Re: The simple essence of Proof Theoretic Semantics

On 7/1/2026 8:23 PM, André G. Isaak wrote:
> On 2026-07-01 19:01, olcott wrote:
>> 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.
> 
> Actually, there's no evidence to support this claim. Can you name a 
> single work on the topic of geography or astronomy which actually 
> asserted that the world was flat? There are religious texts which *can* 
> be interpreted as consistent with a flat earth, but those texts weren't 
> dealing with geography; they were dealing with allegory.
> 
>>> 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.
> 
> You may be. I am not. I'm discussing Q and the Liar Paradox isn't 
> stateable in Q.
> 
>>>>> 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
> 
> That's something that distinguishes Q from PA. By itself, it's not an 
> explanation of why ∀ x, S(x) ≠ x isn't provable in Q. I'm trying to see 
> whether you really even understand this.
> 

It seems to me that I do understand that by itself is the
one and only reason why (∀x, S(x) ≠ x) is unprovable in Q.
That would entail that you would be flat out lying.
Q cannot do the ∀x without an infinite sequence of steps.
Since you know that why do you lie about that?


-- 
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).