## Is this an improved definition of a truth bearer?

What is the basis for reason? And mathematics?

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PeteOlcott
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### Re: Is this an improved definition of a truth bearer?

Skepdick wrote:
Tue Feb 25, 2020 9:18 pm
PeteOlcott wrote:
Tue Feb 25, 2020 9:12 pm
My architectural design of the redefinition of a formal system self-evidently does what it claims to anyone that can understand what I am saying.
I don't want to "understand what you are saying". I want you to show me a working prototype. Practical, not theoretical solution.
Understanding the architectural design makes the feasibility of a prototype self-evident.

This is the second page of the published Incompleteness Theorem proof.

When this bijection is specified:
The chain of symbolic manipulations in the calculus corresponds to
and represents the chain of deductions in the deductive system.
Then formal proof and deduction cannot possibly diverge.

Skepdick
Posts: 4403
Joined: Fri Jun 14, 2019 11:16 am

### Re: Is this an improved definition of a truth bearer?

PeteOlcott wrote:
Tue Feb 25, 2020 9:30 pm
Understanding the architectural design makes the feasibility of a prototype self-evident.
No, it doesn't. Building the prototype makes the feasibility of the prototype self-evident.
What you are promising is a function with a type-signature of f: Any -> Boolean

Either you are a brilliant, or you are a fool. My money is on the latter, but if you show me a prototype you'll change my mind.

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

PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Is this an improved definition of a truth bearer?

Skepdick wrote:
Tue Feb 25, 2020 9:38 pm
PeteOlcott wrote:
Tue Feb 25, 2020 9:30 pm
Understanding the architectural design makes the feasibility of a prototype self-evident.
No, it doesn't. Building the prototype makes the feasibility of the prototype self-evident.
What you are promising is a function with a type-signature of f: Any -> Boolean

Either you are a brilliant, or you are a fool. My money is on the latter, but if you show me a prototype you'll change my mind.

https://en.wikipedia.org/wiki/Binary_classification
I am making my refutation of the Halting Problem concrete by writing the halt decider
in X86 machine language and using a software X86 emulator as the UTM.

None-the-less if you cannot understand what I am saying at most abstract level
you will probably say that you simply don't believe my soon to be fully operational
concrete example of halt deciding even though you could watch every step.

Skepdick
Posts: 4403
Joined: Fri Jun 14, 2019 11:16 am

### Re: Is this an improved definition of a truth bearer?

PeteOlcott wrote:
Tue Feb 25, 2020 10:07 pm
I am making my refutation of the Halting Problem concrete by writing the halt decider
in X86 machine language and using a software X86 emulator as the UTM.
Cool. So if I apply your halt decider to itself, would it be able to tell me if your halt decider will halt for all possible inputs?

PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Is this an improved definition of a truth bearer?

Skepdick wrote:
Tue Feb 25, 2020 10:29 pm
PeteOlcott wrote:
Tue Feb 25, 2020 10:07 pm
I am making my refutation of the Halting Problem concrete by writing the halt decider
in X86 machine language and using a software X86 emulator as the UTM.
Cool. So if I apply your halt decider to itself, would it be able to tell me if your halt decider will halt for all possible inputs?
Itself is only one single input, not many different inputs.
It can decide many different variations of itself each one corresponding to the conventional counter-example that has been presumed to be undecidable.

PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Is this an improved definition of a truth bearer?

This is another way of saying all these same sort of things

https://en.wikipedia.org/wiki/Curry%E2% ... espondence
In other words, the Curry–Howard correspondence is the observation that two families of seemingly unrelated formalisms—namely, the proof systems on one hand, and the models of computation on the other—are in fact the same kind of mathematical objects. I refer to finite string transformations by an algorithm.

From the published Incompleteness Theorem Proof
The chain of symbolic manipulations in the calculus corresponds to
and represents the chain of deductions in the deductive system.

Curry, Haskell 1977. Foundations of Mathematical Logic. New York: Dover Publications, 45
Then the elementary statements which belong to {T} we shall call the elementary theorems of {T}; we also say that these elementary statements are true for {T}. Thus, given {T}, an elementary theorem is an elementary statement which is true. A theory is thus a way of picking out from the statements of {E} a certain subclass of true statements…

When we combine all that together we get finite strings that have been assigned the semantic value of Boolean true as the complete set of true premises. Truth preserving finite string transformations representing deductive inference and conclusions that are necessarily true because the argument is deductively sound.

The above shows how the whole notion of analytical True(X) can be one-and-the-same-thing as a theorem of a formal system, thus eliminating incompleteness. Also the formal system can be the entire body of analytical truth.

Skepdick
Posts: 4403
Joined: Fri Jun 14, 2019 11:16 am

### Re: Is this an improved definition of a truth bearer?

PeteOlcott wrote:
Wed Feb 26, 2020 12:02 am
Itself is only one single input, not many different inputs.
It can decide many different variations of itself each one corresponding to the conventional counter-example that has been presumed to be undecidable.
Pete, you know what instantiation is, yes? Lets say we have two instances of your decider, A and B.

Say we have expression P given to decider A. Expressed as A(P). Can decider B decide whether A(P) halts?

e.g Does B(A(P)) halt ?

