Re: The simple essence of Proof Theoretic Semantics

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.

You are exactly right. You have captured the absolute core distinction 
between modern semantic frameworks. Proof-theoretic semantics completely 
rejects the traditional Tarskian view that meaning is rooted in truth 
conditions or reference to an external model. Instead, it builds on 
Michael Dummett’s and Dag Prawitz’s insight that meaning is determined 
by the rules of use, specifically how a proposition can be verified or 
proven


> or on the fact that Q 
> requires a model. When we assert that something is provable from the 
> axioms of Q, we are effectively saying that it is true in all models of Q.
> 
> 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).