On 1/28/24 19:44, olcott wrote:
>
> Making them incapable of correctly representing the ACTUAL
> self-reference of the Liar Paradox.
>
> LP := ~True(LP)
>
> <is> the correct way to formalize the Liar Paradox
> and most languages cannot do that so I created Minimal
> Type Theory.
>
> https://www.researchgate.net/publication/331859461_Minimal_Type_Theory_YACC_BNF
>

Those languages can formalize the halting problem. If they can't
formalize the Liar Paradox, it proves the halting problem isn't the Liar
Paradox.

On 1/28/24 18:52, olcott wrote:
> On 1/28/2024 10:51 AM, immibis wrote:
>> On 1/28/24 17:08, olcott wrote:
>>> On 1/28/2024 6:44 AM, Mikko wrote:
>>>> On 2024-01-27 15:52:22 +0000, olcott said:
>>>>
>>>>> On 1/27/2024 2:32 AM, Mikko wrote:
>>>>>> On 2024-01-27 00:08:34 +0000, André G. Isaak said:
>>>>>>
>>>>>>> On 2024-01-26 16:07, olcott wrote:
>>>>>>>
>>>>>>>> x = 2*√x is simply a contradiction.
>>>>>>>>
>>>>>>>
>>>>>>> No. It isn't. It is simply an equation. As such, it either has a
>>>>>>> solution or it does not (in this particular case, it does,
>>>>>>> specifically x = 4).
>>>>>>>
>>>>>>> André
>>>>>>
>>>>>> But it cannot be used as a definition because x = 0 is also a
>>>>>> solution.
>>>>>>
>>>>>> Mikko
>>>>>>
>>>>>
>>>>> It is not a logic sentence thus has no truth value until
>>>>> its variables are bound.
>>>>> https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)
>>>>
>>>> It is a logic formula that can be used where a logic formula
>>>> can be used, e.g., as a part of a sentence or definition.
>>>>
>>>> Mikko
>>>>
>>>
>>> It has no truth value until its variables are bound.
>>>
>>
>> Does LP := ¬True(LP) have a truth value until its variables are bound?
>
> Does: ¬True(¬True(¬True(¬True(¬True(¬True(...)))))) have a truth value?
> No it does not have a truth value.
>
not what I asked

On 2024-01-28 18:18:49 +0000, olcott said:

> On 1/28/2024 11:56 AM, Mikko wrote:
>> On 2024-01-28 16:12:05 +0000, olcott said:
>>
>>> On 1/28/2024 7:12 AM, Mikko wrote:
>>>> On 2024-01-27 16:31:33 +0000, olcott said:
>>>>
>>>>> On 1/27/2024 6:38 AM, immibis wrote:
>>>>>> On 1/27/24 09:48, Mikko wrote:
>>>>>>> On 2024-01-26 17:01:58 +0000, olcott said:
>>>>>>>
>>>>>>>>
>>>>>>>> Most other sources simply say that Tarski directly applies
>>>>>>>> his Truth() predicate to the Liar Paradox itself.
>>>>>>>
>>>>>>> It does not really matter what other sources say,
>>>>>>> only what Tarski says.
>>>>>>
>>>>>> Please stop trying to prove truth or falsehood based on a person's authority.
>>>>>
>>>>> When we are determining whether or not Tarski is
>>>>> wrong we must go by what Tarski actually said.
>>>>>
>>>>> On the first line of his proof he contradicts
>>>>> the actual correct truth predicate
>>>>
>>>> It would help if you could quote the first line
>>>> of his proof. But you can't even find it. the
>>>> best you can do is a strawman and not even a
>>>> good strawman:
>>>>
>>>
>>> *The next line is the first line of his proof*
>>
>> What does "next" mean in this context?
>>
>> Mikko
>>
>
> *Line (1) is the next line that I referred to*
>
> *Here is the Tarski Undefinability Theorem proof*
> (1) x ∉ Provable if and only if p // assumption

Not an assumption but a proven in an earlier part of the proof.
Note that the x and p are neither arbitrary nor undefined but
constructed in a particular way that is essential to the proof.

