The Liar Paradox: A proposed resolution

[Dan Christensen]

Some sentences are neither true nor false. We can say that their truth values are _indeterminate_ , not unlike the numerical value of 1/0. Some examples:

- What time is it?

- Wash the dishes.

- This sentence is false.

The latter will take some explaining: Suppose we have a set of sentences s. Define subsets t, f and m of set s such that

t = the subset of "true" sentences
f = the subset of "false" sentences
m = the subset of sentences of indeterminate truth value

Each element of s will be an element of precisely one of these subsets.

Now, "This sentence is false" is problematic. It is assumed to be true if and only if it is false. Therefore, its true value must be indeterminate.

Formal proof: https://dcproof.com/LiarParadox2.htm (only 44 lines in DC Proof format)

Your comments?

Dan

[Mild Shock]

I 100% agree with Olcott, he is the better Logician:

olcott schrieb am Dienstag, 25. Juli 2023 um 15:18:48 UTC+2:
> > On 7/25/2023 12:04 AM, Dan Christensen wrote:
> > We can say that their truth values are INDETERMINATE (like the "numerical value" of 1/0).
> It is not that no one can determine the truth value, it is that the truth value is non-existent.
https://groups.google.com/g/sci.logic/c/KfDliBm1Hb8/m/S4FUx1w_CAAJ

Your usage the word "indeterminate" is completely wrong. According
to these slides the Liar Paradox doesn't have an "indeterminate" truth,
because "indeterminate" is defined as:

A sentence Φ (or set of sentences Σ) is indeterminate if and only if
there is more than one way to coherently assign it a truth value (or to
assign the sentences contained in it truth values)
http://fitelson.org/piksi/piksi_18/cook_notes.pdf

The correct terminology is "paradoxical", defined as:

A sentence Φ (or set of sentences Σ) is paradoxical if and only if
there is no way to coherently assign it a truth value (or to assign the
sentences contained in it truth values).
http://fitelson.org/piksi/piksi_18/cook_notes.pdf

And its relativey easy to prove that the sentence is paradoxical,
just stay in classical logic and stay with proposition you then get:

/* Law of Non-Contradiction */
∀x(~(Tx & Fx)) &
/* Low of Excluded Middle */
∀x(Tx v Fx) =>
/* Paradoxical */
~∃x(Tx <-> Fx)

Its pretty easy. Even Wolfgang Schwartz tree tool can do it:

(∀x¬(Tx ∧ Fx) ∧ ∀x(Tx ∨ Fx)) → ¬∃x(Tx ↔ Fx) is valid.
https://www.umsu.de/trees/#~6x(~3(Tx~1Fx))~1~6x(Tx~2Fx)~5~3~7x(Tx~4Fx)

Now we have Olcotts "non-existent". But we need to go
back to "propositions" and "classical", to have Olcotts
"non-existent". But Olcott was 100% right.

[Mild Shock]

But for a moron like you, you will be not satisfied by:

> (∀x¬(Tx ∧ Fx) ∧ ∀x(Tx ∨ Fx)) → ¬∃x(Tx ↔ Fx) is valid.
> https://www.umsu.de/trees/#~6x(~3(Tx~1Fx))~1~6x(Tx~2Fx)~5~3~7x(Tx~4Fx)

You possibly want to prove, i.e. use a non-empty set s:

/* Law of Non-Contradiction */

(∀x(x e s => ¬(x e t ∧ x e f)) ∧
/* Law of Excluded Middle */
∀x(x e s => (x e t ∨ x e f)) →
/* Paradoxical */
¬∃x(x e s & (x e t ↔ x e f))

But most Logicians will not need this extra detour.
But feel free to proof the above as an exercise.

LoL

[Dan Christensen]

See my reply just now to your identical postings at sci.math

Dan

On Tuesday, July 25, 2023 at 2:40:51 PM UTC-4, Mild Shock wrote:
> I 100% agree with Olcott, he is the better Logician:
>
> olcott schrieb am Dienstag, 25. Juli 2023 um 15:18:48 UTC+2:
> > > On 7/25/2023 12:04 AM, Dan Christensen wrote:
> > > We can say that their truth values are INDETERMINATE (like the "numerical value" of 1/0).
> > It is not that no one can determine the truth value, it is that the truth value is non-existent.
> https://groups.google.com/g/sci.logic/c/KfDliBm1Hb8/m/S4FUx1w_CAAJ
>
> Your usage the word "indeterminate" is completely wrong. According
> to these slides the Liar Paradox doesn't have an "indeterminate" truth,
> because "indeterminate" is defined as:
>

[snip]

[olcott]

On 7/25/2023 1:06 PM, Dan Christensen wrote:
> Some sentences are neither true nor false. We can say that their truth values are _indeterminate_ , not unlike the numerical value of 1/0. Some examples:

Indeterminate is too much like undecidable.
If I ask: How many tons does the color red weigh?

It is not that you cannot determine the correct answer or can't make up
your mind about the correct answer, it is that the question is
incorrect. Likewise with the Liar Paradox.

>
> - What time is it?
>
> - Wash the dishes.
>
> - This sentence is false.
>
> The latter will take some explaining: Suppose we have a set of sentences s. Define subsets t, f and m of set s such that
>
> t = the subset of "true" sentences
> f = the subset of "false" sentences
> m = the subset of sentences of indeterminate truth value
>
> Each element of s will be an element of precisely one of these subsets.
>
> Now, "This sentence is false" is problematic. It is assumed to be true if and only if it is false. Therefore, its true value must be indeterminate.
>
> Formal proof: https://dcproof.com/LiarParadox2.htm (only 44 lines in DC Proof format)
>
> Your comments?
>
> Dan

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

[Dan Christensen]

On Tuesday, July 25, 2023 at 3:36:58 PM UTC-4, olcott wrote:
> On 7/25/2023 1:06 PM, Dan Christensen wrote:
> > Some sentences are neither true nor false. We can say that their truth values are _indeterminate_ , not unlike the numerical value of 1/0. Some examples:
> Indeterminate is too much like undecidable.

[snip]

They have those "indeterminate forms: in calculus. See https://en.wikipedia.org/wiki/Indeterminate_form

It works for me. Thanks anyway.

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

[olcott]

On 7/25/2023 3:09 PM, Dan Christensen wrote:
> On Tuesday, July 25, 2023 at 3:36:58 PM UTC-4, olcott wrote:
>> On 7/25/2023 1:06 PM, Dan Christensen wrote:
>>> Some sentences are neither true nor false. We can say that their truth values are _indeterminate_ , not unlike the numerical value of 1/0. Some examples:
>> Indeterminate is too much like undecidable.
>
> [snip]
>
> They have those "indeterminate forms: in calculus. See https://en.wikipedia.org/wiki/Indeterminate_form
>
> It works for me. Thanks anyway.
>
> Dan

It misleads the whole rest of the world the same way the term
"undecidable" misleads the world.

