Re: The simple essence of Proof Theoretic Semantics

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.

-- 
Mikko