Re: The simple essence of Proof Theoretic Semantics

On 7/3/2026 8:58 PM, dbush wrote:
> On 7/3/2026 9:52 PM, olcott wrote:
>> On 7/3/2026 5:51 PM, André G. Isaak wrote:
>>> On 2026-07-03 16:37, olcott wrote:
>>>> On 7/3/2026 1:47 PM, André G. Isaak wrote:
>>>>> On 2026-07-03 12:36, olcott wrote:
>>>>>> On 7/3/2026 1:18 PM, dbush wrote:
>>>>>
>>>>>>> If an algorithm takes an input and produces an output, that is by 
>>>>>>> definition a mapping. 
>>>>>> That only proves that the definition is incoherent.
>>>>>> The coherent way that it actually works is that
>>>>>> inputs are transformed into outputs by applying
>>>>>> finite string transformation rules to inputs to
>>>>>> derive outputs.
>>>>>
>>>>> Apparently you don't understand the difference between a mapping 
>>>>> and an algorithm. They are two different things.
>>>>>
>>>>> André
>>>>>
>>>>
>>>> A function that ignores its input and only returns 0
>>>> is not any sort of halt function.
>>>
>>> He was defining 'mapping', not 'halt function'.
>>>
>>> André
>>>
>>
>> A actual halt function must compute 
> The mathematical halting function:
> 

When you actually implement this concretely so that
you cannot just "assume away" details then your notion
requires a halt decider to report on the behavior of
its caller having no idea who its caller is.

> 
> Given any algorithm (i.e. a fixed immutable sequence of instructions) X 
> described as <X> with input Y:
> 
> A solution to the halting problem is an algorithm H that computes the 
> following mapping:
> 
> (<X>,Y) maps to 1 if and only if X(Y) halts when executed directly
> (<X>,Y) maps to 0 if and only if X(Y) does not halt when executed directly
> 


-- 
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).