A clerical doctrine such as Christianity has no model

[Belinda]

And it's why Christianity is a better religion than Islam.

As long as Christianity revolves around clerical doctrines invented by its church, it will remain a worthless religion.

ChatGPT: What was the object of the reformation?

The object of the Reformation was to reform the beliefs and practices of the Roman Catholic Church, which many saw as corrupt and inconsistent with biblical teachings.

The only way to get rid of the corruption of church and clergy, is not to have a church at all.

Reformed Christianity still has a church and is as clerical as ever. They did not solve the problem at all.

Islam has no church. That is why Islam is the better religion.

Repeat after me: A religion with a church is a bad religion.

It's human nature to institutionalise codes of conduct. There is a scale of how authoritarian clerics are . The Pope is ,I suppose, pretty near the top of the scale: the Society of Friends (Quakers) at the bottom of the scale. Humanists and Unitarians are like the Quakers as to authoritarianism. Islam has no Pope but Muslims continue to hold a strong attachment to the Koran , Muhmmad , and traditional praxis.

Christianity is saved from undue clericalism by the moving icon of Jesus. Muhammad is not a moving icon for traditional Muslims as Muhammad does not move to serve times and seasons

Iconic Jesus combines the Dionysian and the Apollonian for each successive age and cultural change, so avoiding moral and fallacious aberrations such as Nazism

[Gary Childress]

What about Islam is appealing to you? Or to put it another way, what do you find draws you to Islam? Or why do you find it more appealing than any other religion?

Islam is a fantastically useful counterargument to the fake moralizations of the godless vermin. Most western people have no idea of how incredibly imbecile their views are on right and wrong. It is not even a legitimate system in a mathematical sense. Whether it is Christian doctrine or atheism, their views are not even closed under logical consequence.

I needed a tool to slag off their bullshit, and Islam turned out to be incredibly useful for that purpose. I was looking for a hammer to hit a nail, and after the search, I found one. If Islam did not exist, we would have to invent it. Islam is incredibly fit for purpose.

In the end, my core argument remains mathematical. Islam clearly has a model. Christianity has no model. Atheism does not even attempt to have a model. So, according to Tarski's semantic theory of truth, Christianity and atheism are complete bullshit. Their views are stupid. They do not satisfy the requirements of a legitimate theory. It does not work and it will never work.

To each their own, I guess. I don't worship God because I don't worship evil beings that create evil worlds of pain and suffering. Do you think that if you worship God you will secure a spot in some sort of heavenly afterlife? Is that why you worship God? Or what is the point of worshiping a God who will just as easily put good people through hell as give bad people joy? Or are you one of those who believe only the evil suffer and only the good do not?

[godelian]

Islam has no Pope but Muslims continue to hold a strong attachment to the Koran , Muhmmad , and traditional praxis.

The Koran is a document. It is a Platonic abstraction. It's perfectly fine to hold a strong attachment to an immutable abstract object.

The Pope, on the other hand, is a highly mutable and corruption-prone person. It is absolutely not safe at all to declare the Pope to be infallible or to hold such strong attachment to him.

Muhammad, may he rest in peace, is no longer with us. What we have left, is his legacy, i.e. the prophetic Sunnah. That is again an immutable abstract Platonic object, to which it is perfectly fine to hold a strong attachment.

Traditional praxis is again an immutable abstract Platonic object. It is to living persons that you cannot give too much power. Islam recognizes human authority but within limits. Tradition is perfectly fine and can be trusted because it is a praxis and not a person in authority.

The highly corruption-prone Christian clergy keeps inventing its inconsistent doctrine along the way. I do not trust their inventions. Luther was spot on when he pointed out the following:

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.

In matters of religion, you can only trust immutable abstract Platonic objects. You cannot trust corruption-prone individuals. You cannot trust their inventions.

[Gary Childress]

I'll never understand religious fundamentalists. I don't see how anyone would want to worship the God of this shit world.

[godelian]

I'll never understand religious fundamentalists. I don't see how anyone would want to worship the God of this shit world.

That is actually not what it is about.

We all somehow know that particular behaviors are okay while other behaviors are simply bad.

Even as an atheist, you undoubtedly acknowledge that behavior can be good or evil.

Therefore, the real question that arises, is:

Are your beliefs about good and bad behavior actually closed under logical consequence?

If not, then your beliefs on the matter are mere bullshit.

