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Re: Making the body of knowledge computable

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On 2/12/26 11:06 AM, olcott wrote:
> On 2/12/2026 6:29 AM, Richard Damon wrote:
>> On 2/11/26 8:08 AM, olcott wrote:
>>> On 2/11/2026 6:56 AM, Richard Damon wrote:
>>>> On 2/10/26 11:59 PM, olcott wrote:
>>>>> We completely replace the foundation of truth conditional
>>>>> semantics with proof theoretic semantics. Then expressions
>>>>> are "true on the basis of meaning expressed in language"
>>>>> only to the extent that all their meaning comes from
>>>>> inferential relations to other expressions of that language.
>>>>> This is a purely linguistic PTS notion of truth with no
>>>>> connections outside the inferential system.
>>>>>
>>>>> Well-founded proof-theoretic semantics reject expressions
>>>>> lacking a "well-founded justification tree" as meaningless.
>>>>> ∀x (~Provable(T, x) ⇔ Meaningless(T, x))
>>>>>
>>>>
>>>> The problem is that you new system can't handle mathematics.
>>>>
>>>> The problem, as has been pointed out, is that mathematics, by the 
>>>> axiom of induction, accepts as true statements that can be 
>>>> established by an infinite number of steps as true, and shows a 
>>>> method to solve SOME of them.
>>>>
>>>> Also, "Halting" is a well-founded property of ALL machines, as they 
>>>> MUST either Halt or not, and HALTING is always provable, so those 
>>>> machines that do not halt, must be non-halting.
>>>>
>>>> Your "logic" essentially denies the property of the excluded middle 
>>>> for systems that have infinite members, but some statements are 
>>>> inherently of the class of the excluded middle.
>>>>
>>>> As I have said, TRY to show how your PTS can establish the 
>>>> mathematics of the Natural Numbers.
>>>>
>>>> Try to even fully define ADDITION without the need for allowing 
>>>> unbounded steps.
>>>>
>>>
>>> ∀x ∈ PA (  True(PA, x) ≡ PA ⊢  x )
>>> ∀x ∈ PA ( False(PA, x) ≡ PA ⊢ ¬x )
>>> ∀x ∈ PA ( ¬WellFounded(PA, x) ≡ (¬True(PA, x) ∧ (¬False(PA, x)))
>>
>> So, where is "Addition" in that?
>>
>> How do you determine ~True(PA, x)? in your proof-Theoretic semantics?
>>
> 
> 0=1

So, you start with a false statement?

> equal(successor(0), successor(successor(0))==FALSE

In other words, you just can't do it.

It seems you believe in logic by fallacy.

> 
>>>
>>>    "What is the appropriate notion of truth for sentences whose
>>>    meanings are understood in epistemic terms such as proof or
>>>    ground for an assertion? It seems that the truth of such
>>>    sentences has to be identified with the existence of proofs or
>>>    grounds..." https://doi.org/10.1007/s11245-011-9107-6
>>
>> A question in General Philosophy, not Formal Logic.
>>
>>>
>>> Spend 20 hours carefully studying this and get back to me.
>>> https://plato.stanford.edu/entries/proof-theoretic-semantics/
>>
>> Which is a paper on PHILOSOPHY, not Formal Logic.
>>
> 
> Logic and math choose a notion of truth from philosophy and
> the choose the wrong one.

Nope. I guess you are just admitting that you disagree with what logic 
is, and thus are not using it.

> 
> ∀x (Provable(T, x) ⇔ Meaningful(T, x)) --- (Schroeder-Heister 2024)
> ∀x (Provable(x) ⇒ True(x)) --- Anchored in (Prawitz, 2012)

Your repeating without proper refence just proves your ignorance.

> 
>> Note, Formal Logic BEGINS with its definiton of Truth, which is based 
>> on the, possibly infinite, application of its logical rules.
>>
> 
> As you yourself kept harping on some times true(x)
> exists outside of the domain of knowledge.

Right, things are true that can't be known, but the definition of how 
truth is determined is foundational.

I guess you are just showing you don't understand what the words mean.

> 
>> To change that to a Proof-Theoretical Semantics basis changes the 
>> results of the system. In particular, any system that generates an 
>> infinite domain (like mathemtics) becomes problematic.
>>
> 
> Only in cases where a truth value requires infinite
> inference steps.

So?

Since they exist, your criteria is just a lie.

