Btw, true is an adjective indicative of a quality that true statements have. Good luck with any attempt to comprehensively further define just what that quality exactly is. — tim wood

Analytic TRUE is a constant property of some expressions of language defined in terms of its relation to other expressions of language. An expression of language is TRUE if and only if it is (a) stipulated to be true, or (b) semantically derived from expressions of language that are stipulated to be true.

Let's try this. Suppose you have your "this," call it T. Now suppose you have some expression. Is it or its negation in T? If so, great! You're done. If not, then you have to figure out if it should be or not. And using existing knowledge, you cannot (if you could, it or its negation would already be in T). — tim wood

There is a great debate about whether an expression of language
can be true without a truth maker.

Truthmaker Maximalism defended GONZALO RODRIGUEZ-PEREYRA
https://philarchive.org/archive/RODTMD

A truth without a truthmaker is like a cake without a baker,
non-existent.

True and unprovable is self-contradictory once one understands
how true really works the way that I and Wittgenstein do.
https://www.liarparadox.org/Wittgenstein.pdf

Higher-order logic is the union of first-, second-, third-, ..., nth-order logic; i.e., higher-order logic admits quantification over sets that are nested arbitrarily deeply.
— PL Olcott

Do we need more than first and second order logic in practical uses? — Corvus

For formalizing the entire body of human knowledge that can be expressed using language we need this:

By the theory of simple types I mean the doctrine which says that the objects of thought ... are divided into types, namely: individuals, properties of individuals, relations between individuals, properties of such relations, etc.
https://en.wikipedia.org/wiki/History_of_type_theory#G%C3%B6del_1944 — History of Type Theory

There seems to be a finite limit to the number of orders of logic needed.

↪PL Olcott What is your point? We suppose - that's the best I can do - that a proposition undecidable in L is decidable in L', and one in L' in L'', and so forth. But apparently there is no Lωω...ω that is itself complete. — tim wood

A knowledge ontology inheritance hierarchy capable of formalizing the entire body of human knowledge that can be expressed using language need not be incomplete in the Gödel sense. Such a system would essentially be a type hierarchy from type theory thus isomorphic to HOL.
https://en.wikipedia.org/wiki/Ontology_(information_science)

"[They, (the ideas here presented)] implicitly justify the generalization that every symbolic logic is incomplete...." (316) — tim wood

It would seem that for every n order or logic there would necessarily be an n+1 order of logic provability predicate for this N order of logic.

There are many ways to further extend second-order logic. The most
obvious is third, fourth, and so on order logic. The general principle,
already recognized by Tarski (1933 [1956]), is that in higher order
logic one can formalize the semantics—define truth—of lower order logic.
https://plato.stanford.edu/entries/logic-higher-order/ — Stanford

there'd need to be a single unifying metalanguage in which the formulae of all the levels could be expressed. — fdrake

Yes, definitely.

The truth and provability symbols in the metalanguage would thus apply for theorems applying to the big union logic, rather than having a plethora of distinct symbols in different metalanguages — fdrake

One metalanguage that can refer to one of more elements at any level within the type hierarchy.

Now consider that you're taking the set of all provable statements of all logics up to the nth order. That will then be the set of provable statements of the nth order logic, due to the hierarchy. — fdrake

For every nth order logic that can be shown to be incomplete there is an n+1 order logic that completes it.

seems to follow the principle that every simple idea can be made convoluted enough that it can no longer be understood.
— PL Olcott
That's a convenient principle. Btw, how do you know when an idea is just that simple? — tim wood

When we envision the inherent structure of the set of all knowledge that can be expressed using language then we can see that "incompleteness" and "undecidability" are mere errors in disguise.

A formal system having only one order of logic is like the "C" volume of an encyclopedia only having articles that begin with the letter "C".
— PL Olcott
And a complete set would have everything from A to Z. But in our case, you can't have a complete set. — tim wood

...14 Every epistemological antinomy can likewise be used for a similar undecidability proof...
(Gödel 1931:43-44) Epistemological antinomies (AKA self-contradictory expressions) must be excluded from every formal system of logic because they are not truth bearers. I can't follow your other proof, yet it seems to follow the principle that every simple idea can be made convoluted enough that it can no longer be understood.

I imagine Tarski's indefinability theorem would. AFAIK second order logic already has diagonalization results - so it's either inconsistent or incomplete. You don't need to go above first for it. So long as you put enough arithmetic in, you're going to get the self referential bullshit that sets up these paradoxes. — fdrake

Anything that can't be proved in one order of logical can be proved or refuted in the next. A formal system having every order of logic cannot be incomplete. A formal system having only one order of logic is like the "C" volume of an encyclopedia only having articles that begin with the letter "C".