I just wanted to point out that the Christian beliefs on the matter are bullshit. Same for the atheist beliefs on the matter. They are also complete bullshit. None of that is closed under logical consequence. Therefore, these beliefs are mere crap.

Deductive closure is not optional.

A set of beliefs, i .e. a theory, must be closed under logical consequence, or else it is just worthless nonsense.

[Gary Childress]

A set of beliefs, i .e. a theory, must be closed under logical consequence, or else it is just worthless nonsense.

Not really. But believe what you want.

[godelian]

A set of beliefs, i .e. a theory, must be closed under logical consequence, or else it is just worthless nonsense.

Not really. But believe what you want.

ChatGPT: Is it desirable that a theory is closed under logical consequence?

Yes, it is generally desirable for a theory to be closed under logical consequence, particularly in formal logic and mathematics. Here’s why:

Definition of a Theory: A theory T in logic is typically defined as a set of sentences (axioms) that is closed under logical consequence. This means that if T contains a set of axioms and a sentence logically follows from those axioms, then that sentence should also be in T. This ensures that the theory is deductively complete in the sense that it contains all truths derivable from its assumptions.

Logical Consistency: Closure under logical consequence prevents arbitrary additions to a theory that could lead to inconsistencies. If a theory were not closed under logical consequence, it could fail to account for all the necessary implications of its own axioms.

Predictive Power: In scientific and mathematical contexts, we expect theories to generate reliable predictions or theorems. If a theory were not closed under logical consequence, it might fail to provide all the conclusions that should naturally follow from its axioms.

Formal Systems: In formal logic, a deductively closed theory (sometimes called a "complete" or "maximal" theory in certain contexts) ensures that all provable statements are included. However, in some cases, we work with non-maximal theories (e.g., Peano arithmetic is not complete), but these are still closed under consequence within first-order logic.

ChatGPT: Is it acceptable for a theory to be inconsistent?

In general, an inconsistent theory is problematic because it allows for explosion—in classical logic, if a theory contains a contradiction, then anything can be proven from it (this is known as the principle of ex contradictione quodlibet). This makes the theory useless for reasoning because it fails to distinguish between true and false statements.

[Gary Childress]

You said:
If a theory isn't closed under logical consequence is it just nonsense?
ChatGPT said:
ChatGPT
Not necessarily. If a theory isn't closed under logical consequence, it just means that it isn't deductively complete—there are true statements (within the intended interpretation of the theory) that it fails to entail. That doesn't automatically make the theory nonsense.

Many working scientific, philosophical, and mathematical theories are not closed under logical consequence, either because they are incomplete (they don't include all the true statements that follow from their axioms) or because they are being used in a practical, non-idealized form where full closure would be computationally or epistemically infeasible.

For example:

Peano arithmetic (PA) is not complete—by Gödel’s incompleteness theorems, there are true statements about natural numbers that it does not prove.
Scientific theories often operate with an evolving set of assumptions and are rarely formalized in a way that ensures closure under all logical consequences.
Everyday reasoning is typically not closed under logical consequence because we do not (and cannot) explicitly infer everything that follows from what we know.
So, a theory that isn't closed under logical consequence might still be meaningful, useful, and rationally defensible. It just may need refinement, expansion, or supplementation to capture all the relevant truths in its domain.

[godelian]

Many working scientific, philosophical, and mathematical theories are not closed under logical consequence, either because they are incomplete (they don't include all the true statements that follow from their axioms) or because they are being used in a practical, non-idealized form where full closure would be computationally or epistemically infeasible.

In a deductive system, incompleteness is fine but lack of deductive closure is not:

ChatGPT: Is deductive closure the same as incompleteness?

No, deductive closure and incompleteness are distinct concepts in logic and mathematics, though they relate to each other in some ways.

Deductive Closure

A theory T is deductively closed if, whenever a statement φ is logically implied by T, then φ is already in T. In other words, a deductively closed theory contains all statements that can be derived from its axioms using the rules of inference.

Formally, if T⊢φ (meaning φ is provable from T), then φ∈T.

Incompleteness

A formal system is incomplete (in the Gödelian sense) if there exists a statement φ in the language of the system such that neither φ nor its negation ¬φ is provable within the system. That is, the system cannot determine the truth of every possible statement expressible within its language.

Gödel’s first incompleteness theorem states that any consistent, sufficiently expressive formal system (like Peano Arithmetic) has true statements that are unprovable within the system.