The reason why True(L,x) is thought to be uncomputable is not that
computation is not powerful enough, it is that the misnomer of
"undecidability" misleads people into believing that computations must
be able to correctly answer incorrect questions.

>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com
>
>

[Dan Christensen]

On Tuesday, July 25, 2023 at 4:26:15 PM UTC-4, olcott wrote:
> On 7/25/2023 3:09 PM, Dan Christensen wrote:
> > On Tuesday, July 25, 2023 at 3:36:58 PM UTC-4, olcott wrote:
> >> On 7/25/2023 1:06 PM, Dan Christensen wrote:
> >>> Some sentences are neither true nor false. We can say that their truth values are _indeterminate_ , not unlike the numerical value of 1/0. Some examples:
> >> Indeterminate is too much like undecidable.
> >
> > [snip]
> >
> > They have those "indeterminate forms: in calculus. See https://en.wikipedia.org/wiki/Indeterminate_form
> >
> > It works for me. Thanks anyway.
> >

> It misleads the whole rest of the world the same way the term
> "undecidable" misleads the world.
>

[snip]

Please cite even one of your imagined catastrophes resulting from the world being so misled.

Dan

[Mild Shock]

Indeterminate forms in Math are not 1/0, but 0/0. Because
0/0 = x, respectively 0 = x*0 has many many solutions.

0 = 1*0
0 = 2*0
Etc...

Thats why your usage of the word "indeterminate" is wrong.
Please see the definition here, I capitalized the important
part so that Dumbo Dan-O-Matik understands it:

"A sentence Φ (or set of sentences Σ) is indeterminate if and only if

there is MORE THAN ONE WAY to coherently assign it a truth value (or to

assign the sentences contained in it truth values)"
http://fitelson.org/piksi/piksi_18/cook_notes.pdf

Did you see the MORE THAN ONE WAY? Or do you need to
see a doctor because you need new glasses?

1/0 is undefined
0/0 is indeterminate

[Dan Christensen]

See my reply just now to your identical posting at sci.math

Dan

On Tuesday, July 25, 2023 at 6:48:03 PM UTC-4, Mild Shock wrote:
> Indeterminate forms in Math are not 1/0, but 0/0. Because
> 0/0 = x, respectively 0 = x*0 has many many solutions.
>
> 0 = 1*0
> 0 = 2*0
> Etc...
>

[snip]

[Mild Shock]

Just read Wikipedia for gods sake, 1/0 is undefined you moron!

The expression 1/0 is not commonly regarded as an indeterminate form
https://en.wikipedia.org/wiki/Indeterminate_form#Expressions_that_are_not_indeterminate_forms

Why is this so. Wikipedia argues with limit. We have to side limits:

lim x->0+ 1/x = +oo
lim x->0- 1/x = -oo

Which makes the limit itself undefined:

lim x->0 1/x = undefined

But you can also argue algebraically.
Namely that 1/0 = x respectively 1 = x*0 has no solution.

On the other hand wikipedia also offers a limit explanation why 0/0
is indeterminate, i.e. has many solutions.

But again you can also argue algebraically.
Namely that 0/0 = x respectivaly 0 = x*0 has many many solutions.

[Dan Christensen]

See my reply just now to your identical posting at sci.math

Dan

On Tuesday, July 25, 2023 at 7:05:15 PM UTC-4, Mild Shock wrote:
> Just read Wikipedia for gods sake, 1/0 is undefined ...

[snip]

[Mild Shock]

How long does it take until a) Dan Christensen gets
the math vocabulary right b) admits that he was wrong?

Maybe he needs a little brain massage like here?
https://9gag.com/gag/aEqEVwn

Here is what ChatGPT tells me:

"It's important to distinguish between indeterminate
forms and undefined expressions. Indeterminate
forms imply that the limit exists, but you need
more information or specific techniques to evaluate it.
Undefined expressions, on the other hand, represent
situations where the result cannot be determined at all,
such as "1/0" (division by zero), which is not an
indeterminate form but is considered undefined."
https://chat.openai.com/share/2e5f67dc-2336-435e-84b6-2a73582d2abc

[Dan Christensen]

See my reply just now to your identical and very, very pathetic posting at sci.math, Mr. Collapse.

Dan

On Tuesday, July 25, 2023 at 7:33:03 PM UTC-4, Mild Shock (aka Mr. Collapse) wrote:
> How long does it take until a) Dan Christensen gets
> the math vocabulary right b) admits that he was wrong?
>

[snip childish abuse]

[olcott]

I already told you and you can't remember so you must not care.
It has been over 110 degrees F in Phoenix for 24 consecutive days.

https://armorglass.com/humans-die-at-140-degrees-unless-we-cut-carbon-emissions-your-children-will-face-that-temp-we-can-help/#:~:text=You%20might%20be%20wondering%20about,is%20the%20aforementioned%20heat%20stroke.

https://www.researchgate.net/publication/336568434_Severe_anthropogenic_climate_change_proven_entirely_with_verifiable_facts

[olcott]

*We also cannot determine how many tons the color red weighs*
Calling it indeterminate sounds like this issue is with us.
Calling is an incorrect question places the blame where it belongs.

[olcott]

On 7/25/2023 6:32 PM, Mild Shock wrote:
> How long does it take until a) Dan Christensen gets
> the math vocabulary right b) admits that he was wrong?
>
> Maybe he needs a little brain massage like here?
> https://9gag.com/gag/aEqEVwn
>
> Here is what ChatGPT tells me:
>
> "It's important to distinguish between indeterminate
> forms and undefined expressions. Indeterminate
> forms imply that the limit exists, but you need
> more information or specific techniques to evaluate it.

There you go, good job ChatGPT.

> Undefined expressions, on the other hand, represent
> situations where the result cannot be determined at all,
> such as "1/0" (division by zero), which is not an
> indeterminate form but is considered undefined."
> https://chat.openai.com/share/2e5f67dc-2336-435e-84b6-2a73582d2abc
>
> Dan Christensen schrieb am Mittwoch, 26. Juli 2023 um 01:27:02 UTC+2:
>> See my reply just now to your identical posting at sci.math
>>
>> Dan
>> On Tuesday, July 25, 2023 at 7:05:15 PM UTC-4, Mild Shock wrote:
>>> Just read Wikipedia for gods sake, 1/0 is undefined ...
>>
>> [snip]

[Dan Christensen]

On Tuesday, July 25, 2023 at 10:28:54 PM UTC-4, olcott wrote:
> On 7/25/2023 5:39 PM, Dan Christensen wrote:
> > On Tuesday, July 25, 2023 at 4:26:15 PM UTC-4, olcott wrote:
> >> On 7/25/2023 3:09 PM, Dan Christensen wrote:
> >>> On Tuesday, July 25, 2023 at 3:36:58 PM UTC-4, olcott wrote:
> >>>> On 7/25/2023 1:06 PM, Dan Christensen wrote:
> >>>>> Some sentences are neither true nor false. We can say that their truth values are _indeterminate_ , not unlike the numerical value of 1/0. Some examples:
> >>>> Indeterminate is too much like undecidable.
> >>>
> >>> [snip]
> >>>
> >>> They have those "indeterminate forms: in calculus. See https://en.wikipedia.org/wiki/Indeterminate_form
> >>>
> >>> It works for me. Thanks anyway.
> >>>
> >
> >> It misleads the whole rest of the world the same way the term
> >> "undecidable" misleads the world.
> >>
> > [snip]
> >
> > Please cite even one of your imagined catastrophes resulting from the world being so misled.
> >
> > Dan

> I already told you and you can't remember so you must not care.
> It has been over 110 degrees F in Phoenix for 24 consecutive days.
>

Very evasive. It is not enough to cite some climate catastrophes. You claim WITHOUT PROOF that it is all the result of the application of the standard truth table for logical implication. Do you realize how silly you look?

