Philosophy Now Forum

godelian wrote: ↑Wed Feb 19, 2025 12:52 pm It is in classical logic. I agree, however, that there are constructivist concerns. I am still trying to figure out what to do with this concern. I am currently looking at how the constructivist version of the proof for Gödel's first incompleteness theorem was rephrased away from its original version in classical logic. It may contain a trick as to how to solve the problem.

I’ll tell you why: because, when I teach Gödel to computer scientists, I like to sidestep the nasty details of how you formalize the concept of “provability in F.” (From a modern computer-science perspective, Gödel numbering is a barf-inducingly ugly hack!)

Instead, I simply observe Gödel’s Theorem as a trivial corollary of what I see as its conceptually-prior (even though historically-later) cousin: Turing’s Theorem on the unsolvability of the halting problem.

For those of you who’ve never seen the connection between these two triumphs of human thought: suppose we had a sound and complete (and recursively-axiomatizable, yadda yadda yadda) formal system F, which was powerful enough to reason about Turing machines. Then I claim that, using such an F, we could easily solve the halting problem. For suppose we’re given a Turing machine M, and we want to know whether it halts on a blank tape. Then we’d simply have to enumerate all possible proofs in F, until we found either a proof that M halts, or a proof that M runs forever. Because F is complete, we’d eventually find one or the other, and because F is sound, the proof’s conclusion would be true. So we’d decide whether M halts. But since we already know (thanks to Mr. T) that the halting problem is undecidable, we conclude that F can’t have existed.

“Look ma, no Gödel numbers!”

Skepdick wrote: ↑Wed Feb 19, 2025 12:52 pm You won't find one. The computational formulation of LEM is the CHOICE() function.

Skepdick wrote: ↑Wed Feb 19, 2025 12:52 pm The only reason you choose LEM is because you want to arrive at your conclusion. Which you can't do unless you assume LEM.

godelian wrote: ↑Wed Feb 19, 2025 12:56 pm

Skepdick wrote: ↑Wed Feb 19, 2025 12:52 pm You won't find one. The computational formulation of LEM is the CHOICE() function.

It worked absolutely fine for Gödel's incompleteness theorem. The original version is in classical logic. There are constructivist versions of the proof available in the meanwhile.

Skepdick wrote: ↑Wed Feb 19, 2025 11:26 am

Dr Faustus wrote: ↑Wed Feb 19, 2025 10:45 am Yes, utility function based on the reality that encounter humain kind.

Utility functions are based on the users of reality, not on reality.

Dr Faustus wrote: ↑Wed Feb 19, 2025 4:01 pm

Skepdick wrote: ↑Wed Feb 19, 2025 11:26 am

Dr Faustus wrote: ↑Wed Feb 19, 2025 10:45 am Yes, utility function based on the reality that encounter humain kind.

Utility functions are based on the users of reality, not on reality.

Reality of “users”, “users” are part of reality. there is no such thing as absolute knowledge. knowledge is the work of concrete beings, not of a universal consciousness floating in the air.

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.

Dr Faustus wrote: ↑Wed Feb 19, 2025 4:01 pm

Skepdick wrote: ↑Wed Feb 19, 2025 11:26 am

Dr Faustus wrote: ↑Wed Feb 19, 2025 10:45 am Yes, utility function based on the reality that encounter humain kind.

Utility functions are based on the users of reality, not on reality.

Reality of “users”, “users” are part of reality. there is no such thing as absolute knowledge. knowledge is the work of concrete beings, not of a universal consciousness floating in the air.

Eodnhoj7 wrote: ↑Thu Feb 20, 2025 2:33 am I am not aware of any strict religious or philosophical system that does not contain paradox of the various forms...and this includes the nature of human experience for that matter.

I think, at least for the recent years it has been this way, paradox should be an integral part of rational frameworks to maintain a sense of holistic integrity while being quite resolutely clear that paradox is unavoidable.

