Which is to say you simply do not know the difference between the two. Try this, 2+2=4. Repesenting a fact or a truth? — tim wood

The ONLY way that we know that {cats are animals} is that it is stipulated to be true on the basis of the assigned meaning of the words.

2+2=4 is derived from Peano Arithmetic axioms, thus (2) derived from (1).

The distinction here is between what is a fact and what is true. Notwithstanding that the terms are often not distinguished and used interchangeably, they are not the same thing, and you are stumbling badly over that. — tim wood

This is ALL there is to expressions of language that are true on the basis of their meaning
(1) Some expressions of language are stipulated to be true thus providing semantic meaning to otherwise totally meaningless finite strings. These expressions are the set of facts.
(2) Some expressions of language are derived by applying truth preserving operations to (1).

Epistemological antinomies belong to neither (1) nor (2) thus are merely untrue.

Copyright 2023 and earlier PL Olcott

I got this book as well. It just arrived. It seems Prolog is a great logic program language which is built on FOL and HOL. An ideal PL for AI applications. — Corvus

The key most important thing about Prolog is that Gödel's incompleteness can not be implemented in Prolog. Unprovable simply means untrue. Since this <is> the way that every expression of language that is {true on the basis of its meaning} derives its truth it shows that Gödel's incompleteness is a misconception.

...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:43-44)

In Prolog epistemological antinomies are simply rejected as malformed:

?- LP = not(true(LP)).
LP = not(true(LP)).

?- unify_with_occurs_check(LP, not(true(LP))).
false.

BEGIN:(Clocksin & Mellish 2003:254)
Finally, a note about how Prolog matching sometimes differs from the
unification used in Resolution. Most Prolog systems will allow you to
satisfy goals like:

equal(X, X).
?- equal(foo(Y), Y).

that is, they will allow you to match a term against an uninstantiated
subterm of itself. In this example, foo(Y) is matched against Y,
which appears within it. As a result, Y will stand for foo(Y), which is
foo(foo(Y)) (because of what Y stands for), which is foo(foo(foo(Y))),
and so on. So Y ends up standing for some kind of infinite structure.
END:(Clocksin & Mellish 2003:254)

You know the cows eat house bricks is false from common sense.
https://en.wikipedia.org/wiki/Common_sense
You don't need to generalise it to find out it is false. But generalisation and abstraction is what FOL and HOL are for the computability of ordinary language. — Corvus

That set of facts that comprise the actual model of the real world is the basis.
This includes common sense and also details that almost everyone does not know.

It sounds a weak argument for your point. Some surly confused guy calling an honest man dishonest doesn't make the honest man dishonest. Likewise some obtuse man making totally irrelevant claims wouldn't have anything to do with the fact that analytic truth is true or false. — Corvus

The actual model of the world is the basis. Facts not opinions.

↪PL Olcott I’m puzzled as to why you are posting on this amateur forum. Your ideas are groundbreaking and revolutionary. I urge you to submit your thesis to The American Philosophical Quarterly (or equivalent). If there is any validity to your ideas then of course they will print them and the name PL Olcott will be entered into the pantheon of famous philosophers along side with Aristotle, Kant, etc. Go for it PL! — EricH

When reviewing the actual publications in the field it seems that the greatest experts in the field are incapable of understanding that self-contradictory expressions are not true. This forum right here has proven to be the best forum for these things. People here (on this site) are generally very intelligent and thoughtful.

The world's leading experts all seem to be uniformly indoctrinated into believing that Gödel's words are utterly infallible. No one even bothered to notice that any expression asserting its own unprovability requires a sequence of inference steps that prove that they themselves do not exist.

"This sentence is not provable." — tim wood

To be proven requires a sequence of inference steps that prove that they themselves do not exist.

This is just like this adapted form of René Descartes: "I think therefore I am"
becomes: "I think therefore thoughts do not exist". (AKA nonsense).

But nowhere in that is Godel demonstrated wrong or foolish — tim wood

That he is simply too stupid to reject self-contradictory expressions makes him enormously foolish. On a related note he starved himself to death because he only trusted his wife's cooking. That he did not even trust his own cooking was quite psychotic.

This is helpful. But you have omitted a critical qualification: "[C]annot be proven or refuted" from the axioms of the system. But that the sentence in question is absolutely a truth bearer is established by meta-system argument. — tim wood

We can determine that the Liar Paradox applied to itself is true:
This sentence is not true: "This sentence is not true" is true.

yet we can only do this on the basis that the Liar Paradox is not a truth bearer, thus would have rejected it before it gets to the meta-level of analysis.

"This sentence is false" can be generalised into "Some sentence is false" which is not a contradiction. — Corvus

Likewise we can generalize cows eat house bricks into cows eat something.
Any nonsense sentence can be changed into a different sentence that is not nonsense.

