Re: The simple essence of Proof Theoretic Semantics

On 7/1/2026 6:09 PM, André G. Isaak wrote:
> On 2026-07-01 16:43, olcott wrote:
>> On 7/1/2026 4:10 PM, André G. Isaak wrote:
>>> On 2026-07-01 14:52, olcott wrote:
>>>> On 7/1/2026 3:50 PM, André G. Isaak wrote:
>>>
>>>>>> Any expression X that is unprovable in any formal
>>>>>> system F is untrue in that formal system F
>>>>>>
>>>>>> Any expression X that is irrefutable in any formal
>>>>>> system F is unfalse in that formal system F.
>>>>>
>>>>> So now you have a four-valued logic? (true, false, untrue, unfalse).
>>>>>
>>>>
>>>> Like the expression: "What time is it?"
>>>> we have true, false, not truth apt.
>>>
>>> So a three-valued system. Then the same remarks apply. You need to 
>>> actually define your three-valued system and show how the basic 
>>> logical operators actually work in that system.
>>>
>>
>> It is not a three-valued system as these are commonly
>> understood.
> 
> If it divides sentences into anything other than true and false then it 
> is a three-valued system.
> 
>> When we go with the expressiveness of
>> natural language then construing all sentences as
>> true or false is directly seen to be as stupid as it
>> has always been.
> 
> Natural language tells us nothing about Q.
> 
>>> And your natural language example is entirely unrevealing. Natural 
>>> language distinguishes between interrogative and declarative 
>>> sentences. Q has only declarative sentences and declarative 
>>> sentences, by definition, are sentences which evaluate to a truth value.
>>>
>> Because logic only has propositions that it incorrectly assumed
>> must be true or false it stupidly ignores the third possibility
>> of semantically ill-formed.
>>
>>> And in standard logic there is this thing called the law of the 
>>> excluded middle which states that every declarative sentence is 
>>> either true or false. You can't just introduce some concept like "not 
>>> truth apt" without completely redefining logic from the ground up. 
>>
>> Not truth apt and not a truth bearer already has established
>> well-defined meanings that logic stupidly ignores.
> 
> AFAICT, 'truth bearer' is simply a synonym for 'declarative sentence'. 
> And declarative sentences are the only kind of sentence found in Q. 
> Whatever meaning you intended is not an 'established well-defined 
> meaning'. It is your own private meaning.
> 

Not exactly because most every human has been too stupid
to understand that "This sentence is not true" is a semantically
incoherent declarative sentence. Even the great Saul Kripke
(did better than everyone else) yet did not quite get there.

> And ∀ x, S(x) ≠ x is most definitely a truth bearer.
> 

If is it not provable in Q then it is not a truth
bearer in Q. Because we can see that it is provable
in PA this causes us to screw up and think that this
means that it is true in Q.

>> The law of the excluded middle forces logicians to stupidly
>> classify semantic nonsense as true or false.
> 
> Which is exactly what we want in Boolean logic.
> 
>>> You haven't made even the feeblest attempt at doing this. You simply 
>>> introduce concepts as if they will magically fit into an existing 
>>> system rather than exploring what the consequences of introducing 
>>> such concepts would actually have
>>>
>>>>> If so, you'll need to define what all of these values actually 
>>>>> mean, and you'll need to completely redefine all of the basic 
>>>>> logical operators so that they account for these four values.
>>>>>
>>>>> 5 = 5 is irrefutable in Q. According to what you say above that 
>>>>> makes it 'unfalse'. How is that different from being 'true'?
>>>
>>> No answer?
>>>
>>> André
>>>
>>
>> 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 explained to you not two hours ago why this particular quote carries 
> absolutely no weight with me, so there's really no point in bringing it 
> up again.
> 

Of course woefully fallible humans never give a rat's ass for infallible
truth. They only care if they believe something. You don't believe
Wittgenstein thus can't be bothered to see that he is inherently
correct.

> If true and provable were equivalent, we wouldn't have two different 
> words for them. 'true' is an ontological category; 'provable' is an 
> epistemic category. They don't map onto one another.
> 

Truth as an Epistemic Notion
Truth as an Epistemic Notion
Truth as an Epistemic Notion
https://link.springer.com/article/10.1007/s11245-011-9107-6

For all expressions that are true on the basis of their
meaning expressed in language IT IS ONLY THIS MEANING
EXPRESSED IN LANGUAGE THAT MAKES THEM TRUE.

> André
> 
>> Has been inherently the way that true on the basis
>> of meaning expressed in language HAS ALWAYS WORKED.
>>
>> Expressions of language are ONLY true, or false on
>> the basis of their connections to other Expressions
>> of language.
>>
> 


-- 
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).