Re: There are infallible documents
Posted: Wed Feb 19, 2025 8:48 am
First-order logic is itself an axiomatic system. The proof is valid in first-order logic, assuming the LEM.
For the discussion of all things philosophical.
https://canzookia.com/
First-order logic is itself an axiomatic system. The proof is valid in first-order logic, assuming the LEM.
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 wrote: ↑Wed Feb 19, 2025 6:41 am https://en.wikipedia.org/wiki/Rice%27s_theorem...such as truth-value.all non-trivial semantic properties of programs are undecidable.
Why are you assuming LEM, and not proving it?
Precisely. But you aren't peddling "there are infallible documsnts" as axiomatic.
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.
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.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 prove that the opposite is contradictory.
Good question. If you formalize it, it certainly uses the universal quantifier in first-order logic. So, there is a requirement of language.
Fine.
Non-sequitur.
That's just called "motivated reasoning"
Negation isn't "oppsite". It's just negation. Treating negation as opposite/complementary is LEM.
Probably because English isn't a 1st order language...
There is no need to use a truth predicate to prove this statement.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.
Non-sequitur. A statement that is false in general may still be true in particular.godelian wrote: ↑Wed Feb 19, 2025 9:38 amThere is no need to use a truth predicate to prove this statement.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.
A non-trivial property is one which is neither true for every program, nor false for every program.
It does.
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.
It doesn't need to be generally true.
You are proving my point.
To be completely rigorous, by contradiction we must conclut that there exist at least one document that is not infaillible.godelian wrote: ↑Sat Feb 15, 2025 4:21 am 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.
That is not a valid inference.Dr Faustus wrote: ↑Wed Feb 19, 2025 9:46 am To be completely rigorous, by contradiction we must conclut that there exist at least one document that is not infaillible.
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.godelian wrote: ↑Wed Feb 19, 2025 9:49 amChatGPT: ∀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.
You mean one which is not autoreferenced ?Skepdick wrote: ↑Wed Feb 19, 2025 9:47 amThat is not a valid inference.Dr Faustus wrote: ↑Wed Feb 19, 2025 9:46 am To be completely rigorous, by contradiction we must conclut that there exist at least one document that is not infaillible.
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.
No, I mean one which is necesasrily true.
Not necessarily.Dr Faustus wrote: ↑Wed Feb 19, 2025 10:01 am If i produce a document where i say that, a=a, this is a valid document right ?
With respect to WHICH logic rules. There are infinitely many rule-based systems.
The mystery is why you believe in identity.
Logic doesn't even follow the image you have of it in your own head.Dr Faustus wrote: ↑Wed Feb 19, 2025 10:01 am 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.