Re: The simple essence of Proof Theoretic Semantics

On 7/3/2026 4:22 PM, André G. Isaak wrote:
> On 2026-07-03 14:43, olcott wrote:
>> On 7/3/2026 1:46 PM, André G. Isaak wrote:
>>> On 2026-07-03 12:38, olcott wrote:
>>>> On 7/3/2026 1:21 PM, André G. Isaak wrote:
>>>>> On 2026-07-03 12:12, olcott wrote:
>>>>>> On 7/3/2026 12:17 PM, André G. Isaak wrote:
>>>>>>> On 2026-07-03 10:48, olcott wrote:
>>>>>>>> On 7/3/2026 9:45 AM, André G. Isaak wrote:
>>>>>>>>> On 2026-07-02 23:02, olcott wrote:
>>>>>>>>>> On 7/1/2026 9:03 PM, olcott wrote:
>>>>>>>>>>> Q cannot do the ∀x without an infinite sequence of steps.
>>>>>>>>>>
>>>>>>>>>> So your phrasing is good: Q would need something like an 
>>>>>>>>>> infinite sequence of steps (or a single principle that 
>>>>>>>>>> summarizes them) to get the ∀x. Since formal proofs must be 
>>>>>>>>>> finite, and Q lacks the tool (induction) that would allow a 
>>>>>>>>>> finite proof of the infinite claim, the universal statement 
>>>>>>>>>> remains unprovable.
>>>>>>>>>
>>>>>>>>> I'm not sure why you are responding to yourself nor who 'your 
>>>>>>>>> phrasing' refers to since you don't quote anyone. But, assuming 
>>>>>>>>> we're still talking about ∀ x, S(x) ≠ x in Q, your reasoning is 
>>>>>>>>> simply off.
>>>>>>>>>
>>>>>>>>> You *can* prove universally quantified claims in Q, just not 
>>>>>>>>> that particular claim.
>>>>>>>>>
>>>>>>>>
>>>>>>>> What is the reason that (∀x, S(x) ≠ x) cannot be proved in Q?
>>>>>>>
>>>>>>> Because it isn't true in all models of Q, 
>>>>>> Model theory has been expressly off-topic for
>>>>>> many weeks in every thread. Whenever you ignore
>>>>>> this the rest of your reply will be ignored.
>>>>>
>>>>> The rest of my post which you snipped and (presumably) ignored 
>>>>> explained *why* you are wrong about this. PTS does not reject 
>>>>> models or model theory. It simply doesn't rely on model-theoretic 
>>>>> semantics. Q *requires* a model.
>>>>>
>>>>> André
>>>>>
>>>>
>>>> It replaces Model theory With PTS.
>>>> That you do not understand this is your mistake.
>>>>
>>>> "Is x true" is replaced with something like "Is x provable".
>>>
>>> Which has no bearing on the existence of models
>>
>> Proof theoretic semantics is utterly unconcerned  with true
>> in a model and focuses on the existence of a canonical proof.
> 
> PTS isn't concerned with true at all, 


Truth as an Epistemic Notion
https://link.springer.com/article/10.1007/s11245-011-9107-6
That is Proof Theoretic Semantics being concerned with truth.

> which is why it certainly wouldn't 
> claim that a proposition which can neither be proven nor not proven is 
> not a 'truth bearer'. However, you have made this claim about (∀x, S(x) 
> ≠ x) in Q despite the fact that (∀x, S(x) ≠ x) is *always* either true 
> or false. It cannot be derived as as theorem, but it is still most 
> decidedly a truth-bearer.
> 
> Once you start making claims about things being truth-bhearers/non 
> truth-bearers, you're firmly dealing with a semantics that concerns 
> itself with truth, i.e. not PTS.
> 
> 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).