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

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.

So, you admit that I shows you something that breaks your claim?

 

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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,

 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}

Which doesn't answer the question.

 

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.

fine.

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

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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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?

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

So?

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

So?
It still can create problem that can not be answered, and thus True can't be defined for all statements in your language.
You are just too stupid to understand that Truth goes beyond what is known, or even knowable, because it seems you don't know what either Truth or Knowledge actually mean.

 

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.