Re: How a True(X) predicate can be defined for the set of analytic knowledge (HoTT)

On 3/22/25 2:27 PM, olcott wrote:

On 3/22/2025 12:34 PM, Richard Damon wrote:

On 3/22/25 12:11 PM, olcott wrote:

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:
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On 3/20/2025 8:09 AM, Mikko wrote:

On 2025-03-20 02:42:53 +0000, olcott said:
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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.

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A simple example is the first order group theory.
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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.

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There is no computable predicate that tells whether a sentence
of the first order group theory can be proven.
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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.

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However, it is possible that someone finds a proof of the conjecture
or its negation. Then the predicate True is no longer complete.
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The set of all human general knowledge that can
be expressed using language gets updated.

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And thus your concept of truth breaks.
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Truth, by its definition is an immutable thing, but you just defined it to be mutable.
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How often do we need to re-verify our truths?
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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.

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However, it is possible (and, for sufficiently powerful sysems, certain)
that the provability is not computable.
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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.
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But you aren't begining with basic facts, but with what has been assumed to be the basic facts.

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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.

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But you can't stipulate that you cant' get to things that you can get to.
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If your system can define the Natural Numbers, then we get Godel and Tarski, and you can't stop it.
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The entire semantics of G is defined in the body of human general
knowledge that can be expressed in language henceforth called (BOK).

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Yes, and that is that there does not exist a number that satifies a particular involved Primative Recursive Relationship.
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That you provide reasonable replies that show good
insight some of the time seems to prove that you
are capable of having good insight.

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So, you admit that I shows you something that breaks your claim?
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 Not at all. What I said and you agreed with
it that G is provable in (GKEUL) in the same
way the G is provable in meta-math.

No it isn't as the GKEUL can't have the axioms that enumerate the axioms, and thus doesn't have the information needed to do the proof in the meta-math.
You can't create that enumberation and keep the system finite.
One problem is there are an infinte number of meta-maths that can be derived from the system, each creating a diffferent meaning for the relationship, and perhaps only one of them has the needed information to do the proof.

 

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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.

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Nope, it is clear you just don't understand what the metalanguage does,

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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
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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}

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Which doesn't answer the question.
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 A type hierarchy does accomplish the aspect of Tarki's
metalanguage that I quoted.
    "The truth definition itself was to be a definition of
   True in terms of the other expressions of the metalanguage."

Which you haven't done.

 

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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
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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.

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fine.
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 Great!

And from those we can derive one of the needed metalanguage that let us create the unprovable statement that breaks the True Predicate.

 

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

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Fine, you can answer simple questions, but you still can't handle the tough ones where the truth is established by the infinite chain.
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 The knowledge tree has no infinite chains.
Unless an infinite chain is algorithmically
compressed to a finite sequence it is not
an element of any set of knowledge.

No, but the logic using it can.
You forget that we aren't limited to the axioms of the system, but have a logic system that allows us to combine terms to get new terms, and those are not limited to what was previously in the set of all knowledge at that time.
So the set of STATEMENTS that your predicate needs to be able to answer does include such things.
Of course, if you admit that you logic doesn't allow you to learn anything new, you can have that Truth Predicate as the Knowlege Predicate, but how useful is a logic system that can't be used to learn anything?

 

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https://en.wikipedia.org/wiki/History_of_type_theory#G%C3%B6del_1944
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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.
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    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...

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

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

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So?
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It still can create problem that can not be answered, and thus True can't be defined for all statements in your language.
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 True(X) can be answered for all human general knowledge
that can be expressed using language.

And thus has been misnamed, it should have been called Known(x)
You are just proving you don't knwo what your words actually mean, becuase you made yourself intentionally ignorant becuase you brainwashed yourself into a fear of being brainwashed if you studied what truth was, and thus made you native language to be just lies.

 

You are just too stupid to understand that Truth goes beyond what is known,

 It is not that I am stupid. It is when I define a set
of knowledge you keep forgetting that knowledge never
includes unknowns.

And thus isn't a definiton of TRUTH.
You are just proving that you don't understand what TRUTH is, and thus your whole work is just a FRAUD.

 

or even knowable, because it seems you don't know what either Truth or Knowledge actually mean.
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 Every set of knowledge always excludes every type of unknowns.

And if the logic system it is put into can't learn new things, is worthless.
That is your life, TOTALLY WORTHLESS because you have defined yourself to not be able to learn things, and thus you just speak in lies.

 

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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.

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