Dan

[olcott]

It looks like you don't care about an honest dialogue.

If the Tarski undefinability was overturned then we could have
mathematical proofs of he truth of climate change and election fraud.

Until then it remains a mere difference of opinion of what are
despicable lies and what are verified facts.

[Dan Christensen]

Still no proof??? Oh, well...

Dan

[Mild Shock]

Well you wrote "proposed resolution", if your "proposed resolution"
is based on some mislabeling humbug, I don't know what you want
from us, when you also wrote "Your comments? ".

Usually attempts to "resolve" the Liar Paradox are a typical sign
of mental illness. Once you see such a post on the internet, you
can immediately tell, oh well, here we go again, there is

a fallen angel, they all fall into the category:

"It's like obscenity – you can tell a crank when you see one."
https://en.wikipedia.org/wiki/Mathematical_Cranks

If you would follow these slides closely, you would understand
that there is no "resolution", even 3 partition doesn't resolve anything.
Since there is the "Revenge Paradox" and so on.

Lecture 1: The Liar Paradox
http://fitelson.org/piksi/piksi_18/cook_notes.pdf

[Mild Shock]

The bottom line is, if you read the slides carefully, and if
you also follow the general wisdom concerning the Liar
Paradox all around the globe, reflecting the current

state of the art, in dealing with it, is that against "crank
wisdom" who believe that a "little thinking out of the
box" resolve paradoxes, this is not the case.

The slides not only use the correct terminology,
i.e. "paradoxical" and not "indeterminate", the slides
have also the following benefit over

the gibberish by Dan Christensen:
- The notions "paradoxical", "indeterminate", etc.. are precisely define.
- The "out of the box" idea 3 partition is discussed towards the end:
Revenge Paradox
- The "out of the box" idea 4 partition is discussed towards the end:
Super Revenge Paradox
- The "out of the box" idea n partition is discussed towards the end:
Ultimate Revenge Paradox
- What else?

See for yourself:

Lecture 1: The Liar Paradox
http://fitelson.org/piksi/piksi_18/cook_notes.pdf

[Mild Shock]

I am only a bee performing a Waggle dance:
https://en.wikipedia.org/wiki/Waggle_dance

Dan-O-Matik, you need to move your lazy ass
by yourself, and find the nectar by yourself.

LoL

[Dan Christensen]

On Wednesday, July 26, 2023 at 3:19:55 AM UTC-4, Mild Shock wrote:
> Well you wrote "proposed resolution", if your "proposed resolution"
> is based on some mislabeling humbug, I don't know what you want
> from us, when you also wrote "Your comments? ".
>

Sorry, but I take your comment is a very minor point indeed: an opinion about the informal usage of a single word. Other than that, you have nothing to say about my presentation. I take that a small victory. Thanks anyway for your time and interest.

Dan

[olcott]

On 7/26/2023 10:02 AM, Dan Christensen wrote:
> On Wednesday, July 26, 2023 at 3:19:55 AM UTC-4, Mild Shock wrote:
>> Well you wrote "proposed resolution", if your "proposed resolution"
>> is based on some mislabeling humbug, I don't know what you want
>> from us, when you also wrote "Your comments? ".
>>
>
> Sorry, but I take your comment is a very minor point indeed: an opinion about the informal usage of a single word. Other than that, you have nothing to say about my presentation. I take that a small victory. Thanks anyway for your time and interest.

Whether or not decision problems are undecidable because the problem
itself is incorrect or or computation simply is not powerful enough is a
very important issue.

Objective and Subjective Specifications
Eric C.R. Hehner
Department of Computer Science, University of Toronto
http://www.cs.toronto.edu/~hehner/OSS.pdf

>
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com
>

[Dan Christensen]

On Wednesday, July 26, 2023 at 11:27:59 AM UTC-4, olcott wrote:
> On 7/26/2023 10:02 AM, Dan Christensen wrote:
> > On Wednesday, July 26, 2023 at 3:19:55 AM UTC-4, Mild Shock wrote:
> >> Well you wrote "proposed resolution", if your "proposed resolution"
> >> is based on some mislabeling humbug, I don't know what you want
> >> from us, when you also wrote "Your comments? ".
> >>
> >
> > Sorry, but I take your comment is a very minor point indeed: an opinion about the informal usage of a single word. Other than that, you have nothing to say about my presentation. I take that a small victory. Thanks anyway for your time and interest.

> Whether or not decision problems are undecidable because the problem
> itself is incorrect or or computation simply is not powerful enough is a
> very important issue.
>

How is this relevant? We have a set of objects (sentences) divided into 3 disjoint subsets corresponding to (1) true sentences, (2) false sentences, and (3) sentences of indeterminate truth value. We recognize, for example, the difference between the sentence "It is raining." and "Is it raining?" The former is either true or false. The truth value of the latter can be thought of as, what I call, "indeterminate." (Use another word if you like.) Nothing "paradoxical" about that. Now, if a sentence X is true iff it is false, then X must be of indeterminate truth value. Do you not agree?

[olcott]

It is relevant because computer scientists view that when a computer
system cannot resolve non truth bearers to a value of true or false then
this is a mistake of the computer system.

It is relevant because computer scientists view that when a computer
system cannot resolve non truth bearers to a value of true or false then
this is a mistake of the computer system.

It is relevant because computer scientists view that when a computer
system cannot resolve non truth bearers to a value of true or false then
this is a mistake of the computer system.

It is relevant because computer scientists view that when a computer
system cannot resolve non truth bearers to a value of true or false then
this is a mistake of the computer system.

Whether or not decision problems are undecidable because the problem
itself is incorrect or or computation simply is not powerful enough is a
very important issue.

Objective and Subjective Specifications
Eric C.R. Hehner
Department of Computer Science, University of Toronto
http://www.cs.toronto.edu/~hehner/OSS.pdf

> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com
>
>
>
>
>
>
>

[Mild Shock]

Yes, its the same victory, when Don Quixote
was fighting windmills, thinking they are monsters.

You qualified again for mathematical cranks club.