I cannot give a rigorous answer, but I agree with this. If Tarski's undefinability theorem is basically that "arithmetical truth cannot be defined in arithmetic", or that true Gödel numbers are not definable arithmetically, meaning there’s no first-order formula for this, I think it does go for higher order logics. For those higher order logics there is their own true but unprovable Gödel number. — ssu

There is always an unprovable expression in one order of logic that is provable in the next order. When all orders of logic are included in the same formal system then such a system cannot be incomplete. I posted this in two forums because the more appropriate forum has hardly any views.

↪PL Olcott Which is to say - just between us in case we're both wrong - that each system being itself deficient requires a successor system to fix it, but that simply creating a new deficiency. Ordinal arithmetic being formidable, I don't see an escape. — tim wood

That a formal system can be defined with a single order of logic seems isomorphic to an encyclopedia defined with only the "C" volume. Of course such a system would be incomplete, yet it is a little silly to define systems or encyclopedias this way.

Your question (again, if I understand it), is can there be a super-strong formal system that is not incomplete. I am guessing not. And I'm sure a rigorous discussion would be well, rigorous. — tim wood

It seems that all of the formal systems that these two apply to only have a single order of logic. When we define a formal system that simultaneously has an unlimited number of orders of logic then there is always one more order of logic as needed.

And if nothing else, this is the clue, "next higher order...". It appears you want to get to the point where there is no higher order. And that would seem to lead to a set-of-all-sets type of contradiction. — tim wood

It has always seemed to me that Tarski's Undefinability theorem fails when applied to a knowledge ontology inheritance hierarchy (KOIH). It has only occurred to me recently that KOIH is essentially type theory which is essentially HOL. https://en.wikipedia.org/wiki/Ontology_(information_science)

This is probably hard to believe but I do not have the intuitions necessary to see the “mysteries” of some paradoxes. For example, the liar paradox “this sentence is false” simply appears meaningless to me and I do not enter the logic of: If 'This sentence is false.” is true, then since it is stating that the sentence is false, if it is actually true that would mean that it is false, and so on.
Language conveys information and I can’t extract relevant information from this sentence, this is why I do not understand why people manage to reason logically with it. — Skalidris

I had forgotten that this was also Kripke's analysis. Thanks for reminding me. This same analysis equally applies to an expression of formal language that asserts its own unprovability.

That is always the excuse given by users of that site. It is either "a good summary", when there are often mistakes right in the header and the site itself is rotten to the core, or to "check the sources on the bottom". — Lionino

I examine many original sources and then quote Wikipedia as a verified summation of the technical ideas involved. The key point here is that the Tarski Undefinability theorem is anchored in the Liar Paradox and that by itself is its big mistake. Tarski himself shows that his proof is anchored in the Liar Paradox yet does this in a somewhat convoluted way that would be too difficult for most people to follow.

I directly understand his actual proof.
https://liarparadox.org/Tarski_275_276.pdf

That sounds like a cumbrous task for normal users to go through for using the system. They would want just type in the expressions in their ordinary use of the language or words into the system, and expect to get the correct definitions for their queries. Somehow the Cyc Project must be able to convert the expressions or words into the unique GUID to narrow down and select the correct definitions for them. Would you agree? — Corvus

The Cyc project converts the words of the users into multiple GUIDs and then from context narrows down to list to the intended meanings.

I always go on the basis that his true predicate struggles with the Liar Paradox.
and concludes that no True (Language, Expression) can exists because would
have to handle the Liar Paradox incorrectly.

This is how I propose to handle the Liar Paradox correctly
LP = "This sentence is not true."
Boolean True(English, LP) is false
Boolean True(English, ~LP) is false

↪PL Olcott I have no dog in the fight. I don't know if the article on Tarski is correct or not, I don't have much of a way to judge myself. Mine was just an off-the-cuff comment. — Lionino

Personally I love Wikipedia it always gives me a succinct clear gist of the whole idea in the first couple of sentences. When used this way it seems to be a very high quality standard.

I imagine that that happens because you learn from there. I find nonsense there all the time. The people who run it are oligophrenic. So I avoid it like a plague. — Lionino

Most people can't understand his original proof
https://liarparadox.org/Tarski_275_276.pdf
so I use the Wikipedia simplification.

