Re: The simple essence of Proof Theoretic Semantics

On 7/8/2026 2:42 AM, Mikko wrote:
> On 06/07/2026 17:36, olcott wrote:
>> On 7/6/2026 2:59 AM, Mikko wrote:
>>> On 04/07/2026 20:13, olcott wrote:
>>>> On 7/4/2026 3:49 AM, Mikko wrote:
>>>>> On 03/07/2026 21:38, 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.
>>>>>>
>>>>>> 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".
>>>>>
>>>>> "Is x provable" is there already. Why would one want to lose
>>>>> "Is x true"? If one dosn't need "Is x true" one needn't use
>>>>> it.
>>>>
>>>> The whole focus of most PTS "Is x provable".
>>>> Model theory looks at true in a model and ignores
>>>> the connection between true and provable.
>>>
>>> People rarely care about PTS or model theory. More often they 
>>> careabout what is or is not true about someting they consider important.
>>>
>>
>> Hence we must correct the divergence of logic
>> from correct reasoning if we are to automate
>> correct reasoning.
> 
> Most people would accept as correct any reasoning that produce true
> conclusions from true premises. 

I need a system that is good enough to make disinformation
systems funded by many billions per year look like ridiculous
fools even to themselves.

> If the conclusions are not relevant
> to any real needs they might call the reasoning useless or a waste
> of time but not incorrect. If some of the premises are false or
> obscure someone might call the reasoning incorrect.
> 


-- 
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).