Eliminating undecidability in formal systems

[Logik]

int main()
{
int x = 5;
int y = 3
if (x > y && y > x)
printf("There is no such thing as contradiction!");
else
printf("Logik is a fibber!");
}

The rules of arithmetic are arbitrary...
All logical rules are arbitrary.
All operators and symbols are arbitrary.

https://repl.it/repls/ExcitedSunnyVolume

Appealing to "convention" is a bandwagon fallacy.

Bullshit.

Suit yourself. I am showing you the constructive side of logic.

If you want to impose arbitrary rules (axioms) on yourself - it's your choice.

As soon as you hit Turing-completeness, quite literally spacetime is the limit.

https://en.wikipedia.org/wiki/Computational_complexity

[PeteOlcott]

If you want to impose arbitrary rules (axioms) on yourself - it's your choice.

As soon as you hit Turing-completeness, quite literally spacetime is the limit.

https://en.wikipedia.org/wiki/Computational_complexity

The rules are not arbitrary. They are the set of relations specifying the current set of human knowledge.

[Logik]

The rules are not arbitrary. They are the set of relations specifying the current set of human knowledge.

OK. So you've chosen to conceptualise human knowledge as a graph. Why not a k-way tree?

Lets go with a graph though.

What does a node represent?
What's the highest-weighted node and why?
What do the edges represent?
Are all edges morphic or isomorphic?

[PeteOlcott]

The rules are not arbitrary. They are the set of relations specifying the current set of human knowledge.

OK. So you've chosen to conceptualise human knowledge as a graph. Why not a k-way tree?

Lets go with a graph though.

What does a node represent?
What's the highest-weighted node and why?
What do the edges represent?
Are all edges morphic or isomorphic?

https://en.wikipedia.org/wiki/Tree_(graph_theory)
It looks like a tree is not directed and "k" requires boundaries that make things not work.
The only boundary is that the whole thing is finite.
Since it is an inheritance hierarchy it is directed.

It implements this:
https://en.wikipedia.org/wiki/History_o ... B6del_1944

By the theory of simple types I mean the doctrine which says that the objects of thought (or, in another interpretation, the symbolic expressions) are divided into types, namely: individuals, properties of individuals, relations between individuals, properties of such relations,

Which can be boiled down to relations between things.

Here is a diagram of the graph with {Thing} as the pinnacle node.
http://www.cyc.com/wp-content/uploads/2 ... base-1.png

So this acyclic graph is an inheritance hierarchy of relations each node representing
a single atomic concept. A directed edge links a Relation to its arguments and a child to its parent.

This node would be either a predicate or an argument to a predicate which
itself could be another predicate.

[Logik]

Since it is an inheritance hierarchy it is directed.

What is at the top of the inheritance hierarchy? Things? A thing is a concept.
What is at the bottom of the hierarchy? Domain-specific data?

How many domains are there?

By the theory of simple types I mean the doctrine which says that the objects of thought (or, in another interpretation, the symbolic expressions) are divided into types, namely: individuals, properties of individuals, relations between individuals, properties of such relations,

The above indicates to me that you and I are definitely not on the same page.

Objects of thought? Do you have an ontological model for thought? Where does meta-cognition (thinking about thinking) come into your graph?

Which can be boiled down to relations between things.

Yeah. If you leave the knower out of the equation.

Here is a diagram of the graph with {Thing} as the pinnacle node.
http://www.cyc.com/wp-content/uploads/2 ... base-1.png

So this acyclic graph is an inheritance hierarchy of relations each node representing
a single atomic concept. A directed edge links a Relation to its arguments and a child to its parent.

So, do you recognize that what you are showing is us is just one of infinitely many possible taxonomies?
Taxonomy is the product of the Categorization process
which takes place in the mind.

And you have no model of the mind.... so why is your taxonomy better than my taxonomy?

It seems to me you are allowing the model used by Cyc to influence your thinking.
Have you considered that the model Cyc uses is simply the product of the choices of those who engineered it?
Have you considered that there are more complex, more complete knowledge-bases than Cyc out there? Google....

This node would be either a predicate or an argument to a predicate which
itself could be another predicate.

I can't decide (from your use of the word "would be") is you are talking about the present or future.

