Re: The simple essence of Proof Theoretic Semantics

On 7/1/2026 7:39 PM, André G. Isaak wrote:
> On 2026-07-01 18:05, olcott wrote:
>> On 7/1/2026 6:09 PM, André G. Isaak wrote:
> 
>> Not exactly because most every human has been too stupid
>> to understand that "This sentence is not true" is a semantically
>> incoherent declarative sentence. Even the great Saul Kripke
>> (did better than everyone else) yet did not quite get there.
> 
> Claiming that it is semantically incoherent is *your* view. 

It <is> semantically incoherent in such a way that
anyone disagreeing is ABSOLUTELY INCORRECT.

> It is hardly 
> universally accepted and therefore you are required to actually defend 
> this view rather than simply assert it.
> 

It used to be universally agree that the Earth is flat.

> Also, that isn't the sentence we are considering. We are considering ∀ 
> x, S(x) ≠ x in Q. There is no reason to think that any claim you might 
> make about the LP is also applicable to this sentence.
> 

We are also considering that sentence.

>>> And ∀ x, S(x) ≠ x is most definitely a truth bearer.
>>>
>>
>> If is it not provable in Q then it is not a truth
>> bearer in Q. Because we can see that it is provable
>> in PA this causes us to screw up and think that this
>> means that it is true in Q.
> 
> I never claimed that it was true, nor did I claim that it was false. I 
> simply claimed that it was a truth-bearer without committing to its 
> actual truth value.
> 

Yes that that is the vagueness that prevents much
of what is semantically incoherent to be undecidable.
Q was intentionally defined to be weaker than PA.

> Do you actually understand *why* ∀ x, S(x) ≠ x is not provable in Q? 
> Until you understand this you really don't have a good grasp of what it 
> means for Q to be incomplete.
> 

it lacks the mathematical induction axiom schema
required to generalize this rule to all elements in a domain

> The reason why we cannot prove that ∀ x, S(x) ≠ x in Q is because it is 
> possible in Q to construct a model in which that statement is *false*. 

Not when you make sure to completely and totally toss
model theory out on its ass and replace it with proof
theoretic semantics instead.

> Such a model would not correspond to the natural numbers as commonly 
> understood, but it would be a consistent model. In a model corresponding 
> to the natural numbers as commonly understood, this statement would be 
> *true*.
> 
> For any given model of Q, ∀ x, S(x) ≠ x is either true or it is false. 
> It is never some indeterminate value. But the truth value of this 
> statement cannot be proven solely by considering the axioms of Q. We 
> need to look at the actual model. Thus, Q is incomplete because its 
> axioms don't lead to a single, unique model. And that will hold true for 
> all but the simplest systems.
> 
> 
>>>> The law of the excluded middle forces logicians to stupidly
>>>> classify semantic nonsense as true or false.
>>>
>>> Which is exactly what we want in Boolean logic.
>>>
>>>>> You haven't made even the feeblest attempt at doing this. You 
>>>>> simply introduce concepts as if they will magically fit into an 
>>>>> existing system rather than exploring what the consequences of 
>>>>> introducing such concepts would actually have
>>>>>
>>>>>>> If so, you'll need to define what all of these values actually 
>>>>>>> mean, and you'll need to completely redefine all of the basic 
>>>>>>> logical operators so that they account for these four values.
>>>>>>>
>>>>>>> 5 = 5 is irrefutable in Q. According to what you say above that 
>>>>>>> makes it 'unfalse'. How is that different from being 'true'?
>>>>>
>>>>> No answer?
>>>>>
>>>>> André
>>>>>
>>>>
>>>> Wittgenstein (1937)
>>>> 'True in Russell's system' means, as was said:
>>>> proved in Russell's system; and 'false in Russell's
>>>> system' means: the opposite has been proved
>>>> in Russell's system
>>>
>>> I explained to you not two hours ago why this particular quote 
>>> carries absolutely no weight with me, so there's really no point in 
>>> bringing it up again >
>> Of course woefully fallible humans never give a rat's ass for infallible
>> truth.
> 
> You don't have any special ability to identify 'infallible truth'.
> 
>> They only care if they believe something. You don't believe
>> Wittgenstein thus can't be bothered to see that he is inherently
>> correct.
> 
> That wasn't my point. I claimed that it wasn't clear that *Wittgenstein* 
> actually believed this once he had actually reflected on the problem, 
> and that therefore this quote really cannot be legitimately used to 
> support any particular position.
> 
>>> If true and provable were equivalent, we wouldn't have two different 
>>> words for them. 'true' is an ontological category; 'provable' is an 
>>> epistemic category. They don't map onto one another.
>>>
>>
>> Truth as an Epistemic Notion
>> Truth as an Epistemic Notion
>> Truth as an Epistemic Notion
> 
> Saying it three times doesn't achieve anything. And I would argue that 
> Prawitz is confused here. Citing a single article that makes a claim 
> doesn't validate that claim.
> 

You were sure that provable is epistemic and thus
truth is not. Truth as an Epistemic Notion is the
core of Wittgenstein.

> André
> 
> 


-- 
Copyright 2026 Olcott

My 28 year goal has been to make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
The complete structure of this system is now defined.

The entire body of knowledge expressed in language is
comprised of two types of relations between finite strings:
(a) *Axioms* Expressions of language that are stipulated to be true.

My system bridges the analytic/synthetic distinction by
expressly encoding all empirical "atomic facts" in a formal
language such as CycL of the Cyc project.

(b) *Inference Rules* Expressions of language that are semantically
entailed syntactically from (a) and/or (b).