I'm guessing in your system Godelian self-reference is simply ruled out, which you certainly can do. But that makes Godel neither wrong nor a fool, and to say he is simply means that one of you is both. — tim wood

¬TruthBearer(L,x) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
It does seem ridiculously stupid that a formal system could be construed as incomplete on the basis that it cannot prove self-contradictory expressions. We could analogously say that a baker that cannot bake a perfect angel food cakes using only house bricks for ingredients lacks baking skill.

I'm guessing in your system Godelian self-reference is simply ruled out, which you certainly can do. But that makes Godel neither wrong nor a fool, and to say he is simply means that one of you is both. — tim wood

The way that all self-contradictory sentences are ruled out is simple. Self-contradictory sentences cannot be proven or refuted from axioms thus are tossed out as non-truth bearers.
¬TruthBearer(L, x) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))

Your points seem to be confined in the domain of analytic truth only. In a domain where all in the domain is defined as truth regardless of the real world cases — Corvus

That is not true at all. If someone says that a {dog} <is> a fifteen story office building this is ruled as false because there are no {dogs} that <are> fifteen story office buildings in the actual world.

I am obliged to conclude that what you're writing about has nothing to do with Godel or any of Godel's ideas. But since you claim Godel has made "a ridiculously stupid mistake," it appears you do not know what Godel's ideas are about. Which is too bad, because all you have to do is just read the first section of his 1931 paper, which you cited, and you would have enough of an understanding of his ideas to know that he at least was not mistaken. And as well you might try reading your other citation. — tim wood

In other words you too simply don't understand that epistemological antinomies (AKA self-contradictory expressions) are simply not truth bearers thus have no idea why this statement is pure nonsense:

...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:43-44)

He explicitly limits his argument to systems of sufficient expressive power. He further notes that while his sample expression asserts its own unprovability while at the same time neither itself nor its negation are provable, that it must be true because it asserts its own unprovability. "The proposition undecidable in the system PM is thus decidable by metamathematical arguments." — tim wood

The terrible mistake is that no self-contradictory expression is any kind of propositional at all. Not only is it a mistake it is a ridiculously stupid mistake as if we spent 2000 years years trying to figure out whether this sentence is true or false: "What time is it".

Self-contradictory expressions are not truth bearers in the same way that questions are not truth bearers. Most of the best experts in the world are still trying to "resolve" the truth value of the liar paradox: "This sentence is not true".

A proposition is a central concept in the philosophy of language, semantics, logic, and related fields, often characterized as the primary bearer of truth or falsity. https://en.wikipedia.org/wiki/Proposition

You are not reading what comes after,
Analytic truth can be wrong — Corvus

{correct} is an aspect of the meaning of the term {truth} so analytic truth cannot possibly be wrong in any way what-so-ever. If it cannot possibly be wrong in any way what-so-ever then it cannot possibly be wrong in any specific way.

If it is "if it is wrongly defined" then it never was any sort of truth at all. If it was "applied to an incorrect model of the actual world", then again it never was any sort of truth at all. {Analytic truth} are only the subset of expressions of language that are actually true even if everyone in the universe disagrees about the truth of these expressions.

Analytic truth can be wrong — Corvus

That is like saying the integer five may not be any kind of number at all. Everything that is {incorrect} is excluded from the body of {truth}. That people make mistakes has no actual effect what-so-ever on truth itself. If everyone in the universe is certain that X is true and X is not true their incorrect belief does not change this.

As our Friend PL Olcott points out: "mathematics is incomplete." And that presumption profoundly affects the symbolic language (syntax) of logic in the First and Second Orders. Higher-order logic (HOL) is another animal altogether. — Rocco Rosano

I hypothesize that mathematics is only construed as "incomplete" based on a nonsense meaning of complete. When provable from axioms means true and the opposite is provable from axioms means false then everything else is simply untrue without any {incomplete} in-between.

Analytic truth can be wrong, when it contradicts the reality. The reality has potential possibility to be otherwise from status quo at any moment of time. Therefore AT has potential possibility of being wrong. — Corvus

I am stipulating that analytic truth are only those expressions of language that are a correct model of the actual world. It seems a little nutty to define it any other way.

Analytic truth can be true, but wrong. Can "wrong" be "true", and "right" be "false"? — Corvus

Saying that analytic truth can be wrong it like saying that kittens can be 15 story office buildings it cannot possibly ever happen. The closest thing to analytic truth being wrong is when we changed our mind on calling Pluto a Planet. Pluto was a planet until we changed the definition of the term: "Planet".