Is this how you conceptualise "knowledge"?
Is this a system you intend to build?

Either way - it sure sounds to me like you are oblivious to the analytic/synthetic distinction as it exists in systems theory (N.B NOT the Kantian distinction).

Further observation: you are obsessed with 100% certain truths in a world that has exactly none of those. Time for some probability theory in your episteme?

[PeteOlcott]

Since it is an inheritance hierarchy it is directed.

What is at the top of the inheritance hierarchy? Things? A thing is a concept.
What is at the bottom of the hierarchy? Domain-specific data?

How many domains are there?

By the theory of simple types I mean the doctrine which says that the objects of thought (or, in another interpretation, the symbolic expressions) are divided into types, namely: individuals, properties of individuals, relations between individuals, properties of such relations,

The above indicates to me that you and I are definitely not on the same page.

Objects of thought? Do you have an ontological model for thought? Where does meta-cognition (thinking about thinking) come into your graph?

Which can be boiled down to relations between things.

Yeah. If you leave the knower out of the equation.

Here is a diagram of the graph with {Thing} as the pinnacle node.
http://www.cyc.com/wp-content/uploads/2 ... base-1.png

So this acyclic graph is an inheritance hierarchy of relations each node representing
a single atomic concept. A directed edge links a Relation to its arguments and a child to its parent.

So, do you recognize that what you are showing is us is just one of infinitely many possible taxonomies?
Taxonomy is the product of the Categorization process
which takes place in the mind.

And you have no model of the mind.... so why is your taxonomy better than my taxonomy?

It seems to me you are allowing the model used by Cyc to influence your thinking.
Have you considered that the model Cyc uses is simply the product of the choices of those who engineered it?
Have you considered that there are more complex, more complete knowledge-bases than Cyc out there? Google....

This node would be either a predicate or an argument to a predicate which
itself could be another predicate.

I can't decide (from your use of the word "would be") is you are talking about the present or future.

Is this how you conceptualise "knowledge"?
Is this a system you intend to build?

Either way - it sure sounds to me like you are oblivious to the analytic/synthetic distinction as it exists in systems theory (N.B NOT the Kantian distinction).

Further observation: you are obsessed with 100% certain truths in a world that has exactly none of those. Time for some probability theory in your episteme?

Here is the user's manual. If you want to know more read the book.
https://www.cyc.com/documentation/ontologists-handbook/

Can we get off this extraneous topic and get back to Tarski?
The only semantics that Tarski cares about is formalizing the notions of True and False.
He tries to do this without even having an object of reference.
The key sentence of his whole proof is: "This sentence is not true".

This is the only sentence that he evaluates in his whole proof:

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.

Showing how this one sentence is false and we are done with Tarski
3) x ∉ Pr if and only if x ∈ Tr // page 275

http://liarparadox.org/Tarski_Proof_275_276.pdf

[PeteOlcott]

It seems to me you are allowing the model used by Cyc to influence your thinking.
Have you considered that the model Cyc uses is simply the product of the choices of those who engineered it?
Have you considered that there are more complex, more complete knowledge-bases than Cyc out there? Google....

Over a thirty year period we independently came up with the same model because
there is only one correct way to do this.

[PeteOlcott]

Since it is an inheritance hierarchy it is directed.

What is at the top of the inheritance hierarchy? Things? A thing is a concept.
What is at the bottom of the hierarchy? Domain-specific data?

How many domains are there?

By the theory of simple types I mean the doctrine which says that the objects of thought (or, in another interpretation, the symbolic expressions) are divided into types, namely: individuals, properties of individuals, relations between individuals, properties of such relations,

The above indicates to me that you and I are definitely not on the same page.

Objects of thought? Do you have an ontological model for thought? Where does meta-cognition (thinking about thinking) come into your graph?

Think of it as an acyclic graph that contains every detail of the set of all general knowledge
with each individual concept at its optimal place within the graph.

I have spent 22 years fixing one element of the upper portion of the graph {the truth predicate}.

If we get off topic it will take another 10,000 years to finish this one thing.
If we stay on topic we could have fully formalized the notion of truth today.
It is already fully formalized and succinctly elaborated I only need to get people to understand
its succinct speciation and I am done it will be publishable as soon as it can be understood.

