On 3/22/23 3:25 PM, olcott wrote:
> On 3/19/2023 2:21 PM, olcott wrote:
>> On 3/19/2023 1:22 PM, olcott wrote:
>>> On 3/19/2023 12:55 PM, olcott wrote:
>>>> On 3/19/2023 12:30 PM, olcott wrote:
>>>>> On 3/19/2023 10:52 AM, olcott wrote:
>>>>>> On 3/19/2023 10:45 AM, olcott wrote:
>>>>>>> On 3/19/2023 1:00 AM, olcott wrote:
>>>>>>>> On 3/18/2023 7:32 PM, olcott wrote:
>>>>>>>>> On 3/18/2023 6:17 PM, olcott wrote:
>>>>>>>>>> Just like with syllogisms conclusions a semantically necessary
>>>>>>>>>> consequence of their premises
>>>>>>>>>>
>>>>>>>>>> Semantic Necessity operator: ⊨□
>>>>>>>>>>
>>>>>>>>>> (a) Some expressions of language L are stipulated to have the
>>>>>>>>>> property
>>>>>>>>>> of Boolean true.
>>>>>>>>>>
>>>>>>>>>> (b) Some expressions of language L are a semantically necessary
>>>>>>>>>> consequence of others.
>>>>>>>>>>
>>>>>>>>>> True(L,X) means that a semantic connection exists between (a)
>>>>>>>>>> and X in L. *Axiom(P) ⊨□ X*
>>>>>>>>>>
>>>>>>>>>> Provable(L,P,X) means that a semantic connection exists
>>>>>>>>>> between premises P and X in L.   *P ⊨□ X*
>>>>>>>>>>
>>>>>>>>>> The Moon is made from green cheese ⊨□ The Moon is made from
>>>>>>>>>> cheese
>>>>>>>>>
>>>>>>>>
>>>>>>>> L can be any natural or formal language as long as it has the
>>>>>>>> above interfaces.
>>>>>>>>
>>>>>>>>> When this is called the foundation of correct reasoning
>>>>>>>>> *and indeed is the actual foundation of correct reasoning*
>>>>>>>>> that means that every system of logic either derives all of its
>>>>>>>>> operations on the basis of this system or such a system
>>>>>>>>> diverges from
>>>>>>>>> correct reasoning into incorrect thus erroneous reasoning.
>>>>>>>>>
>>>>>>>>> The kludge of the principle of explosion is eradicated by this
>>>>>>>>> foundation. https://en.wikipedia.org/wiki/Principle_of_explosion
>>>>>>>>>
>>>>>>>>> I am not sure what aspect of logic would be changed by this
>>>>>>>>> system that
>>>>>>>>> is why I opened up this discussion.
>>>>>>>>>
>>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>> The foundation of correct reasoning is just that and applies to
>>>>>>> every
>>>>>>> element of the entire body of analytical truth, whether it be facts
>>>>>>> about the world or mathematical relationships.
>>>>>>>
>>>>>>> (A & ~A) ⊨□ FALSE
>>>>>>> FALSE ⊨□ FALSE
>>>>>>> TRUE  ⊨□ TRUE
>>>>>>> A & B ⊨□ A
>>>>>>> A & B ⊨□ B
>>>>>>> The Moon is made from green cheese ⊨□ The Moon is made from cheese
>>>>>>>
>>>>>>
>>>>>> False(X) ⊨□ True(~X)
>>>>>>
>>>>>> X = "this sentence is not true"
>>>>>> (~True(X) & ~False(X)) ⊨□ ~Truth_Bearer(X)
>>>>>>
>>>>>
>>>>> The predicate True(L,X) is provided to explicitly contradict
>>>>> Tarski's conclusion that no such predicate can possibly exist.
>>>>>
>>>>> True(L,X) means that there is a semantic connection from
>>>>> expressions of language L that are stipulated to be true to X.
>>>>>
>>>>
>>>> expressions of language L that are stipulated to be true correspond to
>>>> Haskell Curry elementary theorems of T.
>>>>
>>>> Let T be such a theory. Then the elementary statements which belong
>>>> to T
>>>> we shall call the elementary theorems of T; we also say that these
>>>> elementary statements are true for T. Thus, given T, an elementary
>>>> theorem is an elementary statement which is true.
>>>> https://www.liarparadox.org/Haskell_Curry_45.pdf
>>>>
>>>> expressions of language L that are stipulated to be true also
>>>> correspond
>>>> to the basic facts of natural language such as cats are animals thus
>>>> cats are not ten story office buildings.
>>>>
>>>> That it is true in F that G is unprovable in F requires a semantic
>>>> connection from elementary theorems of F to G in F, otherwise G is
>>>> untrue in F.
>>>
>>> If a semantic connection exists then this semantic connection can be
>>> specified syntactically.
>>>
>>> If G states that there is no syntactic connection from the elementary
>>> theorems of F to G in F, then G is stating that G is untrue in F.
>>
>> The conventional proof that G is true is not a proof that G is true in
>> F, it is a proof that G is true in meta-F.
>>
>> LP = "This sentence is not true"
>> is not true because LP is not a truth bearer
>>
>> This sentence is not true: "This sentence is not true"
>> is true because LP is not a truth bearer.
>>
>> G = "this sentence cannot be proven in F"
>> cannot be proven in F because G has a vacuous truth object.
>>
>> ?- G = not(provable(F, G)).
>>
>> ?- unify_with_occurs_check(G, not(provable(F, G))).
>> false.
>>
>> Proves that G has a vacuous truth object.
>>
>
>
> We are therefore confronted with a proposition which asserts its own
> unprovability. (Gödel 1931:39-41)

