Re: The simple essence of Proof Theoretic Semantics

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

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.

>> 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.

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.

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*. 
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.

André


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