Mikko

On 1/29/2024 1:54 AM, Mikko wrote:
> On 2024-01-28 18:44:21 +0000, olcott said:
>
>> On 1/28/2024 12:16 PM, Mikko wrote:
>>> On 2024-01-28 16:04:36 +0000, olcott said:
>>>
>>>> On 1/28/2024 6:46 AM, Mikko wrote:
>>>>> On 2024-01-27 21:03:34 +0000, immibis said:
>>>>>
>>>>>> On 1/27/24 16:52, olcott wrote:
>>>>>>> On 1/27/2024 2:32 AM, Mikko wrote:
>>>>>>>> On 2024-01-27 00:08:34 +0000, André G. Isaak said:
>>>>>>>>
>>>>>>>>> On 2024-01-26 16:07, olcott wrote:
>>>>>>>>>
>>>>>>>>>> x = 2*√x is simply a contradiction.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> No. It isn't. It is simply an equation. As such, it either has
>>>>>>>>> a solution or it does not (in this particular case, it does,
>>>>>>>>> specifically x = 4).
>>>>>>>>>
>>>>>>>>> André
>>>>>>>>
>>>>>>>> But it cannot be used as a definition because x = 0 is also a
>>>>>>>> solution.
>>>>>>>>
>>>>>>>> Mikko
>>>>>>>>
>>>>>>>
>>>>>>> It is not a logic sentence thus has no truth value until
>>>>>>> its variables are bound.
>>>>>>> https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)
>>>>>>>
>>>>>>
>>>>>> LP := ¬True(LP) has no truth value until its variables are bound
>>>>>
>>>>> Which includes the predicate True as that is not a standard name
>>>>> of any predicate. In some formalisms True is a sentence but that
>>>>> meaning is not applicable here.
>>>>>
>>>>> Mikko
>>>>>
>>>>
>>>> That <is> the correct way to formalize the actual Liar Paradox.
>>>
>>> Not in languages that require that the expression on the right side
>>> of := must not contain the symbol on the left side. Languages that
>>> have the := often have that restriction.
>>>
>>> Mikko
>>>
>>
>> Making them incapable of correctly representing the ACTUAL
>> self-reference of the Liar Paradox.
>
> And incabable of claiming that the Liar Paradox must be true or false.
>
> Mikko
>

?- LP = not(true(LP)).
LP = not(true(LP)).

?- unify_with_occurs_check(LP, not(true(LP))).
false.

Any system powerful enough to detect cycles in the directed
graph of an expression's evaluation sequence detects and rejects
pathological self-reference(Olcott 2004).

BEGIN:(Clocksin & Mellish 2003:254)
Finally, a note about how Prolog matching sometimes differs from the
unification used in Resolution. Most Prolog systems will allow you to
satisfy goals like:

equal(X, X).
?- equal(foo(Y), Y).

that is, they will allow you to match a term against an uninstantiated
subterm of itself. In this example, foo(Y) is matched against Y,
which appears within it. As a result, Y will stand for foo(Y), which is
foo(foo(Y)) (because of what Y stands for), which is foo(foo(foo(Y))),
and so on. So Y ends up standing for some kind of infinite structure.
END:(Clocksin & Mellish 2003:254)