Key Differences

Deductive closure means a system contains everything that follows from its axioms.
Incompleteness means a system does not prove all true statements expressible in its language.

Are They Related?

Yes, but they are not the same:

A theory can be deductively closed yet incomplete. For example, Peano Arithmetic (PA) is deductively closed (if something is provable in PA, it is in PA) but incomplete (some true statements, like Gödel sentences, are unprovable).

An incomplete theory is not necessarily deductively closed, depending on how it is defined. If a theory is not closed under deduction, it may be missing statements that logically follow from its axioms.

Would you like a more specific example or clarification?

[Gary Childress]

Many working scientific, philosophical, and mathematical theories are not closed under logical consequence, either because they are incomplete (they don't include all the true statements that follow from their axioms) or because they are being used in a practical, non-idealized form where full closure would be computationally or epistemically infeasible.

In a deductive system, incompleteness is fine but lack of deductive closure is not:

ChatGPT: Is deductive closure the same as incompleteness?

No, deductive closure and incompleteness are distinct concepts in logic and mathematics, though they relate to each other in some ways.

Deductive Closure

A theory T is deductively closed if, whenever a statement φ is logically implied by T, then φ is already in T. In other words, a deductively closed theory contains all statements that can be derived from its axioms using the rules of inference.

Formally, if T⊢φ (meaning φ is provable from T), then φ∈T.

Incompleteness

A formal system is incomplete (in the Gödelian sense) if there exists a statement φ in the language of the system such that neither φ nor its negation ¬φ is provable within the system. That is, the system cannot determine the truth of every possible statement expressible within its language.

Gödel’s first incompleteness theorem states that any consistent, sufficiently expressive formal system (like Peano Arithmetic) has true statements that are unprovable within the system.

Key Differences

Deductive closure means a system contains everything that follows from its axioms.
Incompleteness means a system does not prove all true statements expressible in its language.

Are They Related?

Yes, but they are not the same:

A theory can be deductively closed yet incomplete. For example, Peano Arithmetic (PA) is deductively closed (if something is provable in PA, it is in PA) but incomplete (some true statements, like Gödel sentences, are unprovable).

An incomplete theory is not necessarily deductively closed, depending on how it is defined. If a theory is not closed under deduction, it may be missing statements that logically follow from its axioms.

Would you like a more specific example or clarification?

If you want to move the goal posts until you find something that will stick to the wall, then that's fine. If you want to believe this world is a wonderful place created by a wonderful God, then I disagree. I'm sorry.

[godelian]

ChatGPT: mathematical theories are not closed under logical consequence, either because they are incomplete
versus
ChatGPT: A theory can be deductively closed yet incomplete. For example, Peano Arithmetic.
If you want to move the goal posts until you find something that will stick to the wall, then that's fine.

ChatGPT wrote something seemingly inconsistent, related to the subtle distinction between "closed under logical consequence" and "deductively closed":

ChatGPT: Are the terms "deductively closed" and "closed under logical consequence" the same?

The terms "deductively closed" and "closed under logical consequence" are closely related and often used interchangeably, but they have subtle distinctions depending on context.

Similarities

Both terms describe a set of statements S that includes all the statements that logically follow from it. Formally, a set S is:
Deductively closed / Closed under logical consequence  ⟺  (∀P)(S⊢P⇒P∈S).
This means that if P is a theorem (i.e., provable from S), then P is already in S.

Key Differences

Terminology Origins

"Closed under logical consequence" is more commonly used in model theory and philosophical logic, where the focus is on whether all truths that follow from a set of premises are included.

"Deductively closed" is more commonly used in proof theory and mathematical logic, where the emphasis is on formal proof systems.

Implication vs. Proof

"Closed under logical consequence" often refers to semantic entailment (truth in all models), meaning that if a statement is true in every model satisfying the premises, then it should be included in the set.

"Deductively closed" usually refers to syntactic derivability (proofs in a formal system), meaning that if a statement is provable from the axioms, it should be included.

Application in Mathematics

If a system is "closed under logical consequence," it contains every statement that follows from it in any valid interpretation.
If a system is "deductively closed," it contains every statement that can be formally derived from it.
Incomplete formal systems (e.g., Peano Arithmetic) are not deductively closed because there exist true statements that cannot be proven within them.

Are They the Same in Practice?

