Re: The simple essence of Proof Theoretic Semantics

On 7/1/2026 1:40 PM, André G. Isaak wrote:
> On 2026-07-01 12:20, olcott wrote:
>> 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.
> 
> Having no finite sequence of inference steps between the expression and 
> the axioms of Q is *not* the same thing as random gibberish. 

It is closer to an English word such as "cat" that is
defined in English us undefined in Chinese.

> It simply 
> means it is unprovable in Q. 

Which means something entirely different in PTS than
it means in TCS.

> And being ungrounded in the atomic base of 
> Q means that it cannot achieve the PTS meaning of provable but rather 
> remains unprovable. 

What does Boxcar meaning in Chinese?
It has no meaning in Chinese.

When Boxcar is translated into Chinese: 棚车
then is has a Chinese meaning in Chinese.

> There's no way you can get from any of those things 
> to 'random gibberish'.
> 

The main unprovable that I have been working on
for 27 years is cases of pathological self-reference
that have incoherent meaning like this famous sentence:

Colorless green ideas sleep furiously was composed by
Noam Chomsky in his 1957 book Syntactic Structures as
an example of a sentence that is grammatically well-formed,
but semantically nonsensical.

It has no coherent compositional meaning even though
each word has meaning.

> 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).