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 1/29/24 15:35, olcott wrote:
> On 1/29/2024 1:54 AM, Mikko wrote:
>> On 2024-01-28 18:44:21 +0000, olcott said:
>>
>>> On 1/28/2024 12:16 PM, Mikko wrote:
>>>> On 2024-01-28 16:04:36 +0000, olcott said:
>>>>
>>>>> On 1/28/2024 6:46 AM, Mikko wrote:
>>>>>> On 2024-01-27 21:03:34 +0000, immibis said:
>>>>>>
>>>>>>> On 1/27/24 16:52, olcott wrote:
>>>>>>>> On 1/27/2024 2:32 AM, Mikko wrote:
>>>>>>>>> On 2024-01-27 00:08:34 +0000, André G. Isaak said:
>>>>>>>>>
>>>>>>>>>> On 2024-01-26 16:07, olcott wrote:
>>>>>>>>>>
>>>>>>>>>>> x = 2*√x is simply a contradiction.
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> No. It isn't. It is simply an equation. As such, it either has
>>>>>>>>>> a solution or it does not (in this particular case, it does,
>>>>>>>>>> specifically x = 4).
>>>>>>>>>>
>>>>>>>>>> André
>>>>>>>>>
>>>>>>>>> But it cannot be used as a definition because x = 0 is also a
>>>>>>>>> solution.
>>>>>>>>>
>>>>>>>>> Mikko
>>>>>>>>>
>>>>>>>>
>>>>>>>> It is not a logic sentence thus has no truth value until
>>>>>>>> its variables are bound.
>>>>>>>> https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)
>>>>>>>>
>>>>>>>
>>>>>>> LP := ¬True(LP) has no truth value until its variables are bound
>>>>>>
>>>>>> Which includes the predicate True as that is not a standard name
>>>>>> of any predicate. In some formalisms True is a sentence but that
>>>>>> meaning is not applicable here.
>>>>>>
>>>>>> Mikko
>>>>>>
>>>>>
>>>>> That <is> the correct way to formalize the actual Liar Paradox.
>>>>
>>>> Not in languages that require that the expression on the right side
>>>> of := must not contain the symbol on the left side. Languages that
>>>> have the := often have that restriction.
>>>>
>>>> Mikko
>>>>
>>>
>>> Making them incapable of correctly representing the ACTUAL
>>> self-reference of the Liar Paradox.
>>
>> And incabable of claiming that the Liar Paradox must be true or false.
>>
>> Mikko
>>
>
> ?- LP = not(true(LP)).
> LP = not(true(LP)).
>
> ?- unify_with_occurs_check(LP, not(true(LP))).
> false.
>
> Any system powerful enough to detect cycles in the directed
> graph of an expression's evaluation sequence detects and rejects
> pathological self-reference(Olcott 2004).

x=2√x

Contains a cycle.
But it is fine.
You have never responded to this point because you know it proves you
are wrong.

