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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: I just fixed the loophole of the Gettier cases
Date: Wed, 11 Sep 2024 10:18:23 +0300
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On 2024-09-10 13:32:25 +0000, olcott said:

> On 9/10/2024 4:34 AM, Mikko wrote:
>> On 2024-09-09 17:38:04 +0000, olcott said:
>> 
>>> On 9/7/2024 3:46 AM, Mikko wrote:
>>>> On 2024-09-06 23:41:16 +0000, Richard Damon said:
>>>> 
>>>>> On 9/6/24 8:24 AM, olcott wrote:
>>>>>> On 9/6/2024 6:43 AM, Mikko wrote:
>>>>>>> On 2024-09-03 12:49:11 +0000, olcott said:
>>>>>>> 
>>>>>>>> On 9/3/2024 5:44 AM, Mikko wrote:
>>>>>>>>> On 2024-09-02 12:24:38 +0000, olcott said:
>>>>>>>>> 
>>>>>>>>>> On 9/2/2024 3:29 AM, Mikko wrote:
>>>>>>>>>>> On 2024-09-01 12:56:16 +0000, olcott said:
>>>>>>>>>>> 
>>>>>>>>>>>> On 8/31/2024 10:04 PM, olcott wrote:
>>>>>>>>>>>>> *I just fixed the loophole of the Gettier cases*
>>>>>>>>>>>>> 
>>>>>>>>>>>>> knowledge is a justified true belief such that the
>>>>>>>>>>>>> justification is sufficient reason to accept the
>>>>>>>>>>>>> truth of the belief.
>>>>>>>>>>>>> 
>>>>>>>>>>>>> https://en.wikipedia.org/wiki/Gettier_problem
>>>>>>>>>>>>> 
>>>>>>>>>>>> 
>>>>>>>>>>>> With a Justified true belief, in the Gettier cases
>>>>>>>>>>>> the observer does not know enough to know its true
>>>>>>>>>>>> yet it remains stipulated to be true.
>>>>>>>>>>>> 
>>>>>>>>>>>> My original correction to this was a JTB such that the
>>>>>>>>>>>> justification necessitates the truth of the belief.
>>>>>>>>>>>> 
>>>>>>>>>>>> With a [Sufficiently Justified belief], it is stipulated
>>>>>>>>>>>> that the observer does have a sufficient reason to accept
>>>>>>>>>>>> the truth of the belief.
>>>>>>>>>>> 
>>>>>>>>>>> What could be a sufficient reason? Every justification of every
>>>>>>>>>>> belief involves other belifs that could be false.
>>>>>>>>>> 
>>>>>>>>>> For the justification to be sufficient the consequence of
>>>>>>>>>> the belief must be semantically entailed by its justification.
>>>>>>>>> 
>>>>>>>>> If the belief is about something real then its justification
>>>>>>>>> involves claims about something real. Nothing real is certain.
>>>>>>>>> 
>>>>>>>> 
>>>>>>>> I don't think that is correct.
>>>>>>>> My left hand exists right now even if it is
>>>>>>>> a mere figment of my own imagination and five
>>>>>>>> minutes ago never existed.
>>>>>>> 
>>>>>>> As I don't know and can't (at least now) verify whether your left
>>>>>>> hand exists or ever existed I can't regard that as a counter-
>>>>>>> example.
>>>>>>> 
>>>>>>>>> If the belief is not about something real then it is not clear
>>>>>>>>> whether it is correct to call it "belief".
>>>>>>>> 
>>>>>>>> *An axiomatic chain of inference based on this*
>>>>>>>> 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, etc.
>>>>>>>> 
>>>>>>>> ...sentences of the form: " a has the property φ ", " b bears
>>>>>>>> the relation R to c ", etc. are meaningless, if a, b, c, R, φ
>>>>>>>> are not of types fitting together.
>>>>>>>> https://en.wikipedia.org/wiki/History_of_type_theory#G%C3%B6del_1944
>>>>>>> 
>>>>>>> The concepts of knowledge and truth are applicable to the knowledge
>>>>>>> whether that is what certain peple meant when using those words.
>>>>>>> Whether or to what extent that theory can be said to be true is
>>>>>>> another problem.
>>>>>>> 
>>>>>> 
>>>>>> The fundamental architectural overview of all Prolog implementations
>>>>>> is the same True(x) means X is derived by applying Rules (AKA truth 
>>>>>> preserving operations) to Facts.
>>>>> 
>>>>> But Prolog can't even handle full first order logic, only basic propositions.
>>>> 
>>>> The logic behind Prolog is restricted enough that incompleteness cannot
>>>> be differentiated from consistency. It seems that Olcott wants a logic
>>>> with that impossibility.
>>> 
>>> It is not that incompleteness cannot be differentiated
>>> from inconsistency it is that the inconsistency of
>>> self-contradiction has been mistaken for undecidability
>>> instead of invalid input.
>> 
>> Of course incompleteness can be differentiated from incosistency.
> 
> Self-contradictory expressions are incorrect deemed to be
> undecidable expressions instead of invalid expressions.

Invalid expression is a non-expression (i.e., a string that does
not satisfy the syntax rules of an expression) used as if it were
an expression.

> Is this "actual piece of shit" "a rainbow" or "a car engine"?
> I can't decide, therefore the formal system is incomplete.
> (The correct answer is neither, yet the correct answer is not allowed).

Who allows the question but not the correct answer? You?

>> An incosistent theory cannot be incomplete, at least if any ordinary
>> logic is used. If you want to use a paraconsistent logic then you
>> must be very careful with terms of ordinary logic.
>> 
>> The basic theory behind Prolog is Horn Clauses, where incompleteness
>> cannot be differentiated from consistency. Standard Prolog has features
>> that break the logic if used but the terms "incompleteness" and
>> "consistency" are only defined for logic, not programming.
> 
> Tarski's Liar Paradox from page 248
>     It would then be possible to reconstruct the antinomy of the liar
>     in the metalanguage, by forming in the language itself a sentence
>     x such that the sentence of the metalanguage which is correlated
>     with x asserts that x is not a true sentence.
>     https://liarparadox.org/Tarski_247_248.pdf
> 
> Formalized as:
> x ∉ True if and only if p
> where the symbol 'p' represents the whole sentence x
> https://liarparadox.org/Tarski_247_248.pdf
> 
> "this sentence is not true" is not a truth bearer
> that must be rejected as invalid input and not the
> basis for the undecidability theorem.

The string "this sentence is not true" is not a valid arithmetic sentence
and therefore not relevant to definability of arithmetic truth. Arithmetic
truth is about sentences like

    ∀x ∃a ∃b ∃c (x < a ∧ x < b ∧ x < c ∧ a*a*a + b*b*b = c*c*c).

-- 
Mikko