## Eliminating Decision Problem Undecidability

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I forgot to say this: L is the formal language of a formal system

My True(L,x) predicate is defined to return true or false for every
finite string x on the basis of the existence of a sequence of truth
preserving operations that derive x from

A set of finite string semantic meanings that form an accurate
verbal model of the general knowledge of the actual world that
form a finite set of finite strings that are stipulated to have the
semantic value of Boolean true.

False(L,x) is defined as True(L,~x).
Truthbearer(L,x) ≡ (True(L,x) ∨ True(L,~x))

Finite string expressions that are not truth-bearers are rejected
as a type mismatch error for every formal system of bivalent logic.

Truthbearer(English, "This sentence is not true") is false.
Truthbearer(English, "This sentence is true") is false.
Truthbearer(English, "a fish") is false.
Truthbearer(English, "some fish are alive") is true.

Truthmaker Maximalism (is what the above ideas are anchored in)
https://plato.stanford.edu/entries/truthmakers/#Max
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that derive x from

From what?

string semantic meanings

Semantic and meaning mean the same thing. Is the quote above supposed to mean "meaningful strings" as for example "the dog bites the ball" instead of "gorbyr dortug equerxi"?

verbal model of the general knowledge of the actual world

What's "general knowledge" supposed to mean as opposed to just "knowledge"? Also, if the answer to the question in the previous quote is "Yes", "string semantic meanings" and "verbal model" approximately mean the same thing, if they mean anything at all.

that
form a finite set of finite strings that are stipulated to have the
semantic value of Boolean true

?
You are writing things from a train of thought but the purpose of writing is communication, you need to include the train of thought, not just its destination.

A set of finite string semantic meanings that form an accurate
verbal model of the general knowledge of the actual world
that form a finite set of finite strings that are stipulated to have the semantic value of Boolean true.

We have an X that forms a Y that forms a Z, but X and Z seem awfully similar, as if they mean the same thing.

False(L,x) is defined as True(L,x)

Do you mean False(L,x) is defined as True(L,¬x)?

Truthbearer(L,x) ≡ (True(L,x) ∨ True(L,~x))

Yes, something is a truth-bearer if it is true or false.

Finite string expressions that are not truth-bearers are rejected
as a type mismatch error for every formal system of bivalent logic.

Truthbearer(English, "This sentence is not true") is false.
Truthbearer(English, "This sentence is true") is false.
Truthbearer(English, "a fish") is false.
Truthbearer(English, "some fish are alive") is true.

That is the naïve reply to sentences such as "This sentence is a lie". Claiming that it is not a truth-bearer is alike hand-waving, you must give some account as to how it is not a truth bearer.

Another further issue is that by the law of non-contradiction, something is X or it is not-X. Something is true or it is not true. Replying that "a fish" is neither true or false while nevertheless defining "false" as not-true violates the LNC. The "type mismatch" is encompassed in "not true" just like "green" is encompassed in "not salty" when you ask the equally nonsensical question "Is the colour green salty or not salty?" — do we really need to come up with a concept of "salty-bearer" or can't we just say things that taste salty are salty and everything else is not salty?
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That is the naïve reply to sentences such as "This sentence is a lie". Claiming that it is not a truth-bearer is alike hand-waving, you must give some account as to how it is not a truth bearer.

It is far too much to unpack all oat once.
I don't say: "This sentence is a lie",
I refer to the strengthened Liar Paradox: "This sentence is not true."

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

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

The means LP is rejected as not a truth bearer in Prolog because
it has an infinite cycle in its evaluation graph.

This sentence is not true.
What it is not true about?
It is not true about being not true.
• 1.8k
I don't say: "This sentence is a lie",
I refer to the strengthened Liar Paradox: "This sentence is not true."

Those two mean the same.

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

No clue what that means.

The means LP is rejected as not a truth bearer in Prolog because
it has an infinite cycle in its evaluation graph.

