Re: The simple essence of Proof Theoretic Semantics

On 7/1/2026 1:57 PM, dbush wrote:
> On 7/1/2026 2:53 PM, olcott wrote:
>> On 7/1/2026 1:34 PM, dbush wrote:
>>> On 7/1/2026 2:20 PM, 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.
>>>>
>>>
>>> In other words, (∀x, S(x) ≠ x) is not provable in Q.
>>>
>>
>> In PTS that means the expression is undefined.
>> It does not mean that Q has undecidable sentences in PTS.
> 
> In your own words, what do you think it means for an expression to be 
> undefined, and what do you think it means for a formal system to have 
> undecidable sentences?
> 

I am trying to bridge my work to the work of proof
theoretic semantics. I have exactly agreed with
Wittgenstein's view about five years before I ever
heard of him.

What I (and Wittgenstein) call ~True(L,x)
PTS mostly calls ~Defined(L,x).

>>
>>> So once again, you're saying the same thing as everyone else but 
>>> using different words.
>>
>>
> 


-- 
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).