Re: The simple essence of Proof Theoretic Semantics

On 7/1/2026 2:52 PM, dbush wrote:
> On 7/1/2026 3:44 PM, olcott wrote:
>> On 7/1/2026 2:19 PM, dbush wrote:
>>> On 7/1/2026 3:09 PM, olcott wrote:
>>>> On 7/1/2026 1:59 PM, dbush wrote:
>>>>> On 7/1/2026 2:54 PM, olcott wrote:
>>>>>> On 7/1/2026 1:35 PM, dbush wrote:
>>>>>>> On 7/1/2026 2:21 PM, olcott wrote:
>>>>>>>> On 7/1/2026 1:15 PM, dbush wrote:
>>>>>>>>> On 7/1/2026 2:01 PM, olcott wrote:
>>>>>>>>>> On 7/1/2026 12:33 PM, dbush wrote:
>>>>>>>>>>> On 7/1/2026 10:40 AM, olcott wrote:
>>>>>>>>>>>> On 7/1/2026 6:55 AM, dbush wrote:
>>>>>>>>>>>>> On 7/1/2026 12:20 AM, olcott wrote:
>>>>>>>>>>>>>> On 6/30/2026 11:01 PM, dbush wrote:
>>>>>>>>>>>>>>> On 6/30/2026 11:49 PM, olcott wrote:
>>>>>>>>>>>>>>>> On 6/30/2026 10:17 PM, dbush wrote:
>>>>>>>>>>>>>>>>> On 6/30/2026 11:10 PM, olcott wrote:
>>>>>>>>>>>>>>>>>> On 6/30/2026 10:02 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>> On 6/30/2026 10:57 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>> On 6/30/2026 9:34 PM, dbush wrote:
>>>>>>>>>>>>>>>>>>>>> On 6/30/2026 6:04 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>> On 6/30/2026 4:56 PM, André G. Isaak wrote:
>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-30 15:45, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>> On 6/30/2026 4:18 PM, André G. Isaak wrote:
>>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-29 21:06, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>> On 6/29/2026 9:49 PM, André G. Isaak wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>>> On 2026-06-29 20:42, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>>> Those are specific concrete examples of how
>>>>>>>>>>>>>>>>>>>>>>>>>>>> Proof Theoretic Semantics rejects expressions
>>>>>>>>>>>>>>>>>>>>>>>>>>>> as Proof Theoretic Semantically incoherent.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> No, they are not. No author writing in the 
>>>>>>>>>>>>>>>>>>>>>>>>>>> framework of proof- theoretic semantics has 
>>>>>>>>>>>>>>>>>>>>>>>>>>> ever offered those examples or comparable 
>>>>>>>>>>>>>>>>>>>>>>>>>>> examples or made any claims about 'rejecting 
>>>>>>>>>>>>>>>>>>>>>>>>>>> expressions as proof theoretic semantically 
>>>>>>>>>>>>>>>>>>>>>>>>>>> incoherent'. And there's nothing incoherent 
>>>>>>>>>>>>>>>>>>>>>>>>>>> about the statement 'no number is equal to 
>>>>>>>>>>>>>>>>>>>>>>>>>>> its successor' which is the example under 
>>>>>>>>>>>>>>>>>>>>>>>>>>> discussion.
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>> André
>>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>>> None-the-less what I have said remains 
>>>>>>>>>>>>>>>>>>>>>>>>>> completely true.
>>>>>>>>>>>>>>>>>>>>>>>>>> What I have spent 28 years reverse-engineering 
>>>>>>>>>>>>>>>>>>>>>>>>>> from first
>>>>>>>>>>>>>>>>>>>>>>>>>> principles is exactly that. That no one 
>>>>>>>>>>>>>>>>>>>>>>>>>> applied PTS
>>>>>>>>>>>>>>>>>>>>>>>>>> exactly that way before does not mean that it 
>>>>>>>>>>>>>>>>>>>>>>>>>> is not
>>>>>>>>>>>>>>>>>>>>>>>>>> exactly correct PTS.
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>> Unfortunately, your say so carries very little 
>>>>>>>>>>>>>>>>>>>>>>>>> weight.
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>> Yes. That is why I need to carefully find the exact
>>>>>>>>>>>>>>>>>>>>>>>> text that backs me up. Because PTS has their own
>>>>>>>>>>>>>>>>>>>>>>>> private author by author language it must be a
>>>>>>>>>>>>>>>>>>>>>>>> work written for a general audience like this work.
>>>>>>>>>>>>>>>>>>>>>>>> https://plato.stanford.edu/entries/proof- 
>>>>>>>>>>>>>>>>>>>>>>>> theoretic- semantics/
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>> Would you agree that there is a difference 
>>>>>>>>>>>>>>>>>>>>>>>>> between a statement being false and a statement 
>>>>>>>>>>>>>>>>>>>>>>>>> being meaningless?
>>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>> PTS is the way that meaning actually works. We 
>>>>>>>>>>>>>>>>>>>>>>>> can make a
>>>>>>>>>>>>>>>>>>>>>>>> simpler analogy in that English words are 
>>>>>>>>>>>>>>>>>>>>>>>> meaningless until
>>>>>>>>>>>>>>>>>>>>>>>> they are defined. The PTS connection of an 
>>>>>>>>>>>>>>>>>>>>>>>> expression in
>>>>>>>>>>>>>>>>>>>>>>>> Q to its axioms Q is analogous to the connection 
>>>>>>>>>>>>>>>>>>>>>>>> of an
>>>>>>>>>>>>>>>>>>>>>>>> English word to its definition. A proof merely 
>>>>>>>>>>>>>>>>>>>>>>>> looks to
>>>>>>>>>>>>>>>>>>>>>>>> see if a definition exists and if it does not 
>>>>>>>>>>>>>>>>>>>>>>>> then the