This sentence is not true.
What it is not true about?
It is not true about being not true.

It is the criticism of the liar paradox refering to nothing. It was discussed in the thread I linked.
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It is the criticism of the liar paradox refering to nothing. It was discussed in the thread I linked.

I have spent two decades on this. It <is> a truth predicate that would
work because Truthbearer(L,x) ≡ (True(L,x) ∨ True(L,~x)) screens out
epistemological antinomies that Tarski get stuck on.
• 1.8k
I have spent two decades on this.

It <is> a truth predicate that would work because Truthbearer(L,x) ≡ (True(L,x) ∨ True(L,~x)) screens out epistemological antinomies that Tarski get stuck on.

@jgill@fishfry
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What's "general knowledge" supposed to mean as opposed to just "knowledge"?

General knowledge can be expressed in a finite set of finite strings.
Specific knowledge of everything is unmanageably large and infinite.
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It <is> a truth predicate that would work because Truthbearer(L,x) ≡ (True(L,x) ∨ True(L,~x)) screens out epistemological antinomies that Tarski get stuck on.
— PL Olcott

@jgill@fishfry

I'm familiar with Pete's work from other forums.
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True(L,x)
Tell us, how do you know True(L,x) is true?
• 626
True(L,x)
— PL Olcott
Tell us, how do you know True(L,x) is true?

Expressions that are {true on the basis of meaning} are ONLY
(a) A set of finite string semantic meanings that form an accurate
model of the general knowledge of the actual world.

(b) Expressions derived by applying truth preserving operations to (a).

True(English, "a cat is an animal") is true.
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(a) A set of finite string semantic meanings that form an accurate
model of the general knowledge of the actual world.
Ok, but how exactly do you decide what is, or is not, a member of this set?
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(a) A set of finite string semantic meanings that form an accurate
model of the general knowledge of the actual world.
— PL Olcott
Ok, but how exactly do you decide what is, or is not, a member of this set?

It is simply all of the details of every fact of the world. General knowledge is a finite set of axioms.
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The problem is that you have a set. But it is by no means clear how you create that set.
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↪PL Olcott The problem is that you have a set. But it is by no means clear how you create that set.

That is not the point. The set of verified facts of the world is defined to exist, it is merely
not written down all in one place yet. LLM AI models might be able to achieve this
within a few years.

The point is that when we know all of the general knowledge facts of the world then
we can easily screen out every epistemological antinomy (as a type mismatch error
non-truth-bearer) that many of the undecidability proofs depend on. Tarski undefinability
proof depends on this.

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That is not the point.
That is the point. Not only have you not got it; it may not be achievable - that depending on the exact details and definitions. You invoke an oracle, but give no account of it other than some hand-waving. And what do you mean by "verified fact"? Is a verified fact different from just a fact? How do you verify it - what does verified mean? Do you even know what a fact is? Do you know the difference between fact and true?
Of course you can do what you want and claim what you want. But when your claims cross over into nonsense, they amount to an illegitimate appropriation. Make it legitimate.
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That is the point. Not only have you not got it; it may not be achievable - that depending on the exact details and definitions. You invoke an oracle, but give no account of it other than some hand-waving. And what do you mean by "verified fact"? Is a verified fact different from just a fact? How do you verify it - what does verified mean? Do you even know what a fact is? Do you know the difference between fact and true?

If we merely encoded all of the rules of algorithms, logic, and programming in a single
formal system then when when no sequence of truth preserving operations from these
basic axioms derives x or ~x then x can be rejected as a type mismatch error on the basis
that all formal systems of bivalent logic require every expression to be a truth-bearer.
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If we merely encoded all of the rules of algorithms, logic, and programming in a single formal system
All right, a programming language.
truth preserving operations (TPOS)

And the Truths these TPOs are expected to preserve, whence them - your having only a language? And, "no sequence"? How do you define "no sequence"?