In an idealized setting where we assume a complete and sound system, the two terms would be equivalent.
In real mathematical logic, they can differ because truth (logical consequence) and provability (deductive closure) are not always the same—due to incompleteness.

Conclusion

If provability and truth coincide (e.g., in a complete system), then being "deductively closed" and "closed under logical consequence" mean the same thing. If the system is incomplete, then a set may be closed under logical consequence but not deductively closed in terms of what can actually be proven within the system.

A mathematical theory such as Peano Arithmetic would be "closed under logical consequence" but not "deductively closed". But then again, ChatGPT happily flips both terms around as well:

ChatGPT (previously): For example, Peano Arithmetic (PA) is deductively closed (if something is provable in PA, it is in PA) but incomplete (some true statements, like Gödel sentences, are unprovable).

ChatGPT: Is Peano Arithmetic deductively closed?
Yes, Peano Arithmetic (PA) is not deductively closed. // --> (yes or no?)

I asked the question again:

ChatGPT: Is Peano Arithmetic deductively closed?
Yes, Peano Arithmetic (PA) is deductively closed.

ChatGPT: Is Peano Arithmetic closed under logical consequence?
Yes, Peano Arithmetic (PA) is closed under logical consequence in the sense that if a statement can be logically derived from the axioms of PA, it is a logical consequence of PA.

Weird.

[Gary Childress]

Fascinating stuff. The world is still a shit show.

[godelian]

Fascinating stuff.

It is about the subtle interaction between "⊢" (provability) and "⇒" (implication, which is about truth and not about provability) and "∈" (a true statement being an element in a particular set).

While (in PA) all provable statements are true (soundness), not all true statements are provable (incompleteness). This does indeed create some lack of "closure" of sorts, depending on how you look at it.

But then again, in my opinion the terms "closed under logical consequence" and "deductively closed" usually mean the same thing. ChatGPT is exaggerating by insisting that they would be different. ChatGPT does not manage to consistently distinguish between the two terms either.

[1] P ⇒ Q (P implies Q)
[2] P ⊢ Q (P implies Q and you can even prove it)

The term "closed under logical consequence" is used in context [1] while "deductively closed" is used in context [2].

Well, yeah.

When P and Q are true statements that are not provable, you cannot use "P ⊢Q", but given the nature of mathematics, you won't be able to say anything about "P ⇒ Q" either. In theory, it is possible that "P ⇒ Q", but it will be impossible to prove. Hence, the problem of distinguishing between both does not even occur.

[Gary Childress]

Fascinating stuff.

It is about the subtle interaction between "⊢" (provability) and "⇒" (implication, which is about truth and not about provability) and "∈" (a true statement being an element in a particular set).

While (in PA) all provable statements are true (soundness), not all true statements are provable (incompleteness). This does indeed create some lack of "closure" of sorts, depending on how you look at it.

But then again, in my opinion the terms "closed under logical consequence" and "deductively closed" usually mean the same thing. ChatGPT is exaggerating by insisting that they would be different. ChatGPT does not manage to consistently distinguish between the two terms either.

[1] P ⇒ Q (P implies Q)
[2] P ⊢ Q (P implies Q and you can even prove it)

The term "closed under logical consequence" is used in context [1] while "deductively closed" is used in context [2].

Well, yeah.

When P and Q are true statements that are not provable, you cannot use "P ⊢Q", but given the nature of mathematics, you won't be able to say anything about "P ⇒ Q" either. In theory, it is possible that "P ⇒ Q", but it will be impossible to prove. Hence, the problem of distinguishing between both does not even occur.

For what it's worth, I wish you well in your pursuit of Islam. I hope it brings you peace and whatever else you are looking for from it. I have my doubts about pretty much all religions. To me, religions are little more than systems of unjustifiable discrimination. But that's maybe something I am going to perceive out of it that others don't perceive.

[godelian]

For what it's worth, I wish you well in your pursuit of Islam.

I use Islam a lot to slag off fake western morality. I find it very useful for that purpose. It is a fantastic tool for the purpose of criticizing and denouncing western bullshit.

Islam is indeed also supposed to be a goal and not just an instrument. I probably do not spend enough time on that. I probably should. It certainly has benefits to do that but unfortunately it does not bring me the same level of satisfaction.

What brings me the most satisfaction, is to point out that the western take on what is right and what is wrong behavior, is complete bullshit. Islam is mostly an example to provide evidence that it can actually be done correctly. Morality does not need to be just a pile of shit.