Re: The simple essence of Proof Theoretic Semantics

On 2026-07-01 12:50, olcott wrote:
> 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.

That's a completely spurious analogy. 'cat' isn't an expression of the 
Chinese language. ∀ x, S(x) ≠ x *is* an expression of the language of Q.

>> It simply means it is unprovable in Q. 

Then you're either using a completely idiosyncratic definition of 
'gibberish' or a completely idiosyncratic definition of 'provable'
(or both). That's why people keep asking you to provide *your* 
definitions, but you only respond with examples or analogies which fail 
to clarify what you might mean.

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

unprovable means the same thing in both.

>> 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.

Chomsky was making a point about *natural* languages, not Q. And I don't 
think you understand what the purpose of this example was. And, I should 
note that the example isn't nonsensical as it can have a poetic meaning 
(something which doesn't exist in formal languages).

Chomsky's sole point with this example was to stress that you couldn't 
appeal to semantic ill-formedness to explain ungrammaticality in natural 
languages. Rather, syntactic rules had to stand on their own.

-- 
To email remove 'invalid' & replace 'gm' with well known Google mail 
service.