Let's imagine you generate a listing of all possible propositions/statements of 100 or fewer symbols in length - a very long list. Let's further imagine you can test for and eliminate all nonsense strings, leaving only those that are syntactically "healthy." Still a long list. Now to test each for truth with your TPOs. The result for each item on the list being either T for true, or F for everything else. What is your specification for both the TPO and truth itself that the TPO can distinguish what is true from what is not?

And what about strings of longer than 100 symbols? There are lots of true propositions/statements longer than 100 symbols that your TPOs will record as F.

Unless you have already created an encyclopedia of sorts. Then you could test against that, but that would be very far indeed from being either conclusive or exhaustive or in itself interesting.
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If we merely encoded all of the rules of algorithms, logic, and programming in a single formal system
— PL Olcott
All right, a programming language.
truth preserving operations (TPOS)
— PL Olcott

And the Truths these TPOs are expected to preserve, whence them - your having only a language? And, "no sequence"? How do you define "no sequence"?

{All cats are animals}
{All animals are living things}
therefore {All cats are living things}

The principle of explosion is not truth preserving.
{All cats are animals} // axiom
{No cats are animals} // false assumption
therefore FALSE
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{All cats are animals}
You have a programming language - where does a statement about cats come from? How do you know "no cats are animals" is false and not itself an axiom?
• 626
{All cats are animals}
— PL Olcott
You have a programming language - where does a statement about cats come from? How do you know "no cats are animals" is false and not itself an axiom?

Expressions that are {true on the basis of meaning} are ONLY
(a) A set of finite string semantic meanings that form an accurate
model of the general knowledge of the actual world.

(b) Expressions derived by applying truth preserving operations to (a).

The above algorithm specifies True(L,x) and False(L,x) defined
as True(L, ~x).
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Expressions that are {true on the basis of meaning
Ah, meaning. What is that? How does your program assess or even recognize meaning? I am asking the simplest and most basic questions because it seems to me you must have both asked and answered them. But so far I have no evidence of that in this thread, or seen it in your other threads.

Above your syllogism as an example of a TPO
{All animals are living things}
{All cats are animals}
therefore {All cats are living things}

There are many more possible premises/conclusions. The questions here are how does your PTO or your computer know what is true or meaningful? How does it construct the right syllogism from the possible premises?
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Ah, meaning. What is that? How does your program assess or even recognize meaning? I am asking the simplest and most basic questions because it seems to me you must have both asked and answered them. But so far I have no evidence of that in this thread, or seen it in your other threads.

https://en.wikipedia.org/wiki/Ontology_(information_science)
of Rudolf Carnap / Richard Montague {meaning postulates} that stipulate relations
between finite strings as providing the semantic meanings that form an accurate
model of the general knowledge of the actual world.

Rudolf Carnap told Willard Van Orman Quine that the otherwise totally
meaningless finite string of "Bachelor(x)" is defined as the otherwise totally
meaningless finite string "~Married(x)" and Quine just could not get it.

The full definition of "Married(x)" entails (at least) billions of other meaning
postulates defining "Human(x)".
• 8.9k
of Rudolf Carnap / Richard Montague {meaning postulates} that stipulate relations between finite strings as providing the semantic meanings that form an accurate
model of the general knowledge of the actual world.
English, please. Simple sentences are good.

I'll try to make it simpler. Given some string, call it Σ, we can start by supposing that Σ is/is not meaningful, is/is not true. How do you know/decide? Because I infer you have your program do it, the question is really, how does your program decide?
• 626
I'll try to make it simpler. Given some string, call it Σ, we can start by supposing that Σ is/is not meaningful, is/is not true. How do you know/decide? Because I infer you have your program do it, the question is really, how does your program decide?

Facts are sentences that are defined as true. Cats <are> Animals is defined as true.
The otherwise totally meaningless sequence of letters of "cats" and "animals" are defined to have the <is a type of> relation to each other.