[Mild Shock]

Same error again? If the third set would be "indeterminate",
it would mean something like having many values:

s e m <=> s e t & s e f

But you cannot really express indeterminate with your nonsense,
if you would like to express indeterminate, you would

need to distinguish between:
- form of a sentence
- valuation of a sentence

You could then say:

A form f is indeterminate, if we have two valuations v1 and v2, so that:
EXIST(v1):EXIST(v2):[v1(f) coherent & v2(f) coherent & v1 != v2]

But the Liar Paradox is paradoxical, basically:
~EXIST(v):[v(f) coherent]

See also these definitions here:

A sentence Φ (or set of sentences Σ) is indeterminate if and only if

there is more than one way to coherently assign it a truth value (or to

assign the sentences contained in it truth values)
http://fitelson.org/piksi/piksi_18/cook_notes.pdf

A sentence Φ (or set of sentences Σ) is paradoxical if and only if
there is no way to coherently assign it a truth value (or to assign the
sentences contained in it truth values).
http://fitelson.org/piksi/piksi_18/cook_notes.pdf

So how do we correctly model the Liar Paradox?

[Dan Christensen]

On Wednesday, July 26, 2023 at 12:20:44 PM UTC-4, olcott wrote:
> On 7/26/2023 11:15 AM, Dan Christensen wrote:
> > On Wednesday, July 26, 2023 at 11:27:59 AM UTC-4, olcott wrote:
> >> On 7/26/2023 10:02 AM, Dan Christensen wrote:
> >>> On Wednesday, July 26, 2023 at 3:19:55 AM UTC-4, Mild Shock wrote:
> >>>> Well you wrote "proposed resolution", if your "proposed resolution"
> >>>> is based on some mislabeling humbug, I don't know what you want
> >>>> from us, when you also wrote "Your comments? ".
> >>>>
> >>>
> >>> Sorry, but I take your comment is a very minor point indeed: an opinion about the informal usage of a single word. Other than that, you have nothing to say about my presentation. I take that a small victory. Thanks anyway for your time and interest.
> >
> >> Whether or not decision problems are undecidable because the problem
> >> itself is incorrect or or computation simply is not powerful enough is a
> >> very important issue.
> >>
> >
> > How is this relevant? We have a set of objects (sentences) divided into 3 disjoint subsets corresponding to (1) true sentences, (2) false sentences, and (3) sentences of indeterminate truth value. We recognize, for example, the difference between the sentence "It is raining." and "Is it raining?" The former is either true or false. The truth value of the latter can be thought of as, what I call, "indeterminate." (Use another word if you like.) Nothing "paradoxical" about that. Now, if a sentence X is true iff it is false, then X must be of indeterminate truth value. Do you not agree?

[snip]

Please answer the question.

Dan

[olcott]

On 7/26/2023 11:26 AM, Mild Shock wrote:
> Same error again? If the third set would be "indeterminate",
> it would mean something like having many values:
>
> s e m <=> s e t & s e f
>
> But you cannot really express indeterminate with your nonsense,
> if you would like to express indeterminate, you would
>
> need to distinguish between:
> - form of a sentence
> - valuation of a sentence
>
> You could then say:
>
> A form f is indeterminate, if we have two valuations v1 and v2, so that:
> EXIST(v1):EXIST(v2):[v1(f) coherent & v2(f) coherent & v1 != v2]
>
> But the Liar Paradox is paradoxical, basically:
> ~EXIST(v):[v(f) coherent]
>

The Liar Paradox is self-contradictory thus making it semantically
unsound. LP := ~True(LP) has a cycle in its directed graph that Prolog
detects and rejects.

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

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

[olcott]

The computer science misconception of the notion of undecidability
prevents humanity from having a correct understanding of True(L, x).

[Mild Shock]

To resolve the problem that valuations v talk about to many
propositions than only x0 the Liar Paradox, we can approach
the problem differently, we look only at the valuation v at

the point x0, and denote it by w ∈ {0,1}. So
we encode, just the George Boole encoding:

w=0 <=> ~(x0 e v)
w=1 <=> (x0 e v)

The Liar Paradox is now a sentence phi representing its form:

phi(w) <=> w = 1-w

We can now more precisely define indeterminate and also adapt paradoxical:

The form phi is indeterminate:
EXIST(w1):EXIST(w2):[w1 = 1-w1 & w2 = 1-w2 & w1!=w2]

The form phi is paradoxical:
~EXIST(w):w = 1-w

Quizz: Is the Liar Paradox indeterminate or paradoxical?

[Dan Christensen]

On Wednesday, July 26, 2023 at 12:26:41 PM UTC-4, Mild Shock wrote:

> Dan Christensen schrieb am Mittwoch, 26. Juli 2023 um 18:15:36 UTC+2:
> > On Wednesday, July 26, 2023 at 11:27:59 AM UTC-4, olcott wrote:
> > > On 7/26/2023 10:02 AM, Dan Christensen wrote:
> > > > On Wednesday, July 26, 2023 at 3:19:55 AM UTC-4, Mild Shock wrote:
> > > >> Well you wrote "proposed resolution", if your "proposed resolution"
> > > >> is based on some mislabeling humbug, I don't know what you want
> > > >> from us, when you also wrote "Your comments? ".
> > > >>
> > > >
> > > > Sorry, but I take your comment is a very minor point indeed: an opinion about the informal usage of a single word. Other than that, you have nothing to say about my presentation. I take that a small victory. Thanks anyway for your time and interest.
> >
> > > Whether or not decision problems are undecidable because the problem
> > > itself is incorrect or or computation simply is not powerful enough is a
> > > very important issue.
> > >
> > How is this relevant? We have a set of objects (sentences) divided into 3 disjoint subsets corresponding to (1) true sentences, (2) false sentences, and (3) sentences of indeterminate truth value. We recognize, for example, the difference between the sentence "It is raining." and "Is it raining?" The former is either true or false. The truth value of the latter can be thought of as, what I call, "indeterminate." (Use another word if you like.) Nothing "paradoxical" about that. Now, if a sentence X is true iff it is false, then X must be of indeterminate truth value. Do you not agree?

I take it you cannot disagree, Mr. Collapse. Whew!

> Same error again? If the third set would be "indeterminate",
> it would mean something like having many values:
>

[snip]

A minor point. Use another word if you like in your own writing. I prefer "indeterminate." I have made clear its meaning in this context. Deal with it.

[Mild Shock]

You cannot prefer indeterminate. Its nonsense.
Which of the two hold:

The form phi is indeterminate:
EXIST(w1):EXIST(w2):[w1 = 1-w1 & w2 = 1-w2 & w1!=w2]

The form phi is paradoxical:
~EXIST(w):w = 1-w

Quizz: Is the Liar Paradox indeterminate or paradoxical?

[Mild Shock]

Maybe its a problem that you are not native english,
some frog eater essentially, since you french canada?