>>>>>>>>>>>>>>>>>>>>>>>> English Word / Expression of Q remains meaningless.
>>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>>> PTS counts a finite sequence of inference steps 
>>>>>>>>>>>>>>>>>>>>>>>> between
>>>>>>>>>>>>>>>>>>>>>>>> an expression and a set of axioms as the 
>>>>>>>>>>>>>>>>>>>>>>>> definition of
>>>>>>>>>>>>>>>>>>>>>>>> this expression. These are the two papers that 
>>>>>>>>>>>>>>>>>>>>>>>> establish
>>>>>>>>>>>>>>>>>>>>>>>> this Definitional View.
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> None of the above answers my question:
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>> Would you agree that there is a difference 
>>>>>>>>>>>>>>>>>>>>>>> between a statement being false and a statement 
>>>>>>>>>>>>>>>>>>>>>>> being meaningless?
>>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> I don't answer dumb questions.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> Translation:
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> "I don't answer questions that can prove me wrong"
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> That is why I am not responding to any posts
>>>>>>>>>>>>>>>>>>>>>> besides yours. dbush has become a troll again.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> Reminding people that you admitted that disjunction 
>>>>>>>>>>>>>>>>>>>>> intruduction is truth-preserving by your repeated 
>>>>>>>>>>>>>>>>>>>>> dishonest dodging of how P can be true and P ∨ Q 
>>>>>>>>>>>>>>>>>>>>> can be false is not trolling.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> In Robinson Arithmetic (often denoted as Q),
>>>>>>>>>>>>>>>>>>>> the statement "no number is equal to its
>>>>>>>>>>>>>>>>>>>> successor" is not provable. While this statement
>>>>>>>>>>>>>>>>>>>> is true for the standard natural numbers, Robinson
>>>>>>>>>>>>>>>>>>>> Arithmetic is too weak to prove it universally
>>>>>>>>>>>>>>>>>>>> (∀ x, S(x) ≠ x).
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> That you brought this up was a brilliant simplification
>>>>>>>>>>>>>>>>>>>> of the point that I was making proving that you can
>>>>>>>>>>>>>>>>>>>> understand the key ideas.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> What is was is a way to show more easily how you're 
>>>>>>>>>>>>>>>>>>> wrong. That you claim that "no number is equal to its 
>>>>>>>>>>>>>>>>>>> successor" is semantically invalid shows everyone 
>>>>>>>>>>>>>>>>>>> that your ideas are worthless.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> This is more accurate:
>>>>>>>>>>>>>>>>>> The truth value of (∀ x, S(x) ≠ x) 
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> So you admit that the above statement which exists in Q 
>>>>>>>>>>>>>>>>> must be either true or false.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Your lack of reply confirms your above admission.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> I make a very specific statement.
>>>>>>>>>>>>>> You mangle it and ask if I agree.
>>>>>>>>>>>>>
>>>>>>>>>>>>> There was no mangling.  That is the meaning of the words 
>>>>>>>>>>>>> you used.
>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> The truth value of (∀ x, S(x) ≠ x) 
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> You already mangled it.
>>>>>>>>>>>> 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.
>>>>>>>>>
>>>>>>>>> I didn't ask for an example.  I asked for a definition of what 
>>>>>>>>> it mean for the truth value of a statement to not exist in a 
>>>>>>>>> formal system.
>>>>>>>>>
>>>>>>>>
>>>>>>>> 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
>>>>>>>
>>>>>>> I specifically didn't ask for an external quote, as you have 
>>>>>>> demonstrated on countless occasions that you can't understand 
>>>>>>> what others have written.
>>>>>>
>>>>>> Everything that I have said in the last five years
>>>>>> sums up to the above quote.
>>>>>
>>>>> Don't sum it up in someone else's words.  Sum it up *in your own 
>>>>> words*.
>>>>>
>>>>
>>>> True(L, X) ≡ ∃Γ ⊆ BaseFacts(L) (Γ ⊢ X) // copyright Olcott 2018
>>>
>>> So in other words, the truth value of a statement not existing in a 
>>> formal system simply means that the statement is not provable in that 
>>> system.
>>>
>> That is not what I said.
>>
> 
> Then translate the above symbols to English so we can examine that more 
> carefully.

True in the formal language of formal system L for expression X means
that there exists a subset of the axioms of L such that a finite
sequence of inference steps in L reach this subset of axioms of L.

-- 
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).