There are infallible documents

[godelian]

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.

[Skepdick]

Imagine the opposite:

There are no infallible documents.

...

Your entire confusion stems from thinking in "opposites".

Imagine a bag full of balls. There are NO red balls in the bag isn't "the opposite" of "There are red balls in the bag".

You fail to distinguish between constructive and destructive negation.

Having found no infallible documents you construct the claim "there are no infallible documents".
This construction about other documents doesn't go back into the bucket of "documents"; any more than the construction "there are no red balls" falls back into the bucket of balls.

A judgment ABOUT documents is not a document.The confusion arises once you misinterpret it as a document.

Basic category error.

Hence, by contradiction, we must conclude:

There are infallible documents.

Proofs by contradiction are not constructively valid.

The very reason you arrive at the conclusion is because you think in "opposites". Either X is true; or not-X is true. If One of them is false - the other is true.

Learn to distinguish proofs of negation (which are classical proofs by contradiction): Start with A. Find contradiction. Therefore not-A
From this (which is NOT deductively sound): assume not-A. Find contradiction.. Therefore A.

Your reasoning is the latter kind - it's invalid.

Boolean logic is so deeply ingrained in your brain you can't escape fooling yourself.

[godelian]

Boolean logic is so deeply ingrained in your brain you can't escape fooling yourself.

The law of the excluded middle is perfectly viable in this case. There is no reason to believe that the problem is undecidable.

[Skepdick]

Boolean logic is so deeply ingrained in your brain you can't escape fooling yourself.

The law of the excluded middle is perfectly viable in this case. There is no reason to believe that the problem is undecidable.

By insisting that judgments ABOUT documents are themselves documents you've rendered it undecidable.

At best you've introduced a triviality.

The only known infallible document in the set of all documents is this one.

You are just jerrymandering categories.

If the judgment ABOUT documents is outside the set of documents then you get: "There are no infallidble documents"
If the judgment ABOUT documents is inside the set of documents then you get: "The only known infallible document is this one."

Not quote the slam-dunk you were going for, is it?

[godelian]

By insisting that judgments ABOUT documents are themselves documents you've rendered it undecidable.

A judgment about a document is indeed itself also a document. Why would such judgment be undecidable?

[Skepdick]

By insisting that judgments ABOUT documents are themselves documents you've rendered it undecidable.

A judgment about a document is indeed itself also a document. Why would such judgment be undecidable?

Well, what makes the "infalibility" of a document that asserts its own infalibility decidable?

[godelian]

Well, what makes the "infalibility" of a document that asserts its own infalibility decidable?

That would be redundant. Every theory implicitly claims that its axioms are true. In fact, they are, within the context of that theory.

[Skepdick]

Well, what makes the "infalibility" of a document that asserts its own infalibility decidable?

That would be redundant. Every theory implicitly claims that its axioms are true. In fact, they are, within the context of that theory.

Yeah, that's not how it works. You need a meta-theory to reason about; and prove any properties of your base theory.
You can't have a document prove its infallibility property from within.

So which is your theory, and which is your meta-theory?

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

all non-trivial semantic properties of programs are undecidable.

...such as "infallibility"

Also... your perpetual use of "therefore" sure sounded like you considered the statement a theorem, not an axiom.

[godelian]

Yeah, that's not how it works.

That is how it works. A theory considers its own axioms to be true.

[godelian]

...such as "infallibility"

The term "Infallible" is just a synonym for "true".

[Impenitent]

infallible documents? even when folded into paper airplanes, they all fall eventually

-Imp

[godelian]

infallible documents?

The axioms of a theory are considered true by that theory. For example, the axioms of Euclidean geometry are true in Euclidean geometry.

The term "infallible" is just a synonym for "true" in the context of a theory. Truth exists but it is clearly a Platonic abstraction.

[Impenitent]

the best axioms cut down trees with one strike...

-Imp

[Skepdick]

...such as "infallibility"

The term "Infallible" is just a synonym for "true".

Potato/potatoh.

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

all non-trivial semantic properties of programs are undecidable.

...such as truth-value.

[Skepdick]

That is how it works.

No. It isn't.

A theory considers its own axioms to be true.

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.