All of the facts about the world work this same way. You wanted it simple and provided a complex example. In one case "Σ" is a Greek letter. Even this begs the question: What is Greek? and What is a letter?

https://www.mathsisfun.com/algebra/sigma-notation.html requires a whole mathematical infrastructure.
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Facts are sentences that are defined as true.
That's a pretty good definition! But you're missing the whole point. Who or what defines, and on what basis or by what criteria? If it's humans all the way down, I'll take that as an answer, but that will leave the question as to how your whole program will work, in as much as it will have to be preloaded with that which it is supposed to produce.
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That's a pretty good definition! But you're missing the whole point. Who or what defines, and on what basis or by what criteria? If it's humans all the way down, I'll take that as an answer, but that will leave the question as to how your whole program will work, in as much as it will have to be preloaded with that which it is supposed to produce.

Yes it is the case that only humans have a complex language, however, apes have learned a symbolic language known as Yerkish. https://en.wikipedia.org/wiki/Yerkish

There is a data structure known as a knowledge ontology that is based on a directed graph.
https://en.wikipedia.org/wiki/Ontology_(information_science)

A knowledge ontology has a unique integer (such as the CYC project's use of the 128-bit GUIDs) for each sense meaning of every word. https://en.wikipedia.org/wiki/Cyc

A word is a string (AKA sequence) of characters such as "dog". The first sense meaning is the most common one: https://www.merriam-webster.com/dictionary/dog# These differing sense meanings have an integer index in dictionaries. A knowledge ontology might require an ISO standard dictionary so that its unique sense meanings expressed as 128-bit integers can correspond to their sequence of characters in this IOS standard dictionary.

A knowledge ontology is an inheritance hierarchy of these sense meanings. This means that the sense meaning of {dog} (the animal) gets most of its meaning from {animal} and only adds details that distinguish a {dog} from other {animals}.
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Ok, I understand your answer that meaning is assigned by people, the strings of which translated by the machine into a kind of index and score, and that the machine then associates strings on the basis of their index and scores. But these associations are probabilistic only, and neither in themselves truth bearing or producing, Assuming the program is creating some master list of true statements, how is the truth of any particular statement judged, if not by a person?
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But these associations are probabilistic only, and neither in themselves truth bearing or producing

In other words you seem to believe that "a cat is probably an animal" and "a cat is probably not a fifteen story office building". I disagree.
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Not at all - for a person. But all the machine has got is probability and some kind of heuristics.

You seem to be in the position of a carpenter with a hammer who insists that his hammer drives nails, the problem being that the hammer can be used to drive nails, but that there never was nor will be a hammer that itself drives any nails. You have claimed seemingly endlessly and relentlessly claimed that your program/machine does something. I am just attempting to find out from you what exactly it does, and if relevant, how.

So far I think your machine just generates strings of symbols as candidates for inclusion in a list, but that apparently require the judgment of a person for that inclusion.
• 626
So far I think your machine just generates strings of symbols as candidates for inclusion in a list, but that apparently require the judgment of a person for that inclusion.

We simply correctly encode all of the true facts of the world. When the discussion
devolves into "facts according to who" I lose interest because the discussion has
devolved away from actual truth.
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We simply correctly encode all of the true facts of the world.
Then why bother with a machine/program? You have simply gone to the trouble of creating a data-base - in theory because there are significant problems im creating one for real.

Further, as you have defined a fact as a proposition defined as true, then being dismissive of "facts according to whom" is at the least disingenuous. And this in turn implies that while you in part seem to know what a fact is, you do not in fact know what one is. That is, that facts are never true. This a formal distinction that has not much use in the ordinary world, but is critical here. The problem is that in defining it true, you then take it to be true, and that only possible through either a deliberate or ignorant disregard of a distinction between instrumental truth, e.g., gravity is a force, the atom is like a small solar system, and a priori truth. The one being conditionally true and perhaps otherwise untrue, the other being always universally and necessarily true.

So you have created in theory a listing of a lot of your or someone's opinions. Can you demonstrate having done any better than that?
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