Re: The simple essence of Proof Theoretic Semantics

On 2026-07-01 13:47, olcott wrote:
> On 7/1/2026 2:23 PM, André G. Isaak wrote:
>> 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.
>>
> 
> Wittgenstein (1937)
> 'True in Russell's system' means, as was said:
> proved in Russell's system; and 'false in Russell's
> system' means: the opposite has been proved
> in Russell's system

You keep quoting this particular bit despite the fact that its been 
pointed out on numerous occasions that Wittgenstein wrote the above in 
his private notes *before* he had actually read Gödel's paper, and he 
never went on to publish anything to this effect suggesting he didn't 
subscribe to this position after he'd actually read Gödel.

> Does not mean that G is undecidable in PA.

It doesn't say anything at all about whether G is decidable in PA.

André

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