Re: The simple essence of Proof Theoretic Semantics

On 6/30/2026 10:31 PM, André G. Isaak wrote:
> On 2026-06-30 21:10, olcott wrote:
>> On 6/30/2026 10:02 PM, dbush wrote:
> 
>>> 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) does not exist in Q.
> 
> But as soon as you bring up truth values you're no longer talking about 
> PTS which semantically divides expressions into 'provable' and 'not 
> provable'.
> 
> In truth-functional semantics, the law of the excluded middle dictates 
> that ∀x, S(x) ≠ x must be either true or false. It's truth value might 
> be unknown to us but there is a big difference between unknown and 
> nonexistent.
> 
> Within PTS there is absolutely no contradiction about claiming that ∀x, 
> S(x) ≠ x is unprovable while at the same time claiming that its negation 
> is unprovable, but that both are perfectly legitimate and meaningful 
> expressions of Q. If you think otherwise, provide a reference to some 
> author working in PTS who actually asserts something of this sort. I 
> suspect you can't because this simply isn't what PTS claims. It's you 
> imposing your own views on a system developed by others which you do not 
> fully understand.
> 

I am working to get more support for my understandings.
It is difficult when each author has their own terminology
for the same ideas.

> André
> 
> 


-- 
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).