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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic
Subject: Re: How a True(X) predicate can be defined for the set of analytic
 knowledge (HoTT)
Date: Sat, 22 Mar 2025 11:11:26 -0500
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On 3/22/2025 8:37 AM, Richard Damon wrote:
> On 3/21/25 11:14 PM, olcott wrote:
>> On 3/21/2025 9:31 PM, Richard Damon wrote:
>>> On 3/21/25 8:47 PM, olcott wrote:
>>>> On 3/21/2025 6:49 PM, Richard Damon wrote:
>>>>> On 3/21/25 8:49 AM, olcott wrote:
>>>>>> On 3/21/2025 3:57 AM, Mikko wrote:
>>>>>>> On 2025-03-20 15:02:42 +0000, olcott said:
>>>>>>>
>>>>>>>> On 3/20/2025 8:09 AM, Mikko wrote:
>>>>>>>>> On 2025-03-20 02:42:53 +0000, olcott said:
>>>>>>>>>
>>>>>>>>>> It is stipulated that analytic knowledge is limited to the
>>>>>>>>>> set of knowledge that can be expressed using language or
>>>>>>>>>> derived by applying truth preserving operations to elements
>>>>>>>>>> of this set.
>>>>>>>>>
>>>>>>>>> A simple example is the first order group theory.
>>>>>>>>>
>>>>>>>>>> When we begin with a set of basic facts and all inference
>>>>>>>>>> is limited to applying truth preserving operations to
>>>>>>>>>> elements of this set then a True(X) predicate cannot possibly
>>>>>>>>>> be thwarted.
>>>>>>>>>
>>>>>>>>> There is no computable predicate that tells whether a sentence
>>>>>>>>> of the first order group theory can be proven.
>>>>>>>>>
>>>>>>>>
>>>>>>>> Likewise there currently does not exist any finite
>>>>>>>> proof that the Goldbach Conjecture is true or false
>>>>>>>> thus True(GC) is a type mismatch error.
>>>>>>>
>>>>>>> However, it is possible that someone finds a proof of the conjecture
>>>>>>> or its negation. Then the predicate True is no longer complete.
>>>>>>>
>>>>>>
>>>>>> The set of all human general knowledge that can
>>>>>> be expressed using language gets updated.
>>>>>
>>>>> And thus your concept of truth breaks.
>>>>>
>>>>> Truth, by its definition is an immutable thing, but you just 
>>>>> defined it to be mutable.
>>>>>
>>>>> How often do we need to re-verify our truths?
>>>>>
>>>>>>
>>>>>>>> When we redefine logic systems such that they begin
>>>>>>>> with set of basic facts and are only allowed to
>>>>>>>> apply truth preserving operations to these basic
>>>>>>>> facts then every element of the system is provable
>>>>>>>> on the basis of these truth preserving operations.
>>>>>>>
>>>>>>> However, it is possible (and, for sufficiently powerful sysems, 
>>>>>>> certain)
>>>>>>> that the provability is not computable.
>>>>>>>
>>>>>>
>>>>>> When we begin with basic facts and only apply truth preserving
>>>>>> to the giant semantic tautology of the set of human knowledge
>>>>>> that can be expressed using language then every element in this
>>>>>> set is reachable by these same truth preserving operations.
>>>>>>
>>>>>
>>>>> But you aren't begining with basic facts, but with what has been 
>>>>> assumed to be the basic facts. 
>>>>
>>>> That is not what I stipulated.
>>>> When we begin with what actual are the set of basic
>>>> facts and are only allowed to apply truth preserving
>>>> operations to these basic facts then it is self-evident
>>>> that True(X) must always be correct.
>>>
>>> But you can't stipulate that you cant' get to things that you can get 
>>> to.
>>>
>>> If your system can define the Natural Numbers, then we get Godel and 
>>> Tarski, and you can't stop it.
>>>
>>
>> The entire semantics of G is defined in the body of human general
>> knowledge that can be expressed in language henceforth called (BOK).
> 
> Yes, and that is that there does not exist a number that satifies a 
> particular involved Primative Recursive Relationship.
> 

That you provide reasonable replies that show good
insight some of the time seems to prove that you
are capable of having good insight.

>>
>> The whole language metalanguage thing is already taken care of
>> in a hierarchy of types that expresses multiple levels of logic
>> in the same formal system and formal language.
> 
> Nope, it is clear you just don't understand what the metalanguage does, 

The generic term meta-language is this:
   The truth definition itself was to be a definition of
   True in terms of the other expressions of the metalanguage.
https://plato.stanford.edu/entries/tarski-truth/#ObjLanMet

I use Montague Grammar of natural language semantics to
denote the semantic meanings of other terms in this
same language using Rudolf Carnap Meaning postulates.
{cat} <is a type of> {Animal}

> it NUMBERS all the symbols and axioms of the system. A system can not 
> number itself, as the numbering creates axioms that would then need to 
> be numbered, and that makes the system infinite. This allows us to 
> convert ALL logic into mathematics
> 

The basic facts of the body of general knowledge that
can be expressed using language are the finite set of
all facts that cannot be derived from other facts.

Cats <are> {Animals} // basic fact
{Animals} <are> {Living Things} // basic fact
Therefore {cats} <are> {Living Things} // derived fact

>>
>> https://en.wikipedia.org/wiki/History_of_type_theory#G%C3%B6del_1944
>>
> 
> Which has nothing to do with this problem. Your problem is you don't 
> actually understand what this means, and have replaced words with 
> different meanings, and thus invalidated the truths in it.
> 

    By the theory of simple types I mean the doctrine
    which says that the objects of thought (or, in another
    interpretation, the symbolic expressions) are divided
    into types, namely: individuals, properties of individuals,
    relations between individuals, properties of such relations...

{string1 of typeX} has {relation of typeY} to {string2 of typeZ}
{cats} {animals} {living things} are in a type hierarchy.

The entire body of human knowledge that can be expressed using
language can be expressed as different kinds of relations between
finite strings.

> Part of the problem seems to be that you small mind just can't 
> comprehend what infinity does to logic. In fact, your concept VIOLATES 
> this principle, as your "set" of Knowledge, mixes types and is thus 
> excluded from the field. They are excluded, as the theory doesn't hold 
> when such a set is allowed.



-- 
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer