Re: The simple essence of Proof Theoretic Semantics

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, 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

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