This is where Tarski anchors his proof in the actual Liar Paradox
https://liarparadox.org/Tarski_247_248.pdf

Tarski's proof doesn't work the way you describe it. To see that, you just need to read the article that you yourself say is "clear and accurate". — TonesInDeepFreeze

The proof of Tarski's undefinability theorem in this form is again
by reductio ad absurdum. Suppose that an L-formula True(n)

as above existed, i.e., if A is a sentence of arithmetic, then
True(g(A)) holds in N if and only if A holds in N. Hence for all

A, the formula True(g(A)) ⟺ A holds in N. But the diagonal
lemma yields a counterexample to this equivalence, by

giving a "liar" formula S such that S ⟺ ¬True(g(A)) holds
in N. This is a contradiction QED.

https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem

This correctly recognizes that the Liar Paradox is not a truth bearer.
LP = "This sentence is not true."
Boolean True(English, LP) is false
Boolean True(English, ~LP) is false

"Does there exist a proof of T?" is a question. — TonesInDeepFreeze

Is the question associated with every formal proof or lack thereof.

Yet I showed exactly what was amiss in the Wikipedia article recently cited. — TonesInDeepFreeze

That is a degree of detail that is totally irrelevant to my point so I did not
examine it at all. My point is the Tarski anchored his Undefinability
Theorem in the actual Liar Paradox.

So far hardly any modern or ancient philosophers seem to understand
that the Liar Paradox is not a truth bearer thus has no truth value thus
is not in the domain of any decision problem or formal proof.

a good amount of caution is warranted when referencing Wikipedia.
— TonesInDeepFreeze

Huge amount. — Lionino

I have found that it always succinctly and clearly presents an accurate view of
every technical subject that I have ever referenced as measured by its correspondence
with many other sources.

One can couch things as questions. But the proofs themselves do not have questions in them. — TonesInDeepFreeze

Proofs always have the provability question associated with them:
Whether or not a proof exists is.

All steps in proofs are statements, not questions. — TonesInDeepFreeze

The decision problem form of a formal proof <is> a yes/no question about an input.
The formal proof shows the steps of how X is derived from Y in Z.
The decision problem version answers the question: Can X be derived from Y in Z?
When X is a theorem of Z then the premises are empty.

In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values.
Decision problem

The key issue with decision theory is that deciders are required to correctly
answer a self-contradictory (thus incorrect) questions.

The key difficulty with resolving this issue that most modern day philosophers
do not understand that both of these questions are equally incorrect:
(a) Is this sentence true or false: "What time it is?"
(b) Is this sentence true or false: "This sentence is not true."

They do not understand that the Liar Paradox is simply not a truth bearer.

So, if set theory without the axiom of regularity proves a contradiction, then set theory with the axiom of regularity proves a contradiction. — TonesInDeepFreeze

My post is about a single coherent way around all of these issues.
The key mistake of decision theory is that the notion of decidability requires a decider
to correctly answer a self-contradictory (thus incorrect) question otherwise an input
is construed as undecidable.

And, again, as I just explained, disallowing sets from being members of themselves does not avoid inconsistency. — TonesInDeepFreeze

When we ask the question: Does a barber shave everyone that does not shave themselves? is allowed to be rejected as an incorrect question then the paradox goes away. ZFC prevents this question from even being expressed as sets.

Moreover, the purpose of the axiom of regularity is not to avoid inconsistency but rather to facilitate the study of sets as in a hierarchy indexed by the ordinals. — TonesInDeepFreeze

Thereby preventing inconsistency in the same way that type theory prevents inconsistency.

It is a common misconception on Internet forums that ZFC avoids inconsistency by disallowing sets to be members of themselves.

Yes, the axiom of regularity, which is adopted in ZFC, disallows that a set can be a member of itself. — TonesInDeepFreeze

You just contradicted yourself. I will be more precise. ZFC eliminates the possibility
of inconsistency that is caused by allowing sets to be members of themselves.

An isomorphic solution would solve the halting problem. A halt decider is not allowed
to be applied to any input to refers to itself.

(1) The article conflates a language with a theory.

(2) The proof in the article handwaves past the crucial lemma, thus appearing to commit a serious non sequitur. — TonesInDeepFreeze

I have studied these things in my mind continuously for decades. Mathematics uses the term "theory" to mean a set of axioms. Everyone else means a set of ideas that might be true.

My current understanding of a subset of undecidable decision problems is that this undecidability can be easily abolished the same way that ZFC conquered Russell's Paradox. ZFC prevents self-contradictory expressions from coming into existence by forbidding the creation of sets that are members of themselves.

How do the users know the unique ID? How does the Cyc Project know that is the ID it has to select the answer for the query? — Corvus

I would estimate that the users use ordinary English and the Cyc lexical analyzer converts words into GUIDs. The parser can determine which of the multiple GUIDs for the same word is most probably from context. My cat drank of bowl of milk would not refer to Caterpillar Earth moving equipment or Harvey Milk.