The ones I ordered are,
PROLOG ++: The Power of Object-oriented and Logic Programming" by C. Moss.
Prolog Programming for Artificial Intelligence by Bratko. — Corvus

Untrue unless provable from Facts does seem to be the correct model for the entire body of analytic truth. Analytic truth seems to be essentially nothing more than relations between finite strings.

I am waiting for a couple of cheap old Prolog books which are on my way. Prolog seems to be the system for Logic programming. It seems to be a PL which has a long history, but seems to be still very much popular even now especially for AI applications. — Corvus

I have the classic Clocksin and Mellish. https://www.amazon.com/Programming-Prolog-Using-ISO-Standard/dp/3540006788

I see. But my point was, isn't the main point of using HOL (as also mentioned in the OP title) is being able to set TF values to the non-existent truth value sentences or word such as "What time is it?", and make use of them in the real world applications? — Corvus

No, not at all. Nothing like that. I am only talking about HOL because everyone seems to be totally clueless about knowledge ontologies so I am using HOL as the closest analogy that people are familiar with. We can stipulate that {dogs are animals} as an axiom of our model of the actual world. We cannot stipulate that {dogs are fifteen story office buildings} as an axiom of the model of the actual world.

To create a formal system that never gets stuck in any paradox we have the Prolog/Wittgenstein notion of truth. Provable from Facts makes true. The opposite is provable from Facts makes false. Everything else is not a truth bearer.

In PASCAL or C, "What time is it?" can be set as True or False in a variable — Corvus

I am not talking about anything like that. I am referring to the (non-existent truth value of the) actual semantic meaning of the English sentence: What time is it?

It is easer for people to see that sentence has no truth value.
On the other hand it should not have taken humanity 2000 years
to continue to fail to understand that self-contradictory sentences
have no truth value.

The Liar Paradox in C++

int main()
{
bool Liar_Paradox = (Liar_Paradox == true);
}

Yes, it would be good if you could present the Tarski's and Godel's theorems in connection with HOL with your own explanations (the proofs and refutations) in clear English with added formulas too (if needed). — Corvus

This sentence is not true: "This sentence is not true" is true. The inner sentence is in Tarski's theory and the outer sentence is in his meta-theory (the next higher order of logic). Tarski never noticed that "This sentence is not true" is not a truth bearer thus the same ask asking is this sentence true or false: "What time is it?"

In simple English, please, please make clear just what Godel's terrible mistake was. — tim wood

An epistemological antinomy is really just a self-contradictory expression that has no truth value. He might as well of have because "What time is it?" cannot be proved true or false mathematics is incomplete.

Isn't HOL the expanded logical system from the other simpler ones with the relation and operation variables in the formulas? — Corvus

HOL applies to set of things at the next lower order of logic. Prolog works the way that analytic truth really works. If an expression can be proven from facts using rules then it is true. If it cannot be proven from facts using rules then it is untrue. An expression is only actually false when its negation can be proven from facts using rules.

This eliminates this terrible mistake by Gödel:

...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:43-44)

And eliminates the Liar Paradox basis of the Tarski Undefinability Theorem.

Wittgenstein agreed with the above:
https://www.liarparadox.org/Wittgenstein.pdf

Sounds good idea. Only problem with the PLs handling the paradox cases could be the program crash, when the contradicting variables with TF values were encountered during the execution. — Corvus

HOL is simply a bridge so that people that don't have a clue what knowledge ontologies are can think of them using the simpler isomorphism of what they do know. As I already showed Prolog can detect and reject such expressions. A system based on knowledge ontologies could be equivalent to an actual human mind.

From your coding, it seems no problem for HOL dealing with the Liars paradox and also Tarksi's undefinability. — Corvus

Tarski only proved that a truth predicate cannot be applied to a non-truth bearer. He got confused when This sentence is not true: "This sentence is not true" is true. The inner sentence is his theory and the outer one is in his metatheory. The smarter thing to do would be to reject the inner sentence as not a truth bearer.

This sounds a very interesting topic. I was reading on HOL recently, and it seems to be heavily mathematical arithmetic stuff. My question arose with the Liars paradox. How do you convert the Liars paradox sentence into HOL formula? — Corvus

?- LP = not(true(LP)).
LP = not(true(LP)).

?- unify_with_occurs_check(LP, not(true(LP))).
false.