On 1/29/2024 3:19 AM, Mikko wrote:
> On 2024-01-27 16:12:27 +0000, olcott said:
>
>> On 1/27/2024 4:20 AM, Mikko wrote:
>>> On 2024-01-26 16:27:24 +0000, olcott said:
>>>
>>>> On 1/26/2024 6:53 AM, Mikko wrote:
>>>>> On 2024-01-25 16:03:51 +0000, olcott said:
>>>>>
>>>>>> On 1/25/2024 3:08 AM, Mikko wrote:
>>>>>>> On 2024-01-24 15:03:25 +0000, olcott said:
>>>>>>>
>>>>>>>> On 1/24/2024 4:40 AM, Mikko wrote:
>>>>>>>>> On 2024-01-21 20:15:54 +0000, olcott said:
>>>>>>>>>
>>>>>>>>>> On 1/21/2024 2:06 PM, Richard Damon wrote:
>>>>>>>>>>> On 1/21/24 2:22 PM, wij wrote:
>>>>>>>>>>>> I just found an article about the Halting Problem.
>>>>>>>>>>>> https://arxiv.org/pdf/1906.05340.pdf
>>>>>>>>>>>>
>>>>>>>>>>>> In the conclusion section:
>>>>>>>>>>>> The idea of a universal halting test seems reasonable, but
>>>>>>>>>>>> cannot be
>>>>>>>>>>>> for-
>>>>>>>>>>>> malised as a consistent specification. It has no model and
>>>>>>>>>>>> does not
>>>>>>>>>>>> exist as
>>>>>>>>>>>> a conceptual object. Assuming its conceptual existence leads
>>>>>>>>>>>> to a
>>>>>>>>>>>> paradox.
>>>>>>>>>>>> The halting problem is universally used in university
>>>>>>>>>>>> courses on
>>>>>>>>>>>> Computer
>>>>>>>>>>>> Science to illustrate the limits of computation. Hehner
>>>>>>>>>>>> claims the
>>>>>>>>>>>> halting
>>>>>>>>>>>> problem is misconceived......
>>>>>>>>>>>>
>>>>>>>>>>>> It looks like what olcott now is claiming. Am I missing
>>>>>>>>>>>> something?
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> I think the problem he is seeing is that the property of
>>>>>>>>>>> "Halting" can not be uniformly determined in Finite Time.
>>>>>>>>>>>
>>>>>>>>>>> That is all that I can get from his statement of:
>>>>>>>>>>>
>>>>>>>>>>> The idea of a universal halting test seems reasonable, but
>>>>>>>>>>> cannot be formalised as a consistent specification.
>>>>>>>>>>>
>>>>>>>>>>> There certainly CAN be defined formal test that define
>>>>>>>>>>> Halting, the issue is that non-halting is defined by the
>>>>>>>>>>> non-existence of a number N for the number of steps needed to
>>>>>>>>>>> reach a final state.
>>>>>>>>>>>
>>>>>>>>>>> Some people just don't like the fact that it can be
>>>>>>>>>>> absolutely provable what the answer is (and thus unknowable),
>>>>>>>>>>> even if we know from the definition, that it must be one or
>>>>>>>>>>> the other.
>>>>>>>>>>>
>>>>>>>>>>> This leads us to a great divide in logics. The classical
>>>>>>>>>>> branch accepts that some truth is only established by
>>>>>>>>>>> infinite chains of connections, and thus can not be proven
>>>>>>>>>>> with a finite proof, and thus is unknowable.
>>>>>>>>>>>
>>>>>>>>>>> Others don't accept that, and require Truth to be only
>>>>>>>>>>> established by Finite chains. The problem then is, such logic
>>>>>>>>>>> system need to greatly limit the domain they attempt to
>>>>>>>>>>> cover, as otherwise you get into endless chains of asking if
>>>>>>>>>>> a question can be asked, at which point you need to ask if
>>>>>>>>>>> you can even ask about asking the questions. Only when the
>>>>>>>>>>> domain is restricted in a way that the answer MUST be
>>>>>>>>>>> determinable with finite work, can we break the cycle.
>>>>>>>>>>>
>>>>>>>>>>> For instance, if we limit ourselves to Finite State Machines
>>>>>>>>>>> (which could be Turing Machines with a fixed finite tape, or
>>>>>>>>>>> a classical program in a computer with limited memory) then
>>>>>>>>>>> we can be sure that the answer is determinable with a finite
>>>>>>>>>>> amount of work.
>>>>>>>>>>
>>>>>>>>>> Tarski did not understand that the Liar Paradox is not a truth
>>>>>>>>>> bearer
>>>>>>>>>> thus cannot possibly be true or false.
>>>>>>>>>
>>>>>>>>> It is a sin to lie about other people. Tarski obviously
>>>>>>>>> unnderstood that,
>>>>>>>>> as he could see an opportunity to exloit the fact.
>>>>>>>>>
>>>>>>>>> Mikko
>>>>>>>>>
>>>>>>>>
>>>>>>>> If Tarski understood that self-contradictory expressions
>>>>>>>> have no truth value and did not have his truth predicate
>>>>>>>> reject such expressions as invalid then Tarski was stupid.
>>>>>>>
>>>>>>> If you could show some exmples of self-contradictory arithmetic
>>>>>>> expressions we perhaps could understand what you are trying to say.
>>>>>>>
>>>>>>> Mikko
>>>>>>>
>>>>>>
>>>>>> Tarski did not use PA in this proof:
>>>>>
>>>>> Tarski's discussion is fairly generic. It is applicable to PA and
>>>>> many other theories.
>>>>>
>>>>> Mikko
>>>>>
>>>>
>>>> No conventional language encodes self-reference correctly
>>>> thus No conventional language encodes the Liar Paradox
>>>> correctly. Thus No conventional language can directly
>>>> see the infinite recursive structure of the actual Liar
>>>> Paradox.
>>>
>>> What do you mean by "conventional"? Is the l anguage of ordinary
>>> formal first order logic "conventional"? Is ordinary Enlish
>>> "conventional"?
>>>
>>> Mikko
>>>
>>
>> Every conventional formal language incorrectly formalizes the Liar
>> Paradox as something like this LP ↔ ~True(LP) rather than like this
>> LP := ~True(LP) thus the infinitely recursive structure of the Liar
>> Paradox is invisible when formalized incorrectly.
>
> That does not actually answer the question but apparently
> you mean 'convetional formal laguage' so yes about ordinary
> first order longic and no about ordinary English.
>
> Mikko
>