> 
>>>
>>> It makes "true on the basis of meaning expressed in language"
>>> reliably computable for the entire body of knowledge.
>>>
>>
>> Nope.
>>
>> IT makes your definition of "true" just a lie.
>>
> 
> ∀x (Provable(T, x) ⇔ Meaningful(T, x)) --- (Schroeder-Heister 2024)
> ∀x (Provable(x) ⇒ True(x)) --- Anchored in (Prawitz, 2012)



> 
> That you are woefully ignorant of PTS does not entail
> that I am incorrect. What you call an intentional falsehood
> has always been your own ignorance.

Why do you think I am ignorant of it, and not yourself?

I have pointed out the problem with your interpreation, and you best 
answer seems to be agreeing with me that these cases occur.

> 
>> How can you "compute" if a number exist that satisfies the 
>> relationship that Godel developes in his proof?
>>
> 
> PTS has no notion of satisfies.

Sure it does. Maybe you don't.

The expression x - 2 = 0 is statisfied by the value of x = 2.

A number satisfies a relationship, if it results in a "true" result in 
evaluating it.

I guess you don't think PTS can evaluate statements of know values to 
evaluate if they are true.

> 
>> Why does that relationship, which is just built from the fundamental 
>> operation of mathematics in PA not have "meaning"?
>>
> 
> When any expression of language lacks a semantic connection
> to the expressions that define it this expression remains
> undefined.
> 

But a "semantic connection" can involve an infinite number os semantic 
steps.

>> Where is the line between that relationship, and the statement that we 
>> can assert that 1 + 1 = 2?
>>
> 
> successor(0) + successor(0) = successor(successor(0))

So, where is the line between that and evaluating the relationship in 
Godel's G.

It seems you just divert to trivalities when presented with problems 
that require you to think.

Your claim is that PTS can't evalute if a number makes the relationship 
in G true, but it could evaluate 1 + 1 = 2.

Where is the line between the two?

At what point does an expression get "too complicated" to evaluate?

Your problem is you just don't understand what you are talking about.

> 
>> Proof-Thoeretic Semantics in sets with infinite members just severly 
>> limits what you can do in that field.
>>
>> And that is why real Proof-Theoretic Semantics doesn't assert that 
>> what we haven't proven is meaningless, just that we don't yet know the 
>> meaning, because it knows that all we can know is that we haven't yet 
>> proven something, and not that it can't be proven, unless we can 
>> actually find a proof of that (like proving its opposite).
>>
> 
> It could be meaningless or it could be unknown either
> way is is outside of the domain of knowledge.

But you aren't talking about KNOWLEDGE, but TRUTH.

The fact that we don't know the answer, doesn't mean that the answer 
doesn't exist.

That is your problem, you can't TELL if something is actually 
"meaningless" unless you can make it part of knowledge that it really 
can't be proven or disproven.

Thus, statements like G can't be called "meaningless" in PA, because you 
can't actually PROVE in PA that you can't prove or disprove it.

In fact, you CAN prove in PA that if G isn't true, you must be able to 
prove that. Thus, G can't be Proof-Therotically non-well founded, as a 
proof that it can't be disproven, a necessary requirement to prove it 
non-well-founded, turns out to be a proof that it is true.

Thus, your PTS is by necessity, needs to accept Truth-Conditional 
semantics for determining its PTS, and accept non-well-foundedness that 
can't be proven.

> 
>> Your system becomes a lie, because while you assert you are using just 
>> Proof-Theoretical Semantics, you need to actully have Truth- 
>> Conditional logic to determine those semantics, as there ARE things 
>> not provablable unprovable, and thus your tri-valued system (true, 
>> false, non-well- founded) can't have values exstablished by proof- 
>> thoeretic logic.
>>
> 
> Like I already said carefully study the article written by
> the guy that coined the term: Proof-Theoretic Semantics
> https://plato.stanford.edu/entries/proof-theoretic-semantics/
> 

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Making the body of knowledge computable olcott <polcott333@gmail.com> - 2026-02-10 22:59 -0600
  Re: Making the body of knowledge computable Richard Damon <Richard@Damon-Family.org> - 2026-02-11 07:56 -0500
    Re: Making the body of knowledge computable olcott <polcott333@gmail.com> - 2026-02-11 07:08 -0600
      Re: Making the body of knowledge computable Richard Damon <Richard@Damon-Family.org> - 2026-02-12 07:29 -0500
        Re: Making the body of knowledge computable olcott <polcott333@gmail.com> - 2026-02-12 10:06 -0600
          Re: Making the body of knowledge computable Richard Damon <Richard@Damon-Family.org> - 2026-02-12 23:11 -0500

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