To use the word "indeterminate", you have to exhibit
at least two valuations v1 and v2, that satisfy a given form.

Its extremly easy:

"A sentence Φ (or set of sentences Σ) is indeterminate if and only if

there is MORE THAN ONE WAY to coherently assign it a truth value (or to

assign the sentences contained in it truth values)"
http://fitelson.org/piksi/piksi_18/cook_notes.pdf

Read the MORE THAN ONE WAY.

[Mild Shock]

Example of an indeterminate self referencing sentence is:

"This sentences is true"

It is coherent that it is true, and it is coherent that it is false.
Or symbolically with George Boole where w ∈ {0,1}:

"This sentences is true"
w = w

How many solutions w does it have?
Now take the Liar Paradox, its not indeterminate:

"This sentences is false"
w = 1 - w

How many solutions w does it have?

[Dan Christensen]

On Wednesday, July 26, 2023 at 12:48:48 PM UTC-4, Mild Shock wrote:

> Dan Christensen schrieb am Mittwoch, 26. Juli 2023 um 18:46:41 UTC+2:
> > On Wednesday, July 26, 2023 at 12:26:41 PM UTC-4, Mild Shock wrote:
> >
> > > Dan Christensen schrieb am Mittwoch, 26. Juli 2023 um 18:15:36 UTC+2:
> > > > On Wednesday, July 26, 2023 at 11:27:59 AM UTC-4, olcott wrote:
> > > > > On 7/26/2023 10:02 AM, Dan Christensen wrote:
> > > > > > On Wednesday, July 26, 2023 at 3:19:55 AM UTC-4, Mild Shock wrote:
> > > > > >> Well you wrote "proposed resolution", if your "proposed resolution"
> > > > > >> is based on some mislabeling humbug, I don't know what you want
> > > > > >> from us, when you also wrote "Your comments? ".
> > > > > >>
> > > > > >
> > > > > > Sorry, but I take your comment is a very minor point indeed: an opinion about the informal usage of a single word. Other than that, you have nothing to say about my presentation. I take that a small victory. Thanks anyway for your time and interest.
> > > >
> > > > > Whether or not decision problems are undecidable because the problem
> > > > > itself is incorrect or or computation simply is not powerful enough is a
> > > > > very important issue.
> > > > >
> > > > How is this relevant? We have a set of objects (sentences) divided into 3 disjoint subsets corresponding to (1) true sentences, (2) false sentences, and (3) sentences of indeterminate truth value. We recognize, for example, the difference between the sentence "It is raining." and "Is it raining?" The former is either true or false. The truth value of the latter can be thought of as, what I call, "indeterminate." (Use another word if you like.) Nothing "paradoxical" about that. Now, if a sentence X is true iff it is false, then X must be of indeterminate truth value. Do you not agree?

> > I take it you cannot disagree, Mr. Collapse. Whew!

Thanks for again confirming your cannot disagree.

> > > Same error again? If the third set would be "indeterminate",
> > > it would mean something like having many values:
> > >
> > [snip]
> >
> > A minor point. Use another word if you like in your own writing. I prefer "indeterminate." I have made clear its meaning in this context. Deal with it.

[snip repetitive nonsense]

> Quizz: Is the Liar Paradox indeterminate or paradoxical?

Now that I have resolved it, it is no longer a paradox. The Liar Theorem???

[olcott]

That you resolved it as not having a truth value is the only correct way
to resolve it.

> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com
>

[Dan Christensen]

Yes. Some sentences have truth values. Some do not, e.g . (1) "Is it raining?" (2) "This sentence is false."

[olcott]

Computer science blames the software for this lack of a truth value.

[Dan Christensen]

I'm not into theoretical computer science, but that seems unlikely. Citing climate catastrophes alone will convince no one.

Dan

[olcott]

This is a case where the software gets blamed instead of the input that
does the opposite of whatever Boolean value that its halt decider
returns. https://en.wikipedia.org/wiki/Halting_problem

[Dan Christensen]

Is this suppose to convince us that climate catastrophes were caused the application material implication (vacuous truth, etc.)?

Dan

[olcott]

Since you don't having any understanding of any of the background
information I must provide that as a basis before you can begin to
understand the rest.

The bottom line is that the Tarski Undefinability theorem gets confused
by the Liar Paradox and does not understand that it is neither true nor
false and concludes that there is no such thing as objective truth on
the basis that it cannot prove that the Liar Paradox is true.

[Dan Christensen]

I am still not convinced that climate catastrophes were caused by the application of material implication. Is that still your claim?

Dan

[olcott]

You seem to be proving that you are dishonest.

[olcott]

That the Tarski undefinability theorem "proves" that there is no
objective difference between verified facts and despicable lies prevents
lies about climate change from being discerned from the objective facts
about climate change.

[Dan Christensen]

No one, not even Donald Trump, believes that, in the real world, there is no objective difference between verified facts and despicable lies. It is NOT the basis for ANY science or legitimate legal system. So, move on from this. It is a dead end. What else have you got?

Dan

[olcott]

That you reject the verified facts about what the Tarski Undefinability
theorem states leaves us at a dead end. Maybe despite your DC proof you
are merely a troll.

[Dan Christensen]

> That you reject the verified facts about what the Tarski Undefinability
> theorem states leaves us at a dead end. Maybe despite your DC proof you
> are merely a troll.

A "troll" because I reject your claim that, in the real world, there is no objective difference between verified facts and despicable lies???? Just who is the troll here, Ollie? I see no point in continuing this preposterous line of "reasoning."

[olcott]

Yes you are a troll because of that. According to that principle
2 + 3 = 5 is literally false if one does not happen to "believe in"
numbers.

[Mild Shock]

If you could prove something being indeterminate,
then one could use the axioms here, and the approach
how your formulate the theorem here:
https://dcproof.com/LiarParadox2.htm

To prove this sentence indeterminate:

/* Indeterminate Example */
"this sentence is true"

Did you ever try whether your approach works? For
the Liar Paradox you use:

/* Liar Paradox */
"this sentence is false"
[b e t <=> b e f]

Which is the same like:

/* Liar Paradox */
"this sentence is false"
[b e t => b e f] & [b e f => b e t]

For the Indeterminate Example you could use:

/* Indeterminate Example */
"this sentence is true"
[b e t => b e t] & [b e f => b e f]

Can you prove "indeterminacy" of Indeterminate Example?
Can you prove this here:

=> ALL(b):[b e s => [[b e t => b e t] & [b e f => b e f] => b e m]]]]

if yes, congratulation, your LiarParadox2.htm is indeed related
to "indeterminacy". If no, do you know why it doesn't work?

