Re: The simple essence of Proof Theoretic Semantics

On 7/1/2026 5:04 PM, olcott wrote:
> On 7/1/2026 3:57 PM, dbush wrote:
>> On 7/1/2026 4:50 PM, olcott wrote:
>>> On 7/1/2026 3:37 PM, dbush wrote:
>>>> On 7/1/2026 4:29 PM, olcott wrote:
>>>>> On 7/1/2026 3:13 PM, André G. Isaak wrote:
>>>>>> On 2026-07-01 13:53, olcott wrote:
>>>>>>> On 7/1/2026 2:31 PM, André G. Isaak wrote:
>>>>>>>> On 2026-07-01 12:51, olcott wrote:
>>>>>>>>> On 7/1/2026 1:45 PM, André G. Isaak wrote:
>>>>>>>>>> On 2026-07-01 12:15, dbush wrote:
>>>>>>>>>>> On 7/1/2026 2:01 PM, olcott wrote:
>>>>>>>>>>
>>>>>>>>>>>> The same thing as: "cats are animals" expressed in
>>>>>>>>>>>> English has no English meaning in Chinese.
>>>>>>>>>>>
>>>>>>>>>>> I didn't ask for an example.  I asked for a definition of 
>>>>>>>>>>> what it mean for the truth value of a statement to not exist 
>>>>>>>>>>> in a formal system.
>>>>>>>>>>
>>>>>>>>>> I'm actually not convinced that Olcott understands what a 
>>>>>>>>>> definition is. I've frequently asked him for definitions and 
>>>>>>>>>> he invariably responds with an example or an analogy (assuming 
>>>>>>>>>> he responds at all). He doesn't get that examples don't take 
>>>>>>>>>> the place of definitions. Examples can be useful for 
>>>>>>>>>> clarifying definitions, but they aren't particularly useful on 
>>>>>>>>>> their own.
>>>>>>>>>>
>>>>>>>>>> André
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> You want a definition look-it-up.
>>>>>>>>
>>>>>>>> Until someone publishes an Olcott to Standard English 
>>>>>>>> dictionary, this isn't really an option.
>>>>>>>>
>>>>>>>> André
>>>>>>>>
>>>>>>>
>>>>>>> True(L, X) ≡ ∃Γ ⊆ BaseFacts(L) (Γ ⊢ X) // copyright Olcott 2018
>>>>>>> has been updated to this
>>>>>>>
>>>>>>> True(L, X):= ∃Γ ⊆ AtomicFacts(L) (Γ ⊢ X)
>>>>>>
>>>>>> That claims what it means to have the truth value 'true' (or at 
>>>>>> least it would if you defined AtomicFacts in a coherent way). It 
>>>>>> doesn't in any way clarify what you think it means for something 
>>>>>> to not have a truth value.
>>>>>>
>>>>>> André
>>>>>>
>>>>>
>>>>> When I define a term hundreds of times and you did
>>>>> not bother to pay attention that is your mistake
>>>>> and your fault.
>>>>>
>>>>
>>>> You gave no such definition of what it means for the truth value of 
>>>> a statement to not exist in a formal system.
>>>>
>>>> A valid answer would look something like this:
>>>>
>>>> "The truth value of a statement does not exist in a formal system 
>>>> when ..."
>>>>
>>>> Now complete the sentence.
>>>
>>> It is neither provable nor refutable in F.
>>
>> Good.  So when you say "The truth value of (∀ x, S(x) ≠ x) does not 
>> exist in Q", you mean "(∀ x, S(x) ≠ x) is unprovable in Q", which is 
>> commonly known.
>>
>> So once again, you're saying the same thing as everyone else but using 
>> different words.
>>
> 
> Not really. It is normally thought of as undecidable
> meaning that Q is incomplete meaning that Q is deficient.

False.  It means that there are statements in the language of Q that 
have *only* an infinite connection to the axioms of the system.

> 
> The Halting Problem counter-example input 

Which starts with the assumption that an algorithm H exists that meets 
the following requirements:

Given any algorithm (i.e. a fixed immutable sequence of instructions) X 
described as <X> with input Y:

A solution to the halting problem is an algorithm H that computes the 
following mapping:

(<X>,Y) maps to 1 if and only if X(Y) halts when executed directly
(<X>,Y) maps to 0 if and only if X(Y) does not halt when executed directly


> is more like
> the Liar Paradox. For HHH(DD) 

Which must return 1 to meet the above requirements, since finite string 
DD is stipulated to be the description of algorithm DD which halts when 
executed directly.

> DD is bad input 

which is just another way of saying it gets the wrong answer.

> not at all
> the same thing as saying that computation is fundamentally
> limited.

Sure it does, as no algorithm exists that can compute the above mapping


> 
>> And just to remind everyone, the following terms mean the same as 
>> "unprovable" as per the definitions given by olcott:
>>
>> - truth value doesn't exist
>> - out-of-scope
>> - not semantically grounded
>> - not grounded in the atomic base
>> - not a confirmed statement
>>
>>
>>
>>
>>> I have said this so many thousands of times over
>>> the years that it did not occur to me that I did
>>> not fully spell this out this time.
>>>
>>>
>>
> 
>