Re: The simple essence of Proof Theoretic Semantics

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.

-- 
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).