[PeteOlcott]

So, do you recognize that what you are showing is us is just one of infinitely many possible taxonomies?
Taxonomy is the product of the Categorization process
which takes place in the mind.

And you have no model of the mind.... so why is your taxonomy better than my taxonomy?

Within an optimization criteria this taxonomy becomes unique.
The taxonomy <is> the model of the mind AI does not realize this.

AI is modeling brains an deriving machines that are great at playing video games
yet utterly clueless about the meaning of words.

[Logik]

Here is the user's manual. If you want to know more read the book.
https://www.cyc.com/documentation/ontologists-handbook/

Way to dodge all my points. Manuals do not contain any of the design/decision-making behind the system.

Intent is a vital piece of the puzzle for all man-made things. And "Truth" is not exempt from this.

Can we get off this extraneous topic and get back to Tarski?
The only semantics that Tarski cares about is formalizing the notions of True and False.
He tries to do this without even having an object of reference.
The key sentence of his whole proof is: "This sentence is not true".

And I already offered you a system which interprets it as true and a system which interprets it as false.

Which is back to my point.

IF you want the sentence to be true - use system A.
IF you want the sentence to be false - use system B.

Showing how this one sentence is false and we are done with Tarski
3) x ∉ Pr if and only if x ∈ Tr // page 275

OK, So construct a logical system that gives you the answer you want.

[PeteOlcott]

Can we get off this extraneous topic and get back to Tarski?
The only semantics that Tarski cares about is formalizing the notions of True and False.
He tries to do this without even having an object of reference.
The key sentence of his whole proof is: "This sentence is not true".

And I already offered you a system which interprets it as true and a system which interprets it as false.

Which is back to my point.

IF you want the sentence to be true - use system A.
IF you want the sentence to be false - use system B.

Showing how this one sentence is false and we are done with Tarski
3) x ∉ Pr if and only if x ∈ Tr // page 275

OK, So construct a logical system that gives you the answer you want.

Even Prolog's standard inference model directly refutes the Third
line of Tarski's proof:

http://liarparadox.org/Tarski_Proof_275_276.pdf
(3) x ∉ Pr if and only if x ∈ Tr

~Provable(x) ↔ True(x)
Because of the way that the Prolog inference model works the above
expression would be false, and its negation: Provable(x) ↔ True(x)
would be true. So the Prolog inference model already refutes Tarski's
line (3) which in turn causes the whole rest of Tarski's proof to fail.

[PeteOlcott]

With a very slight change to how formal systems are defined undecidability can
no longer be expressed in any of these formal systems.

(1) True(x) ↔ (⊢x)
A set of facts adds up to X being TRUE.

(2) False(x) ↔ (⊢~x)
A set of facts adds up to X being FALSE.

(3) ~True(x) ↔ ~(⊢ x)
There is no set of facts that add up to X being TRUE.

In other words it it is impossible to prove True or False then it is decided to have
a truth value of ~True making every closed WFF of every formal system decidable.

Replaced with a continually updated PDF
https://www.researchgate.net/publicatio ... al_Systems

[Logik]

With a very slight change to how formal systems are defined undecidability can
no longer be expressed in any of these formal systems.

(1) True(x) ↔ (⊢x)
A set of facts adds up to X being TRUE.

(2) False(x) ↔ (⊢~x)
A set of facts adds up to X being FALSE.

(3) ~True(x) ↔ ~(⊢ x)
There is no set of facts that add up to X being TRUE.

In other words it it is impossible to prove True or False then it is decided to have
a truth value of ~True making every closed WFF of every formal system decidable.

Replaced with a continually updated PDF
https://www.researchgate.net/publicatio ... al_Systems

*yawn*

Here is an algorithm for generating a set:
https://repl.it/repls/VerticalPlasticImplementation

Code: Select all

from numpy.random import choice

SAMPLE_SIZE=100**100**100**100
P_TRUE = 1/( SAMPLE_SIZE-1 )

class undecidable(object):
  def __eq__(self, other):
    p_true = P_TRUE
    p_false = 1 - p_true
    return choice( [True, False], p = [p_true, p_false])

sample_set = []
for i in range(SAMPLE_SIZE):
  sample.append(undecidable())

Please help me decide the truth-value of: for all x in sample_set: x == x