I think the "anti-paradox" mentality of the west has been a self defeating mindset.

godelian wrote: ↑Thu Feb 20, 2025 3:15 am

Eodnhoj7 wrote: ↑Thu Feb 20, 2025 2:33 am I am not aware of any strict religious or philosophical system that does not contain paradox of the various forms...and this includes the nature of human experience for that matter.

I think, at least for the recent years it has been this way, paradox should be an integral part of rational frameworks to maintain a sense of holistic integrity while being quite resolutely clear that paradox is unavoidable.

I think the "anti-paradox" mentality of the west has been a self defeating mindset.

Quite a few paradoxes in mathematics, such as the liar paradox or Russell's paradox, are actually not paradoxical, if you allow the tetralemma:

- true
- false
- not true nor false <--------
- true and false

The Buddhist Catuskoti allows for two extra truth values, which allows to venture into seemingly paradoxical territory.

The liar paradox and Russell's paradox are "not true nor false". In my opinion, that is even intuitively perfectly fine.

If the truth value "not true nor false" is allowed, however, then for matters of deductive closure, its negation, "true and false", should be allowed as well. That is pretty much inevitable. However, this value truly is contradictory.

The "anti-paradox" mindset is actually also understandable.

The tetralemma leads to the similar concerns as constructivist logic does. The resulting arithmetic is fraught with obstacles. For example, when ¬¬p is no longer automatically equal to p, the calculation may prematurely come to a halt. So, it may lead to fewer solvable problems instead of more.

Most software is not purely boolean. Programming languages typically allow for (true, false, null) with null meaning "neither true nor false".

However, they usually evaluate "not null" as "true". In my opinion, this is wrong. The Buddhist answer is that "not null" is "both true and false ". No programming language will, however, allow this value. Their solution, to map it to the wrong value, is in fact not a solution either.

If the value is not null, it is necessarily one of the other values, i.e. true or false. However, "true or false" is technically also "true". According to the Buddhist Catuskoti, however, the opposite of "neither true nor false " is "true and false", which is technically false.

So, all of this has degenerated into a widely implemented software bug.

In my opinion, they should rather treat "not null" as an error, in a manner similar to a division by zero. They currently just suppress the error by silently turning it into "true". That would amount to silently converting a division by zero as if it were zero instead of raising an error.

I think that this approach is plain wrong.

In my opinion, the solution is to allow "neither true nor false" by means of "null" and to treat "not null" as an error.

An alternative would be to leave"not null" unevaluated. The answer to a yes/no question would then be allowed to be: not null, i.e. we know that there is an answer to your question but we don't know this answer. This probably sounds like eastern mysticism. There are actually very valid logical reasons why eastern mysticism sounds the way it does.

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.

My ChatGpt wrote
The core argument in the passage attempts to demonstrate, by contradiction, that infallible documents must exist. However, this reasoning is flawed due to an equivocation on the term "infallible."

Analysis of the "Contradiction" Argument
The argument presented runs as follows:
Assume there are no infallible documents.
Write this claim in a document: "There are no infallible documents."
This document itself would then be fallible.
If the document is fallible, then its claim may be false.
If its claim is false, then there must be at least one infallible document.
Thus, the assumption that there are no infallible documents leads to contradiction.
Therefore, infallible documents must exist.
This argument is a version of a self-referential paradox, but it does not actually prove the existence of infallible documents. Here’s why:

The conclusion that "this document is fallible" does not logically entail that its single claim is false. A document being "fallible" means that it could contain falsehoods, not that it necessarily does.
The leap from fallibility (i.e., the possibility of error) to falsehood (i.e., actual error) is unwarranted.
The paradoxical structure does not force us to accept that an infallible document exists, only that the notion of an absolute negative claim about infallibility leads to a self-referential issue.
This is akin to the "Liar Paradox" and does not prove the positive existence of an infallible document any more than "This sentence is false" proves the existence of true contradictions.