Right, which isn't G itself, but something derived from it in Meta-F

Read the proof you have posted, keep track of what system he is talking
about.

Your just PROVING that you don't actually understand what the proof is
about.

>
> Thus Gödel's G is simplified to this:
> G = ¬(F ⊢ G)

Nope, it says that G is true if and only if it is not true that F proves
G, as proven in Meta-F

>
> Translated into Prolog like this:
> ?- G = not(provable(F, G)).

Nop,e as that ISN'T G, only the statment PROVEN to have and equivalent
truth value ot G.

Note, Prolog is incapable of handling this level of Logic.

Can you use Prolog to prove the Pythagorean Theorem?

>
> Found to be incorrect by this:
> ?- unify_with_occurs_check(G, not(provable(F, G))).
> false

Which just means that it is beyond Prolog

Also, since you LIED to Prolog, doesn't mean anything.

>
> Because the Prolog G has an “uninstantiated subterm of itself” we can
> know that unification will fail because it specifies “some kind of
> infinite structure.”The quotes come from: (Clocksin and Mellish 2003:255)

Right, because the logic of the system exceeds the capabilities of Prolog.

>
> So G is unprovable in F because G is incorrect, thus not because F is
> incomplete.

Nope, Since you have shown you don't understand what G actually is, your
logic is incorret.

IF G is incorrect, then there must exist a number that matches the
Primative Recursive Relationship, and thus from the proof in Meta-F, we
know that G is provable, so by your logic, you logic system can prove an
incorrect statement, and thus is shown to be inconsistent.

Of course, since you don't understand what G is, even though you have
presented the paper (translated) of the proof, you are showing that this
is above your head, just shows how little you understand about logic.

>
>> This sentence cannot be proven: "this sentence cannot be proven in F"
>> is true because G has a vacuous truth object.
>>
>> TT = "This sentenced is true"
>> is not true because TT has a vacuous truth object.
>>
>> This sentence is true.
>> What is it true about?
>> It is true about being true.
>> What is it true about being true about?
>> It is true about being true about being true...
>>
>>
>>
>