible theory refutes every possible proposition.

To find a counterexample, we need to provide a theory T and a proposition P such that T⊭¬P. This means T does not necessarily refute P, implying that there is at least one model of T where P holds.

Counterexample:

Consider the theory T={}, i.e., the empty theory, which contains no axioms. Take any proposition P, say P itself as a single atomic formula.

Since T is empty, it imposes no constraints on P. In particular, there exist models where P is true and models where P is false. Thus, T⊭¬P because there exists at least one model of T in which P holds.

Since we found a case where the universal quantification fails, the given logical statement is false.

ChatGPT can't do any of your reasoning for you.

See, just like you decided to mix up judgments and documents; so I will mix up theories and propositions.

Consider the proposition/theory: ∀T∀P (T⊨¬P)
Is this theory/proposition true or false?

You are treating it as true/axiomatic. I am treating it as neither true nor false e.g undecidable in general.

For some T and P (T⊨¬P)
For some T and P (T⊭¬P)

[Dr Faustus]

To be completely rigorous, by contradiction we must conclut that there exist at least one document that is not infaillible.

That is not a valid inference.

To claim "at least one document that is not infallible" you must produce at least one such document.

Not all balls in the bag are blue doesn't mean any are green.

You mean one which is not autoreferenced ?

If i produce a document where i say that, a=a, this is a valid document right ?

Valid document are just those which respect logic rules, rules of identity, there is no big mistery inside it.

Real things does not always follow the images that we have of its in mind. That's a problem that logic can not really solve.

[Skepdick]

You mean one which is not autoreferenced ?

No, I mean one which is necesasrily true.

If i produce a document where i say that, a=a, this is a valid document right ?

Not necessarily.

There exists a logic such that: For some x: x = x, for some y: MAYBE y = y; and for some z: z != z

Equality/identity is simply not well-defined.

https://en.wikipedia.org/wiki/Schr%C3%B6dinger_logic

Valid document are just those who respect logic rules

With respect to WHICH logic rules. There are infinitely many rule-based systems.

rules of identity, there is no big mistery inside it.

The mystery is why you believe in identity.

Real things does not always follow the images that we have of them in mind. That's a problem that logic can not really solve.

Logic doesn't even follow the image you have of it in your own head.

That's a problem you can't solve.