Does Christianity Reject Infallible Documents?
The claim that Christianity "does not believe in the existence of infallible documents but in the existence of infallible Popes" is inaccurate. Catholic doctrine holds that the Bible is divinely inspired and inerrant in matters of faith and morals. Protestant traditions, like that of Martin Luther, affirm sola scriptura, the belief that the Bible is the sole infallible authority.

The counter-quote from the Catholic side—"The Bible itself is the arsenal whence each heresiarch from the past has drawn his deceptive arguments"—does not mean that Catholicism denies biblical infallibility. It simply acknowledges that interpretations of the Bible can vary, leading to doctrinal disputes.

"Cannot Possibly Be Closed Under Logical Consequence"
The assertion that "Christian doctrine cannot possibly be closed under logical consequence" is unclear and needs unpacking. If it means:

Christian doctrine is inconsistent—this is a strong claim that would require demonstrating an explicit contradiction within dogmatic teachings.
Christian doctrine does not include all its logical consequences—this is likely true in the sense that theology often maintains mysteries or paradoxes that are not fully resolved within human logic.
Christian doctrine does not follow from purely logical axioms—this is trivially true, as religious belief is based on revelation, faith, and tradition, not just logical derivations.

Papal Infallibility and Framework Dependence
The doctrine of Papal Infallibility is not a claim that the Pope is always infallible in all statements. Rather, within the Catholic faith-system (theistic-catholic-FS), infallibility is a conditionally applied principle—when the Pope speaks ex cathedra on matters of faith and morals, he is considered infallible.

Since this is a doctrine internal to a specific theological framework, it does not violate logical rules per se. Logical validity applies within frameworks; if one rejects the Catholic FS, then Papal Infallibility is simply not a binding rule.

Conclusion
The contradiction argument for infallible documents is flawed due to a misapplication of fallibility.
Christianity does recognize infallible texts (e.g., the Bible) in various traditions.
The claim that Christian doctrine is "not closed under logical consequence" is ambiguous and needs specification.
Papal Infallibility, as a framework-dependent condition, does not violate logic—it holds only within the Catholic FS.

Eodnhoj7 wrote: ↑Thu Feb 20, 2025 2:33 am I think the "anti-paradox" mentality of the west has been a self defeating mindset.

Eodnhoj7 wrote: ↑Thu Feb 20, 2025 2:39 am If there is no absolute knowledge, then the knowledge that there is "no absolute knowledge" is not absolute thus relative and as relative is sometimes true and sometimes false thus necessitating absolute truth as relative as absolute truth is contextual.

godelian wrote: ↑Thu Feb 20, 2025 3:15 am In my opinion, the solution is to allow "neither true nor false" by means of "null" and to treat "not null" as an error.

Skepdick wrote: ↑Thu Feb 20, 2025 6:45 am

godelian wrote: ↑Thu Feb 20, 2025 3:15 am In my opinion, the solution is to allow "neither true nor false" by means of "null" and to treat "not null" as an error.

Why would "not null" be an error? If "null" is an acceptable value e.g you can compare it using the equality operator, then surely you can assert whether a given value is; or isn't null.

You don't seem to distinguish between an expression (neither true nor false) and its value (null).

The negation of "neither true nor false" should indeed be "both true and false" in the tetralemma's logic.
But NOT(null) is simply asking "is this value not null?" - which is a different kind of operation. It's checking for the presence/absence of a value, not performing logical negation on the semantic meaning that null represents.

https://en.wikipedia.org/wiki/Three-valued_logic

In these truth tables, the unknown state can be thought of as neither true nor false in Kleene logic, or thought of as both true and false in Priest logic. The difference lies in the definition of tautologies.

godelian wrote: ↑Thu Feb 20, 2025 7:19 am ...

#include <stdbool.h>
#include <stdlib.h>

bool trinary(void) {
    while (1) {}
    return rand() % 2 == 0;
}