Re: The simple essence of Proof Theoretic Semantics

On 7/1/2026 1:10 PM, André G. Isaak wrote:
> On 2026-07-01 12:01, olcott wrote:
>> On 7/1/2026 12:33 PM, dbush wrote:
>>> On 7/1/2026 10:40 AM, olcott wrote:
> 
>>>> The truth value of (∀ x, S(x) ≠ x) does not exist in Q.
>>>
>>> In your own words, what does it mean for the truth value of statement 
>>> to not exist in a formal system?
>>>
>>
>> The same thing as: "cats are animals" expressed in
>> English has no English meaning in Chinese.
>>
>> Until "cats are animals" is translated into Chinese
>> it is just random gibberish that has no meaning or
>> truth value in Chinese.
> 
> But ∀ x, S(x) ≠ x *isn't* random gibberish in Q. It is a well-formed 
> expression of Q that has a well-defined meaning. It just happens to be 
> unprovable. If it were random gibberish no one would have entertained 
> the question of whether it could or could not be proven in Q.
> 
> André
> 

It has no finite sequence of inference steps between
the expression and the axioms of Q. This seems to
mean that (∀x, S(x) ≠ x) is ungrounded in the atomic
base of Q in many of the different ways that this
can be expressed by different PTS authors.

-- 
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).