This does correctly formalize the Liar Paradox
LP := ~True(LP)
expands to ~True(~True(~True(~True(~True(...)))))
thus proving the cycle in the directed graph of the evaluation
sequence of the Liar Paradox, thus proving that the Liar Paradox
(and other isomorphic self-referential sentences) are not
truth bearers.

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 1/29/2024 8:56 AM, immibis wrote:
> On 1/29/24 15:35, olcott wrote:
>> On 1/29/2024 1:54 AM, Mikko wrote:
>>> On 2024-01-28 18:44:21 +0000, olcott said:
>>>
>>>> On 1/28/2024 12:16 PM, Mikko wrote:
>>>>> On 2024-01-28 16:04:36 +0000, olcott said:
>>>>>
>>>>>> On 1/28/2024 6:46 AM, Mikko wrote:
>>>>>>> On 2024-01-27 21:03:34 +0000, immibis said:
>>>>>>>
>>>>>>>> On 1/27/24 16:52, olcott wrote:
>>>>>>>>> On 1/27/2024 2:32 AM, Mikko wrote:
>>>>>>>>>> On 2024-01-27 00:08:34 +0000, André G. Isaak said:
>>>>>>>>>>
>>>>>>>>>>> On 2024-01-26 16:07, olcott wrote:
>>>>>>>>>>>
>>>>>>>>>>>> x = 2*√x is simply a contradiction.
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> No. It isn't. It is simply an equation. As such, it either
>>>>>>>>>>> has a solution or it does not (in this particular case, it
>>>>>>>>>>> does, specifically x = 4).
>>>>>>>>>>>
>>>>>>>>>>> André
>>>>>>>>>>
>>>>>>>>>> But it cannot be used as a definition because x = 0 is also a
>>>>>>>>>> solution.
>>>>>>>>>>
>>>>>>>>>> Mikko
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> It is not a logic sentence thus has no truth value until
>>>>>>>>> its variables are bound.
>>>>>>>>> https://en.wikipedia.org/wiki/Sentence_(mathematical_logic)
>>>>>>>>>
>>>>>>>>
>>>>>>>> LP := ¬True(LP) has no truth value until its variables are bound
>>>>>>>
>>>>>>> Which includes the predicate True as that is not a standard name
>>>>>>> of any predicate. In some formalisms True is a sentence but that
>>>>>>> meaning is not applicable here.
>>>>>>>
>>>>>>> Mikko
>>>>>>>
>>>>>>
>>>>>> That <is> the correct way to formalize the actual Liar Paradox.
>>>>>
>>>>> Not in languages that require that the expression on the right side
>>>>> of := must not contain the symbol on the left side. Languages that
>>>>> have the := often have that restriction.
>>>>>
>>>>> Mikko
>>>>>
>>>>
>>>> Making them incapable of correctly representing the ACTUAL
>>>> self-reference of the Liar Paradox.
>>>
>>> And incabable of claiming that the Liar Paradox must be true or false.
>>>
>>> Mikko
>>>
>>
>> ?- LP = not(true(LP)).
>> LP = not(true(LP)).
>>
>> ?- unify_with_occurs_check(LP, not(true(LP))).
>> false.
>>
>> Any system powerful enough to detect cycles in the directed
>> graph of an expression's evaluation sequence detects and rejects
>> pathological self-reference(Olcott 2004).
>
> x=2√x
>
> Contains a cycle.
> But it is fine.
> You have never responded to this point because you know it proves you
> are wrong.