BEGIN:(Clocksin & Mellish 2003:254)
Finally, a note about how Prolog matching sometimes differs from the
unification used in Resolution. Most Prolog systems will allow you to
satisfy goals like:

equal(X, X).
?- equal(foo(Y), Y).

that is, they will allow you to match a term against an uninstantiated
subterm of itself. In this example, foo(Y) is matched against Y,
which appears within it. As a result, Y will stand for foo(Y), which is
foo(foo(Y)) (because of what Y stands for), which is foo(foo(foo(Y))),
and so on. So Y ends up standing for some kind of infinite structure.
END:(Clocksin & Mellish 2003:254)

The HOL book by Bacon must be an introduction to the subject, and doesn't seem to discuss anything about Liars paradox or Tarski's truth. I will try to finish it first, and then look at the Tarski's truth and Godel numbers. — Corvus

I have been working on self-referential paradox for two decades. I have understood that a
https://en.wikipedia.org/wiki/Ontology_(information_science) knowledge ontology would correct these issues for about five years. No one else that understands the math of the things has the slightest clue what a knowledge ontology is. It just occurred to me much more recently that HOL is isomorphic to a knowledge ontology.

I don't believe anything in the propositional sense is known in the kinds of altered states of consciousness people refer to as "enlightenment". But such altered states are a kind of knowledge: a kind of familiarity or know-how. — Janus

It never was any altered state of consciousness. It is actually noticing a very well hidden aspect to cause-and-effect that is impossible to see until after one fully appreciates the actual limit to logically justified certainty.

Such skepticism based on mere imaginable possibilities seem toothless and irrelevant to me. I see a dog in the room, I have no cogent reason to doubt its existence. And that is exactly why I say that where there is no possibility of genuine, as opposed to merely feigned, doubt, then speaking in terms of belief is inapt. — Janus

The key thing about this limit to logically justified certainty is that it opens the mind sufficiently for things as a Buddhist enlightenment to occur. If not for this the true nature of reality would always be dismissed out-of-hand as ridiculously implausible and never even cross the mind as a remote possibility.

I think there is a valid distinction between knowledge and belief, although I also think that much of what is generally considered to be knowledge might be more accurately classed as belief. It may well turn out that I am sympathetic to Chet's belief. Let's see... — Janus

When "knowledge is defined as: "justified true belief", the "belief" aspect means truths that one is aware of, not expressions of language that are tentatively held as true. We can easily overcome the {Gettier problem} by adding that "Knowledge is justified true belief such that the justification guarantees the truth of the belief.

Because of the brain in a bottle thought experiment we cannot be logically certain of any empirical truth. Instead of saying {there is} a dog in my living room right now we must qualify this {there appears to be} a dog in my living room right now. It is not 100% logically impossible that all of reality is not a mere figment of the imagination. We can be logically certain that 2 + 3 = 5;

What is PA? — Corvus

https://en.wikipedia.org/wiki/Peano_axioms PA = (Peano Arithmetic)

↪PL Olcott Sure, I am currently reading on "High-Order Logics" by Andrew Bacon, and this is a really nice supporting thread for the reading. Thanks. — Corvus

https://andrew-bacon.github.io/papers/Front%20matter.pdf

It sounds like Tarski's indefinability theorem is only applicable to arithmetical truths according to the dictionary. — Corvus

You have to look at his actual proof to see otherwise:
It would
then be possible to reconstruct the antinomy of the liar in the
metalanguage, by forming in the language itself a sentence x
such that the sentence of the metalanguage which is correlated
with x asserts that x is not a true sentence.
https://liarparadox.org/Tarski_247_248.pdf

The full actual proof: https://liarparadox.org/Tarski_275_276.pdf

Is the actual Liar Paradox in PA ?

It's been observed were truth so definable, then the usual reading of Godel's sentence, unprovable but true, would have been instead untrue but true. — tim wood

It is true that it is provable in meta-F that G is unprovable in F. "G is unprovable in F" has its truth-maker in Meta-F.

One can formalize the semantics—define truth—of lower order logic in high-order logic. Under that fact, isn't it the case HOL defeats Tarsky Undefinability in the formalization, because TU only applies to the domain of Algebraic statements? — Corvus

The Liar Paradox basis of the proof: https://liarparadox.org/Tarski_247_248.pdf
The full actual proof: https://liarparadox.org/Tarski_275_276.pdf

It seems that the Tarski proof concludes this:
This sentence is not true: "This sentence is not true" is true.

A knowledge ontology https://en.wikipedia.org/wiki/Ontology_(information_science) is essentially an inheritance hierarchy of types from type theory which is apparently the same thing as HOL.
— PL Olcott

Great link with much useful info to learn. Thank you PL. — Corvus

I am trying to see whether or not HOL actually defeats Tarski Undefinability.

I was under impression that higher than 3rd-order logic would be for the multiple set theories and advanced calculus applications, therefore they wouldn't be used for describing the empirical world cases. — Corvus

A knowledge ontology https://en.wikipedia.org/wiki/Ontology_(information_science) is essentially an inheritance hierarchy of types from type theory which is apparently the same thing as HOL.