But so far I can't make any sense of what you're saying - this is why I'm trying to get some basic terminology clear. I'll ask again. What is the difference between a "sense input" and a "sense meaning"? — EricH

That you can hear dogs actually barking with your ears is a sense input from your
ears to your mind. Hearing dogs bark is the sense meaning of "dogs bark". These
things cannot be expressed using words thus are not Analytic(Olcott).

“Analytic” sentences, such as “Pediatricians are doctors,” have historically been characterized as ones that are true by virtue of the meanings of their words alone and/or can be known to be so solely by knowing those meanings. The Analytic/Synthetic Distinction

When we exclude those aspects of meanings such as the actual sound of dogs barking and include every element of human general knowledge that can be expressed using words expressly including that some “Pediatricians are rich” then we have the subset of human knowledge that can be processed by a computer program.

I still can't make any sense out of this. What is the difference between a "sense input" and a "sense meaning"? The only way we can even know that there are such things as dogs is through sense input. — EricH

I am providing the means for a computer to compute Boolean True(L, x) where L is a language such as English and x is an expression of that language. When I show how this can be coherently accomplished then the Tarski Undefinability Theorem is refuted.

(1) The article conflates a language with a theory.

(2) The proof in the article handwaves past the crucial lemma, thus appearing to commit a serious non sequitur. — TonesInDeepFreeze

The whole Idea as I present it is whether or not the computable function
Boolean True(String Language, String Expression) can correctly and
consistently determine the truth value of every element of human general
knowledge on the basis of an accurate model of the current world.

When we construe that Tarski's Undefinability Theorem got stuck
on the Liar Paradox here is the resolution to that:

LP = "This sentence is not true"
Boolean True(English, LP) returns false for not true.
Boolean True(English, ~LP) returns false for not true.

The generic issue across many undecidable decision problems is that
epistemological antinomies (AKA self-contradictory expressions) are
not excluded from the problem domain.

Another issue with this is that most modern philosopher's do not understand
that self-contradictory expressions are not truth bearers thus have no associated
truth value.

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

Gödel, Kurt 1931. On Formally Undecidable Propositions of Principia Mathematica And Related Systems

↪Count Timothy von Icarus

That article doesn't properly state the subject matter. — TonesInDeepFreeze

I think that it does a decent job.
Here is the most relevant part
As always diagonalization shows THAT an expression is unprovable and ignores
the reason WHY it is unprovable is that it is self-contradictory.

Here is his actual proof
Here is where he anchors it in the Liar Paradox

The use of "analytic" here bears little resemblance to the normal usage. As far as I can tell, any fact is "analytic" so long as it can be defined as true by definition by some string. The analytic normally is "what is true by definition," and apparently non-analytic facts like "Moscow is the current capital of Russia," can become analytic despite the fact that "Moscow" is not synonymous with "the capital of Russia," by simply stipulating an axiom that says "Moscow is the capital of Russia, by definition." — Count Timothy von Icarus

The current model of the actual world (that can be periodically updated) that includes all knowledge of mathematics and logic and is able to perform any mathematical and logical operations is defined to be Analytic(Olcott). This is the foundation of analytical knowledge that is used as the basis to define a True(Language, Expression_of_Language) predicate thus refuting Tarski that incorrectly "proved" this cannot be done.

I still can't make any sense of this. Does the Cyc project identifier refer to
- a conceptual object
- a collection of conceptual objects (i.e., how do we know that one person's conception of a dog is the same as another's)
- a particular existing living animal that happens to be a dog
- all living animals that happen to be dogs
- other? — EricH

A 128-bit integer GUID refers to a single unique sense meaning, thus the class living animal {dog} has its own unique GUID. A particular individual {dog} could have its own unique GUID within the discourse context knowledge ontology, yet not a part of the general knowledge ontology.

In one person believes that the living animal {dog} has an elevator because it is a {fifteen story office building} then they are simply incorrect.

Something is true or false always in relation to some respect. Dogs are animals is false in case of the robot AI dogs. Dogs can be tools in wood carving toolbox. Dogs are pieces of the wooden material that get inserted in the holes of the workbenches to secure a plank of wood to be carved. In this case dogs are animals is false again. — Corvus

3ab2c577-7d38-4a3c-adc9-c5eff8491282 stands for the living animal dog, this is the same way that the Cyc project identifies unique sense meanings,

Dogs exist as conceptual objects even if all of reality is a mere figment of the imagination.
— PL Olcott

So this whole project is merely the embodiment of people's imagination. — EricH

Not at all, exactly the opposite. Dogs are animals is absolutely true no matter what.
It is true in the same sort of way that we know that 5 is numerically greater than 2.
5 > 2 remains true even after the heat death of the universe when zero minds exist.