Dan Christensen schrieb am Mittwoch, 26. Juli 2023 um 22:26:57 UTC+2:
> Dan

[Dan Christensen]

On Wednesday, July 26, 2023 at 5:05:55 PM UTC-4, Mild Shock wrote:
> If you could prove something being indeterminate,
> then one could use the axioms here, and the approach
> how your formulate the theorem here:
> https://dcproof.com/LiarParadox2.htm
>
> To prove this sentence indeterminate:
>
> /* Indeterminate Example */
> "this sentence is true"
>
> Did you ever try whether your approach works? For
> the Liar Paradox you use:
>
> /* Liar Paradox */
> "this sentence is false"
> [b e t <=> b e f]
>

Set s is not necessarily the set of ALL sentences. It is just a set of sentences each of which can be classified as precisely one of: true, false, or indeterminate such that trichotomy holds. Every sentence in s that is true iff it is false (e.g. The Liar) must be classed as indeterminate.

> Which is the same like:
>
> /* Liar Paradox */
> "this sentence is false"
> [b e t => b e f] & [b e f => b e t]
>
> For the Indeterminate Example you could use:
>
> /* Indeterminate Example */
> "this sentence is true"
> [b e t => b e t] & [b e f => b e f]
>

This will true of every sentence b in s.

> Can you prove "indeterminacy" of Indeterminate Example?
> Can you prove this here:
>
> => ALL(b):[b e s => [[b e t => b e t] & [b e f => b e f] => b e m]]]]
>

1. Set(s) & Set(t) & Set(f)
Axiom

2. x in s
Premise

3. x in t
Premise

4. x in t => x in t
Conclusion, 3

5. x in f
Premise

6. x in f => x in f
Conclusion, 5

7. [x in t => x in t] & [x in f => x in f]
Join, 4, 6

8. ALL(b):[b in s => [b in t => b in t] & [b in f => b in f]]
Conclusion, 2

> if yes, congratulation, your LiarParadox2.htm is indeed related
> to "indeterminacy". If no, do you know why it doesn't work?
>

Dan

[Mild Shock]

What happend to m?

8. ALL(b):[b in s => [b in t => b in t] & [b in f => b in f]]
Conclusion, 2

The goal was to prove:

> ALL(b):[b e s => [[b e t => b e t] & [b e f => b e f] => b e m]]]]

After all you claim that "m" expresses indeterminate. You litteraly wrote:
> Some sentences are neither true nor false.
> We can say that their truth values are _indeterminate_

LoL

[Mild Shock]

BTW you can use a truth table generator to find out
whether a sentence is paradoxical or indeterminate,
using the official definitions from here:
http://fitelson.org/piksi/piksi_18/cook_notes.pdf

A sentences is indeterminate if the truth table of
its form has a least two rows that are marked true T.

Example "Indeterminate":
"this sentences is true"

A (A ↔ A)
F T /* There are at least two Rows with "T" */
T T
https://web.stanford.edu/class/cs103/tools/truth-table-tool/

A sentences is paradoxical if the truth table of
its form has no row marked true T:

Example "Paradoxical":
"this sentences is false"

A (A ↔ ¬A)
F F /* There is no Row with "T" */
T F
https://web.stanford.edu/class/cs103/tools/truth-table-tool/

[Mild Shock]

Can you formalize these two notions? Can
you formalize truth tables? What does a truth table
show, does it not show the "graph" of a truth functional?

Don't you have Function Spaces now in DC Proof?
Lets say B = {0,1} is the Boolean Domain. A truth functional
that does a propositional form, is nothing else than:

f : B x ... x B -> B

It maps the cartesian product of the Boolean Domain
to the Boolean Domain. The cartesian product is B^n
= B x ... x B where n is the number of propositional variables

in the propositional form. So how would one express
paradoxical/indeterminate using set theory and function spaces?
Do we need to become rocket scientists for that?

[Dan Christensen]

> What happend to m?
> 8. ALL(b):[b in s => [b in t => b in t] & [b in f => b in f]]
> Conclusion, 2

> The goal was to prove:
> > ALL(b):[b e s => [[b e t => b e t] & [b e f => b e f] => b e m]]]]
> After all you claim that "m" expresses indeterminate.

We have:

ALL(b):[b in s => [b in t => b in t] & [b in f => b in f]]

Your criteria is too general. It is satisfied by EVERY element of s:

In https://dcproof.com/LiarParadox2.htm I proved:

42. ALL(b):[b e s => [[b e t <=> b e f] => b e m]]

This claim is restricted to sentences that are true iff they are false, e.g. "This sentence is false."

[Mild Shock]

What criteria? I don't use criterias. You are crazy. I model sentences.
The [b in t => b in t] & [b in f => b in f] is not the same model
like [b e t <=> b e f]. This is because these are two different sentences:

"This sentence is true"
"This sentence is false"

Why do not both sentences land in "m"?

[Dan Christensen]

On Friday, July 28, 2023 at 8:46:19 AM UTC-4, Mild Shock wrote:
> BTW you can use a truth table generator to find out
> whether a sentence is paradoxical or indeterminate,

[snip]

Not sure. I just know that the truth value of A is indeterminate if A<=>~A. See https://dcproof.com/LiarParadox2.htm

[Dan Christensen]

On Friday, July 28, 2023 at 12:51:36 PM UTC-4, Mild Shock wrote:
> What criteria? I don't use criterias. You are crazy. I model sentences.
> The [b in t => b in t] & [b in f => b in f] is not the same model
> like [b e t <=> b e f]. This is because these are two different sentences:
>
> "This sentence is true"
> "This sentence is false"
>
> Why do not both sentences land in "m"?

I only know that "This sentence is false" is indeterminate. See https://dcproof.com/LiarParadox2.htm

[Mild Shock]

But indeterminate would mean that it has multiple truth values.
But fact is that it has no truth value that satisfies the form.

So I guess you proved nonsense. You can easily check:

A (A ↔ ¬A)
F F /* There is no Row with "T" */
T F
https://web.stanford.edu/class/cs103/tools/truth-table-tool/

No truth value. Its not indeterminate. Its only paradoxical.

See also here:
http://fitelson.org/piksi/piksi_18/cook_notes.pdf

[Mild Shock]

Also there is a logical error in your arguing, that it would
"resolve" something going from propositions to expressions.
In propositions we learn from the Liar Paradox, that for

for functions with boolean formal parameters and boolean
return values, i.e. functions where B={0,1}:

f : B x ... x B -> B

There are functions which are constantly zero 0. Even if
they use some arguments, like here, a single argument A:

A (A ↔ ¬A)
F F /* There is no Row with "T" */
T F
https://web.stanford.edu/class/cs103/tools/truth-table-tool/

The Revenge Paradox is the simple observation, that if
we go from propositions to expressions, and extend the
domain from boolean B={0,1}, lets say to K={0,1,u},

we would then need to look at truth functionals, that
work with this new domain:

f : K x ... x K -> K

Now its easy to see that there are again functions wicht
are constantly zero 0. Even if the domain is K={0,1,u}.

