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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic,comp.theory
Subject: Re: Undecidability based on epistemological antinomies V2 --H(D,D)--
Date: Wed, 24 Apr 2024 10:33:12 -0500
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On 4/24/2024 3:35 AM, Mikko wrote:
> On 2024-04-23 14:31:00 +0000, olcott said:
> 
>> On 4/23/2024 3:21 AM, Mikko wrote:
>>> On 2024-04-22 17:37:55 +0000, olcott said:
>>>
>>>> On 4/22/2024 10:27 AM, Mikko wrote:
>>>>> On 2024-04-22 14:10:54 +0000, olcott said:
>>>>>
>>>>>> On 4/22/2024 4:35 AM, Mikko wrote:
>>>>>>> On 2024-04-21 14:44:37 +0000, olcott said:
>>>>>>>
>>>>>>>> On 4/21/2024 2:57 AM, Mikko wrote:
>>>>>>>>> On 2024-04-20 15:20:05 +0000, olcott said:
>>>>>>>>>
>>>>>>>>>> On 4/20/2024 2:54 AM, Mikko wrote:
>>>>>>>>>>> On 2024-04-19 18:04:48 +0000, olcott said:
>>>>>>>>>>>
>>>>>>>>>>>> When we create a three-valued logic system that has these
>>>>>>>>>>>> three values: {True, False, Nonsense}
>>>>>>>>>>>> https://en.wikipedia.org/wiki/Three-valued_logic
>>>>>>>>>>>
>>>>>>>>>>> Such three valued logic has the problem that a tautology of the
>>>>>>>>>>> ordinary propositional logic cannot be trusted to be true. For
>>>>>>>>>>> example, in ordinary logic A ∨ ¬A is always true. This means 
>>>>>>>>>>> that
>>>>>>>>>>> some ordinary proofs of ordinary theorems are no longer valid 
>>>>>>>>>>> and
>>>>>>>>>>> you need to accept the possibility that a theory that is 
>>>>>>>>>>> complete
>>>>>>>>>>> in ordinary logic is incomplete in your logic.
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> I only used three-valued logic as a teaching device. Whenever an
>>>>>>>>>> expression of language has the value of {Nonsense} then it is
>>>>>>>>>> rejected and not allowed to be used in any logical operations. It
>>>>>>>>>> is basically invalid input.
>>>>>>>>>
>>>>>>>>> You cannot teach because you lack necessary skills. Therefore you
>>>>>>>>> don't need any teaching device.
>>>>>>>>>
>>>>>>>>
>>>>>>>> That is too close to ad homimen.
>>>>>>>> If you think my reasoning is incorrect then point to the error
>>>>>>>> in my reasoning. Saying that in your opinion I am a bad teacher
>>>>>>>> is too close to ad hominem because it refers to your opinion of
>>>>>>>> me and utterly bypasses any of my reasoning.
>>>>>>>
>>>>>>> No, it isn't. You introduced youtself as a topic of discussion so
>>>>>>> you are a legitimate topic of discussion.
>>>>>>>
>>>>>>> I didn't claim that there be any reasoning, incorrect or otherwise.
>>>>>>>
>>>>>>
>>>>>> If you claim I am a bad teacher you must point out what is wrong with
>>>>>> the lesson otherwise your claim that I am a bad teacher is 
>>>>>> essentially
>>>>>> an as hominem attack.
>>>>>
>>>>> You are not a teacher, bad or otherwise. That you lack skills that
>>>>> happen to be necessary for teaching is obvious from you postings
>>>>> here. A teacher needs to understand human psychology but you don't.
>>>>>
>>>>
>>>> You may be correct that I am a terrible teacher.
>>>> None-the-less Mathematicians might not have very much understanding
>>>> of the link between proof theory and computability.
>>>
>>> Sume mathematicians do have very much understanding of that. But that
>>> link is not needed for understanding and solving problems separately
>>> in the two areas.
>>>
>>>> When I refer to rejecting an invalid input math would seem to construe
>>>> this as nonsense, where as computability theory would totally 
>>>> understand.
>>>
>>> People working on computability theory do not understand "invalid input"
>>> as "impossible input".
>>
>> The proof then shows, for any program f that might determine whether
>> programs halt, that a "pathological" program g, called with some input,
>> can pass its own source and its input to f and then specifically do the
>> opposite of what f predicts g will do. No f can exist that handles this
>> case, thus showing undecidability.
>> https://en.wikipedia.org/wiki/Halting_problem#
>>
>> So then they must believe that there exists an H that does correctly
>> determine the halt status of every input, some inputs are simply
>> more difficult than others, no inputs are impossible.
> 
> That "must" is false as it does not follow from anything.
> 

Sure it does. If there are no "impossible" inputs that entails
that all inputs are possible. When all inputs are possible then
the halting problem proof is wrong.

*Termination Analyzer H is Not Fooled by Pathological Input D*
https://www.researchgate.net/publication/369971402_Termination_Analyzer_H_is_Not_Fooled_by_Pathological_Input_D 


Everyone that objects to the statement that H(D,D) correctly determines 
the halt status of its inputs say that believe that H(D,D) must report 
on the behavior of the D(D) that invokes H(D,D).

They say this knowing full well that computable functions only operate 
on their inputs. This also violates the definition of a decider that 
only computes the mapping from its inputs. Thus expecting H(D,D) to 
report on the behavior of the D(D) that invokes H(D,D) violates two core 
principles of of computer science.

Finally the behavior of the simulated D(D) before H aborts its 
simulation is different than the behavior of the executed D(D) after H 
has aborted its simulation. H(D,D) must report on the behavior that it 
actually sees.

>>> They understand it as an input that must be
>>> handled differently from ordinary input. Likewise, mathematicians do
>>> understand that some inputs must be considered separately and 
>>> differently.
>>> But mathematicians don't call those inputs "invalid".
> 
>> It is so dead obvious that the whole world must be wired with a short
>> circuit in their brains. Formal bivalent mathematical systems of logic
>> must reject every expression that cannot possibly have a value of true
>> or false as a type mismatch error.
> 
> Gödel's completeness theorem proves that every consistent first order
> theory has a model, i.e., there is an interpretation that assigns a
> truth value to every formula of the theory. No such proof is known for
> second or higher order theories.
> 

By switching from model theory to proof theory we need no
interpretations. Every system of logic is simply relations
between finite strings.

To get rid of undecidability and incompleteness we simply encode all of
the facts of the general knowledge of the actual world as axioms of a
formal system of logic.

True(L, x)  ≡ ∃x ∈ L (L ⊢ x)
False(L, x) ≡ ∃x ∈ L (L ⊢ x)
Truth_Bearer(L, x) ≡ ∃x ∈ L (True(L, x) ∨ False(L, x))

>> 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
> 
> In formal logic the corresponding concept is sentence.
> 

-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer