Re: The philosophy of logic reformulates existing ideas on a new basis --- infallibly correct

On 11/16/24 10:24 AM, olcott wrote:

On 11/16/2024 8:32 AM, Richard Damon wrote:

On 11/16/24 9:21 AM, olcott wrote:

On 11/16/2024 3:11 AM, Mikko wrote:

On 2024-11-15 23:49:17 +0000, olcott said:
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On 11/15/2024 3:03 AM, Mikko wrote:

On 2024-11-14 23:40:19 +0000, olcott said:
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When you start with truth and only apply truth preserving
operations then you necessarily end up with truth.

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And if you don't you prove nothing.
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That is the basic model of all correct proofs.
https://en.wikipedia.org/wiki/Syllogism#Basic_structure

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No, it is not. There are truth preserving transformations that do
not follow that pattern.

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There are zero truth preserving operations that are not truth
preserving operations. The principle of explosion is not a
truth preserving operation. The full semantics of natural
can be extended to only apply truth preserving operations
to its own statement of basic fact.

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But Syllogism is not the only form of "Truth Preserving Operations".
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 It is the foundation of necessarily correct reasoning.

Which you haven't been able to define, and if that is all you allow, then you are just admitting you logic system is too weak to handle many of the problems we want to handle.

 True(L,x) ≡ Haskell_Curry_Elementary_Theorems(L) □ x
x is a necessary consequence of the expressions of the
language of L that have been stipulated to be true.
 False(L,x) ≡ Haskell_Curry_Elementary_Theorems(L) □ ~x
~x is a necessary consequence of the expressions of the
language of L that have been stipulated to be true.
 The above provides the basis for LLM AI systems to
distinguish facts from fictions.
 That the provability operator has been replaced
with the necessity operator seems to require semantic
relevance. This prevents logic from diverging from
correct reasoning in many different ways such as
the principle of explosion.
 

Which just shows you don't understand what KNOWLEDGE is about, as the necessity operator, since is allows for infinite chains, isn't something that can be always KNOWN.

 

IF that is all you accept, then be prepared for a very limited logic system.
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For example, the reduction rule: if A,
B, and C are formulas, the recution rule permits that from
A ∨ B and ¬A ∨ C can be inferred B ∨ C.
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That is the way the contradiction is supposed to work
A ∧ ¬A cancel each other out leaving B ∨ C.
A ∧ ¬A ∴ Trump is the Christ is proven (is nuts)
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Just shows you are the one that is NUTS.
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Your problem is it seems you only understand the most elementary of logic, but presume everyone one else is just using that most elementary of logic.
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Yes, With the most restricted set of rules, you can't get to incompleteness, but that is because you can't create the system with the power needed for the proof.
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The problem is that having the fullness of the logic of Natual Numbers is enough to cross the line, so your "Complete" Logic system can't have that, but you just are too stupid to undetstand that limit, because you don't know how any of your tools actually work.

 When True(L,x) and False(L,x) are defined as above then
Truth_Bearer(L,x) ≡ (True(L,x) ∨ False(L,x))
eliminating the notion of Incomplete(L) previously defined by
Incomplete(L) ≡ ∃x ∈ Language(L) ((L ⊬ x) ∧ (L ⊬ ¬x))
 

Nope, because Incompleteness isn't defined in terms of necessisty, but of PROVABILITY, something you are admitting you are just throwing out the window.
In WORDS, so we have clarity, System L is incomplete if there exist statements accepted in the language of L that can neither be proven or refuted. Note, statements accepted in the Lanaguage L means statements the language accepts as having semantic meaning, and thus assumed to be truth bearers.