[Mild Shock]

In set theory you can even prove the ultimate revenge
paradox, for arbitrary domains J = {0,...}. There is the
following set theory theorem, relatively trivial:

Theorem Constant Function:
Assume an arbitrary domain J = {0,...} which is non-empty
and has zero 0, i.e. 0 e J. Then looking at the function space,
i.e. functions for an arity n:

f : J^n -> J

We find that there is always a constant function:

ALL(x1):[x1 e J => .... ALL(xn):[xn e J => f(x1,..,xn) = 0] ...]

Proof:
Just use the Subset Axiom, and construct this function:
f = { (x1,...,xn,0) | x1 e J & ... & xn e J }
Q.E.D:

Corrolary Ultimate Revenge:
In a language that is at least as expressive as set theory,
there is always a Liar Paradox like form, even if we allow
to range expressions over more than B = {0,1}.

Proof:
We had J arbitrary in the previous theorem, so it can be also
different or larger than B. Even excelling the super revenge
paradox which transcends only K = {0,1,u}.
Q.E.D.

[Mild Shock]

A tricky construction of an ultimate revenge paradox, possibly
totally out of Dan Christensens mental radar, is the Diagonal
Argument, for example used in Cantors Theorem. One also

constructs a function which is constant, but make it only
constant along a diagonal, i.e. we have a function g : A x A -> B,
but we only make the function h : A -> B, which is h(x) = g(x,x),

constant, and then can use it to prove the following:

Cantor's theorem
The cardinality of the power set is larger then
the cardinality of its base set
https://en.wikipedia.org/wiki/Cantor%27s_theorem

Or symbolically, |P(A)| > |A|. It can also be used to show
that there is no universal set. The Russel paradox is not
the only way to show there is no universal set.

The Russel paradox was predated by the Cantor paradox.
Assue the universal class V were a set. Since P(V) consists
of sets, and since V is universal, we must have P(V) ⊆ V,

or by definition of cardinality |P(V)| =< |V|, a contradiction.
Q.E.D.

But DC Proof is possibly the only tool that claims to provide
set theory, but has blogged about set theory practically nothing.
Dan Christensen already claimed he is not

interested in proving things about ordinals. He is also not
interesting in proving things about cardinals. He really misses a
fine piece of history in logic and mathematics.

[olcott]

I bet you also cannot determine how to bake an angel food cake using
only a pile of shit. Indeterminate means that you haven't figured it out
yet, not that it is impossible.

Completeness of a theory T is equivalent to the following condition:
For any closed formula ϕ, precisely one of the two assertions applies:
either ϕ is derivable from T or ¬ϕ is derivable from T.
https://encyclopediaofmath.org/wiki/Completeness_(in_logic)

Thus self-contradictory expression ϕ is incorrectly assumed to be a
semantically correct expression that T is simply too weak to resolve to
a truth value.

[Dan Christensen]

On Saturday, July 29, 2023 at 6:53:37 AM UTC-4, Mild Shock wrote:

> Dan Christensen schrieb am Samstag, 29. Juli 2023 um 04:30:41 UTC+2:
> > On Friday, July 28, 2023 at 12:51:36 PM UTC-4, Mild Shock wrote:
> > > What criteria? I don't use criterias. You are crazy. I model sentences.
> > > The [b in t => b in t] & [b in f => b in f] is not the same model
> > > like [b e t <=> b e f]. This is because these are two different sentences:
> > >
> > > "This sentence is true"
> > > "This sentence is false"
> > >
> > > Why do not both sentences land in "m"?
> > I only know that "This sentence is false" is indeterminate. See https://dcproof.com/LiarParadox2.htm

> But indeterminate would mean that it has multiple truth values.

I take it to mean that no truth value can be inferred for the sentence in question.

> But fact is that it has no truth value that satisfies the form.
>
> So I guess you proved nonsense. You can easily check:
> A (A ↔ ¬A)
> F F /* There is no Row with "T" */
> T F

Here, A is logical proposition that is either true or false. Again, in the case of a sentence, it may be true or false or of indeterminate truth value. Some examples of the latter:

1. What time is it?
2. Wash the dishes.
3. This sentence is false.

[Dan Christensen]

On Saturday, July 29, 2023 at 7:23:00 AM UTC-4, Mild Shock wrote:
> A tricky construction of an ultimate revenge paradox, possibly
> totally out of Dan Christensens mental radar, is the Diagonal
> Argument, for example used in Cantors Theorem. One also
>
> constructs a function which is constant, but make it only
> constant along a diagonal, i.e. we have a function g : A x A -> B,
> but we only make the function h : A -> B, which is h(x) = g(x,x),
>
> constant, and then can use it to prove the following:
>
> Cantor's theorem
> The cardinality of the power set is larger then
> the cardinality of its base set
> https://en.wikipedia.org/wiki/Cantor%27s_theorem
>

[snip]

See my proof of Cantor's Theorem: http://www.dcproof.com/CountableL3.htm (only 54 lines)

[Mild Shock]

You are confusing pragmatics with semantics.

In linguistics none of your examples
are called "indeterminate". They have
different pragmatic functions, especially

in relation to their performativity:

> 1. What time is it?

~~> Thats a question

> 2. Wash the dishes.
~~> Thats an imperative

> 3. This sentence is false.

~~> Thats a declarative

See for yourself you moron. The most
precise theory of pragmatic functions
is speech act theory which sees effects
on both ends of a speech act.

In linguistics and related fields,
pragmatics is the study of how context
contributes to meaning.
https://en.wikipedia.org/wiki/Pragmatics

Performativity is the concept that
language can function as a form of social
action and have the effect of change.
https://en.wikipedia.org/wiki/Performativity

In the philosophy of language and linguistics,
speech act is something expressed by an individual
that not only presents information but performs
an action as well.
https://en.wikipedia.org/wiki/Speech_act

Dan Christensen schrieb:

[olcott]

On 7/29/2023 2:07 PM, Mild Shock wrote:
> You are confusing pragmatics with semantics.
>
> In linguistics none of your examples
> are called "indeterminate". They have
> different pragmatic functions, especially
>
> in relation to their performativity:
>
> > 1. What time is it?
> ~~> Thats a question
>
> > 2. Wash the dishes.
> ~~> Thats an imperative
>
> > 3. This sentence is false.
> ~~> Thats a declarative
>
> See for yourself you moron. The most
> precise theory of pragmatic functions
> is speech act theory which sees effects
> on both ends of a speech act.
>
> In linguistics and related fields,
> pragmatics is the study of how context
> contributes to meaning.
> https://en.wikipedia.org/wiki/Pragmatics
>

Yes this is crucial.
The situational context of who is asked a question changes the meaning
of the question.