Skepdick
Posts: 4403
Joined: Fri Jun 14, 2019 11:16 am

### Re: Is this an improved definition of a truth bearer?

PeteOlcott wrote:
Wed Feb 26, 2020 6:20 am
https://en.wikipedia.org/wiki/Curry%E2% ... espondence
In other words, the Curry–Howard correspondence is the observation that two families of seemingly unrelated formalisms—namely, the proof systems on one hand, and the models of computation on the other—are in fact the same kind of mathematical objects. I refer to finite string transformations by an algorithm.
The Curry-Howard-Lambek correspondence is about the syntactic equivalence between proofs and programs. Syntax is not semantics. Not to a person who subscribes to denotational semantics anyway.

And so "provable mathematical statement" is exactly the same thing as "halting algorithm", sure. But "provable" does not mean "true".

On the meaning of logical rules I: syntax versus semantics

PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Is this an improved definition of a truth bearer?

Skepdick wrote:
Wed Feb 26, 2020 7:30 am
PeteOlcott wrote:
Wed Feb 26, 2020 12:02 am
Itself is only one single input, not many different inputs.
It can decide many different variations of itself each one corresponding to the conventional counter-example that has been presumed to be undecidable.
Pete, you know what instantiation is, yes? Lets say we have two instances of your decider, A and B.

Say we have expression P given to decider A. Expressed as A(P). Can decider B decide whether A(P) halts?

e.g Does B(A(P)) halt ?
I merely show that it is impossible to construct any H such that H(H, H) is undecidable.
I do that by creating the conventional counter-example and show how it correctly decides itself.

PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Is this an improved definition of a truth bearer?

Skepdick wrote:
Wed Feb 26, 2020 7:35 am
PeteOlcott wrote:
Wed Feb 26, 2020 6:20 am
https://en.wikipedia.org/wiki/Curry%E2% ... espondence
In other words, the Curry–Howard correspondence is the observation that two families of seemingly unrelated formalisms—namely, the proof systems on one hand, and the models of computation on the other—are in fact the same kind of mathematical objects. I refer to finite string transformations by an algorithm.
The Curry-Howard-Lambek correspondence is about the syntactic equivalence between proofs and programs. Syntax is not semantics. Not to a person who subscribes to denotational semantics anyway.

And so "provable mathematical statement" is exactly the same thing as "halting algorithm", sure. But "provable" does not mean "true".

On the meaning of logical rules I: syntax versus semantics
If you carefully examine all of what I said you will see how a formal system can be defined
such that provable does mean true and true means provable.

The exact same semantic interconnected relationships that cause a conclusion to be deduced
from its true premises can be transformed into an algorithm operating on finite strings.

Skepdick
Posts: 4403
Joined: Fri Jun 14, 2019 11:16 am

### Re: Is this an improved definition of a truth bearer?

PeteOlcott wrote:
Wed Feb 26, 2020 7:46 am
I merely show that it is impossible to construct any H such that H(H, H) is undecidable.
Pete, show me the implementation of the halts() function below.

https://repl.it/repls/CreativeSecondhandPreprocessor

Code: Select all

def halts(x):
// Fix this function
return False

def f():
if halts(f):
while True:
continue
else:
print('Function f does not halt.')

f()
Last edited by Skepdick on Wed Feb 26, 2020 7:57 am, edited 1 time in total.

Skepdick
Posts: 4403
Joined: Fri Jun 14, 2019 11:16 am

### Re: Is this an improved definition of a truth bearer?

PeteOlcott wrote:
Wed Feb 26, 2020 7:50 am
If you carefully examine all of what I said you will see how a formal system can be defined such that provable does mean true and true means provable.
Pete, pay attention.

If provable means true, and your proof-algorithm returns Boolean:False, then you have literally proven false to be true..

That's the definition of an inconsistent system!

PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Is this an improved definition of a truth bearer?

Skepdick wrote:
Wed Feb 26, 2020 7:56 am
PeteOlcott wrote:
Wed Feb 26, 2020 7:50 am
If you carefully examine all of what I said you will see how a formal system can be defined such that provable does mean true and true means provable.
Pete, pay attention.

If provable means true, and your proof-algorithm returns Boolean:False, then you have literally proven false to be true..

That's the definition of an inconsistent system!
There are some expressions of language that because of their structure cannot possibly be resolved to True or False.
These are rejected as ill-formed truth bearers.

If you can't prove that it is true then it is not true. If you can't prove that it is false then it is not false.
If you can't prove either one then it is not a truth bearer which must be true or false.

PeteOlcott
Posts: 970
Joined: Mon Jul 25, 2016 6:55 pm

### Re: Is this an improved definition of a truth bearer?

Skepdick wrote:
Wed Feb 26, 2020 7:54 am
PeteOlcott wrote:
Wed Feb 26, 2020 7:46 am
I merely show that it is impossible to construct any H such that H(H, H) is undecidable.
Pete, show me the implementation of the halts() function below.

https://repl.it/repls/CreativeSecondhandPreprocessor

Code: Select all

def halts(x):
// Fix this function
return False

def f():
if halts(f):
while True:
continue
else:
print('Function f does not halt.')

f()
I use the Linz version linked above because is specifies key details about state transitions.
I created a simplification of the Linz version using X86 machine code that analyses itself
in an X86 emulator.

Skepdick
Posts: 4403
Joined: Fri Jun 14, 2019 11:16 am

### Re: Is this an improved definition of a truth bearer?

PeteOlcott wrote:
Wed Feb 26, 2020 3:28 pm
There are some expressions of language that because of their structure cannot possibly be resolved to True or False.
These are rejected as ill-formed truth bearers.
So what?

Just because you can't resolve any Russian sentences to True or False it doesn't mean the sentences are ill-formed.

It only means you lack comprehension of the language.

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