Re: Proving the consistency of the body of knowledge expressed in language

On 4/1/25 10:13 PM, olcott wrote:

On 4/1/2025 8:03 PM, Richard Damon wrote:

On 4/1/25 7:22 PM, olcott wrote:

On 4/1/2025 5:30 PM, Richard Damon wrote:

On 4/1/25 1:56 PM, olcott wrote:

On 4/1/2025 1:33 AM, Mikko wrote:

On 2025-03-31 18:33:26 +0000, olcott said:
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Anything the contradicts basic facts or expressions
semantically entailed from these basic facts is proven
false.

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Anything that follows from true sentences by a truth preserving
transformations is true. If you can prove that a true sentence
is false your system is unsound.
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Ah so we finally agree on something.
What about the "proof" that detecting inconsistent
axioms is impossible? (I thought that I remebered this).
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No, the proof is that it is impossible to prove that a system is consistant. (sort of the opposite of what you are thinking of).
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Proving inconsistancy is easy, you just need one example.
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Proving the non-existance isn't as easy, and for a complicated enough system, can't be done, as you need to search an infinite space for the problem, which we can't be sure we have finished,
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I have always only been referring to the consistency
of a finite set of axioms. Just test each one against
all the others. When we use a type hierarchy we only
have to do this for axioms with compatible types.

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And, if they can support the needed level of logic, Godel has shown that they can not prove their own consistancy.
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 How is it that each element of a finite set of axioms
can not simply be tested against all of the others?

Because you can combine the axioms in various combinations.
The possible combinations can become like a finite string composed of all the axioms as symbols, which can be infinite.
Note, part of the requirements to reach that point of not being able to prove consistancy, is the ability to create an infinite number of items in the system (like the natural numbers).

 

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If we are only allowed to apply the single truth
preserving operation of semantic logical entailment
then we know the whole system must be consistent.
https://en.wikipedia.org/wiki/Logical_consequence#Semantic_consequence

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You mean your logic has no "And" or "Or" or "Not" operations?
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 Those are part of its semantics.

So, what keeps us from using Godel's or Tarski's proof in the system?
Or are you admitting you can't have the full properties of the Natural Numbers?

 

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We bypass any need for model theory by having the full
semantics embedded directly in the formal language.

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Not sure you can do that. You haven't been very good at being right in the past.
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 Spend few years carefully studying Montague grammar
and you might get it.

Which is just a way to encode the meaning of a sentence.
Doesn't do anything about the logic system.
He also doesn't handle all the inherent ambiguity of Natural Language words, which is sometimes used INTENTIONALLY as a special from of meaning.

 

So, which step of Tarski's proof doesn't follow that requirement?
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 He started with falsehoods as his basic facts.

Which one?
Remember, the point you keep on pointing out is late in the proof, and follows from the previous steps.

 

(Not that you disagree with his conclusion, but is logical operation violated this rule).
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 True(X) inherently exists for the entire body of
knowledge that can be expressed in language.

Which isn't a logic system, and is just a knowledge predicate.

 For Pete's sake it is like you don't understand
that 3 > 2.

Sure I do, you just don't undetstand how logic works, or what truth actually is.

 

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Sort of like we can easily prove that a machine halts, but simulating it to that point (like a real emulator can do for DDD), but showing that a machine is non-halting can be more of a problem. Sometimes we can find an induction property to let us prove it, but not always.

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