Other people have responded that you are wrong.

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 1/29/24 19:15, olcott wrote:
> On 1/29/2024 8:56 AM, immibis wrote:
>> On 1/29/24 15:35, olcott wrote:
>>> On 1/29/2024 1:54 AM, Mikko wrote:
>>>
>>> ?- LP = not(true(LP)).
>>> LP = not(true(LP)).
>>>
>>> ?- unify_with_occurs_check(LP, not(true(LP))).
>>> false.
>>>
>>> Any system powerful enough to detect cycles in the directed
>>> graph of an expression's evaluation sequence detects and rejects
>>> pathological self-reference(Olcott 2004).
>>
>> x=2√x
>>
>> Contains a cycle.
>> But it is fine.
>> You have never responded to this point because you know it proves you
>> are wrong.
>
> Other people have responded that you are wrong.
>

What is unify_with_occurs_check(x, timestwo(squareroot(x)))?

On 2024-01-29 18:00:07 +0000, olcott said:

> On 1/29/2024 3:19 AM, Mikko wrote:
>> On 2024-01-27 16:12:27 +0000, olcott said:
>>
>>> On 1/27/2024 4:20 AM, Mikko wrote:
>>>> On 2024-01-26 16:27:24 +0000, olcott said:
>>>>
>>>>> On 1/26/2024 6:53 AM, Mikko wrote:
>>>>>> On 2024-01-25 16:03:51 +0000, olcott said:
>>>>>>
>>>>>>> On 1/25/2024 3:08 AM, Mikko wrote:
>>>>>>>> On 2024-01-24 15:03:25 +0000, olcott said:
>>>>>>>>
>>>>>>>>> On 1/24/2024 4:40 AM, Mikko wrote:
>>>>>>>>>> On 2024-01-21 20:15:54 +0000, olcott said:
>>>>>>>>>>
>>>>>>>>>>> On 1/21/2024 2:06 PM, Richard Damon wrote:
>>>>>>>>>>>> On 1/21/24 2:22 PM, wij wrote:
>>>>>>>>>>>>> I just found an article about the Halting Problem.
>>>>>>>>>>>>> https://arxiv.org/pdf/1906.05340.pdf
>>>>>>>>>>>>>
>>>>>>>>>>>>> In the conclusion section:
>>>>>>>>>>>>> The idea of a universal halting test seems reasonable, but cannot be
>>>>>>>>>>>>> for-
>>>>>>>>>>>>> malised as a consistent specification. It has no model and does not
>>>>>>>>>>>>> exist as
>>>>>>>>>>>>> a conceptual object. Assuming its conceptual existence leads to a
>>>>>>>>>>>>> paradox.
>>>>>>>>>>>>> The halting problem is universally used in university courses on
>>>>>>>>>>>>> Computer
>>>>>>>>>>>>> Science to illustrate the limits of computation. Hehner claims the
>>>>>>>>>>>>> halting
>>>>>>>>>>>>> problem is misconceived......
>>>>>>>>>>>>>
>>>>>>>>>>>>> It looks like what olcott now is claiming. Am I missing something?
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> I think the problem he is seeing is that the property of "Halting" can
>>>>>>>>>>>> not be uniformly determined in Finite Time.
>>>>>>>>>>>>
>>>>>>>>>>>> That is all that I can get from his statement of:
>>>>>>>>>>>>
>>>>>>>>>>>> The idea of a universal halting test seems reasonable, but cannot be
>>>>>>>>>>>> formalised as a consistent specification.
>>>>>>>>>>>>
>>>>>>>>>>>> There certainly CAN be defined formal test that define Halting, the
>>>>>>>>>>>> issue is that non-halting is defined by the non-existence of a number N
>>>>>>>>>>>> for the number of steps needed to reach a final state.
>>>>>>>>>>>>
>>>>>>>>>>>> Some people just don't like the fact that it can be absolutely provable
>>>>>>>>>>>> what the answer is (and thus unknowable), even if we know from the
>>>>>>>>>>>> definition, that it must be one or the other.
>>>>>>>>>>>>
