Re: The simple essence of Proof Theoretic Semantics

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?
Are the universally quantified claims that can be proven like
this one (∀x, x = x) ?

> And there isn't an infinite sequence of steps that will get you from the 
> axioms of Q to ∀ x, S(x) ≠ x. There's *no* sequence of steps, finite or 
> infinite.
> 

So trying every element of the set of natural numbers
would not derive the truth value after am infinite
number of steps (that never complete)?

> The issue here is that there are models of Q 

Which do not exist in PTS thus are off topic in this thread.
All of the rest is off-topic in this thread.

> in which ∀ x, S(x) ≠ x is 
> true, but there are also models of Q in which it is false.
> 
> For any given model of Q, it will either be true or false, so your claim 
> that ∀ x, S(x) ≠ x is somehow 'not a truth bearer' is simply ludicrous. 
> It's simply the case that this particular statement cannot be derived as 
> a theorem of Q nor can its negation. Thus Q is incomplete.
> 
> 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).