>>>>>>>>>>>> This leads us to a great divide in logics. The classical branch accepts
>>>>>>>>>>>> that some truth is only established by infinite chains of connections,
>>>>>>>>>>>> and thus can not be proven with a finite proof, and thus is unknowable.
>>>>>>>>>>>>
>>>>>>>>>>>> Others don't accept that, and require Truth to be only established by
>>>>>>>>>>>> Finite chains. The problem then is, such logic system need to greatly
>>>>>>>>>>>> limit the domain they attempt to cover, as otherwise you get into
>>>>>>>>>>>> endless chains of asking if a question can be asked, at which point you
>>>>>>>>>>>> need to ask if you can even ask about asking the questions. Only when
>>>>>>>>>>>> the domain is restricted in a way that the answer MUST be determinable
>>>>>>>>>>>> with finite work, can we break the cycle.
>>>>>>>>>>>>
>>>>>>>>>>>> For instance, if we limit ourselves to Finite State Machines (which
>>>>>>>>>>>> could be Turing Machines with a fixed finite tape, or a classical
>>>>>>>>>>>> program in a computer with limited memory) then we can be sure that the
>>>>>>>>>>>> answer is determinable with a finite amount of work.
>>>>>>>>>>>
>>>>>>>>>>> Tarski did not understand that the Liar Paradox is not a truth bearer
>>>>>>>>>>> thus cannot possibly be true or false.
>>>>>>>>>>
>>>>>>>>>> It is a sin to lie about other people. Tarski obviously unnderstood that,
>>>>>>>>>> as he could see an opportunity to exloit the fact.
>>>>>>>>>>
>>>>>>>>>> Mikko
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> If Tarski understood that self-contradictory expressions
>>>>>>>>> have no truth value and did not have his truth predicate
>>>>>>>>> reject such expressions as invalid then Tarski was stupid.
>>>>>>>>
>>>>>>>> If you could show some exmples of self-contradictory arithmetic
>>>>>>>> expressions we perhaps could understand what you are trying to say.
>>>>>>>>
>>>>>>>> Mikko
>>>>>>>>
>>>>>>>
>>>>>>> Tarski did not use PA in this proof:
>>>>>>
>>>>>> Tarski's discussion is fairly generic. It is applicable to PA and
>>>>>> many other theories.
>>>>>>
>>>>>> Mikko
>>>>>>
>>>>>
>>>>> No conventional language encodes self-reference correctly
>>>>> thus No conventional language encodes the Liar Paradox
>>>>> correctly. Thus No conventional language can directly
>>>>> see the infinite recursive structure of the actual Liar
>>>>> Paradox.
>>>>
>>>> What do you mean by "conventional"? Is the l anguage of ordinary
>>>> formal first order logic "conventional"? Is ordinary Enlish
>>>> "conventional"?
>>>>
>>>> Mikko
>>>>
>>>
>>> Every conventional formal language incorrectly formalizes the Liar
>>> Paradox as something like this LP ↔ ~True(LP) rather than like this
>>> LP := ~True(LP) thus the infinitely recursive structure of the Liar
>>> Paradox is invisible when formalized incorrectly.
>>
>> That does not actually answer the question but apparently
>> you mean 'convetional formal laguage' so yes about ordinary
>> first order longic and no about ordinary English.
>>
>> Mikko
>>
>
> This does correctly formalize the Liar Paradox
> LP := ~True(LP)

This does not actually answer my question but it does not disagree
with my interpretation so we can assume that you accept it.

--
Mikko