> Performativity is the concept that
> language can function as a form of social
> action and have the effect of change.
> https://en.wikipedia.org/wiki/Performativity
>
> In the philosophy of language and linguistics,
> speech act is something expressed by an individual
> that not only presents information but performs
> an action as well.
> https://en.wikipedia.org/wiki/Speech_act
>
> Dan Christensen schrieb:
>> On Saturday, July 29, 2023 at 6:53:37 AM UTC-4, Mild Shock wrote:
>
>> Here, A is logical proposition that is either true or false. Again, in
>> the case of a sentence, it may be true or false or of indeterminate
>> truth value. Some examples of the latter:
>>
>> 1. What time is it?
>> 2. Wash the dishes.
>> 3. This sentence is false.
>>
>> Dan
>>
>> Download my DC Proof 2.0 freeware at http://www.dcproof.com
>> Visit my Math Blog at http://www.dcproof.wordpress.com
>>
>>
>

[Mild Shock]

As a rule of thumb and dumb,
just look at the sentence closing:

When it has a question mark "?":
> 1. What time is it ?
~~> Thats a question

When it has an exclamation mark "!":
> 2. Wash the dishes!
~~> Thats an imperative

When it has a full stop ".":

> 3. This sentence is false.
~~> Thats a declarative

But I have sad news for you, its not indeterminate the Liar
Paradox, its only paradoxical. See for yourself moron, or
are you not able to read? To stupid to even read?

A sentence Φ (or set of sentences Σ) is indeterminate if and only if ..
A sentence Φ (or set of sentences Σ) is paradoxical if and only if ..
http://fitelson.org/piksi/piksi_18/cook_notes.pdf

But there is a simple trick to find out whethe a sentences is
indeterminate or paradoxical. Just change the full stop "."
into a question mark "?" and count the solutions:

This sentence is false?

If you have multiple answers its indeterminate.
If you have no answer its paradoxical. Even a complete
dumbo like you could use this trick.

[Mild Shock]

The answer to this question is "Yes and No":
This sentence is true?
~~> Therefore the sentence is indeterminate

The answer to this question is "neither Yes nor No":
This sentence is false?
~~> Therefore the sentence is paradoxical

[Mild Shock]

Beware of Dan Christensen, the DC Proof fraudster. He doesn't
understand indeterminate and paradoxical. Lets assume there
are coolers that can only hold one turkey. Guess what,

imperatives can be also indeterminate and paradoxical.

Example: Indeterminate imperative,
situation there are two empty coolers:

- Put the turkey in a cooler!

Usually one would respond a question, which cooler?
Or it could be also open to the asked person to
choose which cooler, could be also an expectation.

Example: Paradoxical imperative,
situation there is one non-empty cooler:

- Pu the turkey in a cooler!

Usually one would respond with the declaration, this
is impossible, the cooler is already full. Maybe followed by
a question do you have another cooler somewhere.

[Dan Christensen]

On Saturday, July 29, 2023 at 3:07:28 PM UTC-4, Mild Shock wrote:
> You are confusing pragmatics with semantics.
>
> In linguistics none of your examples
> are called "indeterminate". They have
> different pragmatic functions, especially
>
> in relation to their performativity:
> > 1. What time is it?
> ~~> Thats a question
>

It's truth value (true or false) cannot be determined, i.e. its truth value is indeterminate.

> > 2. Wash the dishes.
> ~~> Thats an imperative

As above.

> > 3. This sentence is false.
> ~~> Thats a declarative
>

[snip]

As above. Also see https://dcproof.com/LiarParadox2.htm

[Mild Shock]

Here is a simpler set of example, showing that quantifiers might
go with imperatives, quantifying an aspect of the imperative:

- Scratch your thumb!
Indeterminate: People have usually two Thumbs.

- Scratch your head!
Determinate: People have usually only one Head.

- Scratch your tail!
Paradoxical: People usually don't have a Tail.

[Dan Christensen]

On Wednesday, August 2, 2023 at 8:34:35 PM UTC-4, Mild Shock wrote:

[snip]

> Also you didn't show yet that this here:
>
> this sentence is true
>
> Falls into your category "m".

For some reason, that sentence has not generated nearly as much interest as "This sentence is false" (the Liar Paradox).

My proof is a proposed resolution of the Liar Paradox. See: https://dcproof.com/LiarParadox2.htm

The essential property of "This sentence is false," the property on which I based my proof is that this sentence is true if and only if it is false. I prove it must be of indeterminate truth value.

What do you suggest is the essential property of "This sentence is TRUE?" Maybe you can invent a new paradox--the Truth-teller Paradox? You could even introduce your own "paradoxical" terminology!

[Mild Shock]

It seems your cherry picking has turned you
into an annoying complete lazy dumb ass crank.

[Mild Shock]

Explain us why category does or does not fit into set theory?
And why this would be relevant? Related to this question,
how far do you have to go in category to even bang your

head in set theory. With cherry picking approach, just
looking at mappings, giving no structure to the objects,
it all seems to fit into set theory:

id : A -> A
f : A -> B & g : B -> C => comp(f,g) : A -> C

With set theory we can show id and comp exist. But things
get interesting when objects become some structure. Lets
say for two objects A and B there is product object A x B and

a power object B^A. And taking the n-th root:

ev : B x A^B -> A

How define „obj“ in set theory, especially in DC Proof?

[Mild Shock]

Hint the cherry picking approach for „obj“ does not anymore
work, there is no set that contains all function spaces. You
might find this omega tower in set theory:

F_0 = some urelements
F_n+1 = F_n u (F_n -> F_n)
F_omega = U_n e omega F_n

But then the function space (F_omega -> F_omega) is
not contained in F_omega. You can use Cantors Theorem
to show that. Have to check which innocuous category

theory expression would express that. Maybe we need
also some limits. Category theory would not crumble, but
the modelling of „obj“ as a set would crumble, since

it would amount to the same contradictory idea, contradictory
in set theory, of having a universal set.

[Dan Christensen]

On Thursday, August 3, 2023 at 4:31:10 AM UTC-4, Mild Shock wrote:
> Explain us why category does or does not fit into set theory?

[snip]

I made use of set theory so that I could prove (84 lines) that for every set there exists 3 disjoint subsets that satisfy the trichotomy rule.

ALL(s):[Set(s) => EXIST(t):EXIST(u):EXIST(v):[Set(t) & Set(u) & Set(v)

& ALL(a):[a in t => a in s]
& ALL(a):[a in u => a in s]
& ALL(a):[a in v => a in s]

& ALL(a):[a in s => [a in t | a in u | a in v]
& ~[a in t & a in u]
& ~[a in t & a in v]
& ~[a in u & a in v]]]]

Hint: Use t=s, u=v={ }

[Mild Shock]

You are using the wrong terminology. The Liar Paradox is
considered paradoxical, its not considered indeterminate.
You can check yourself:

A sentence Φ (or set of sentences Σ) is paradoxical if and only if

there is no way to coherently assign it a truth value (or to assign the
sentences contained in it truth values).
http://fitelson.org/piksi/piksi_18/cook_notes.pdf

[Mild Shock]

Why not prove something nice in DC Proof related
to constructive mathematics. Can you prove that

π is transcendental

in DC Proof. What does that even mean? Do you
have some clue. Whats the history of this question?

[Mild Shock]

I don't know Dan-O-Matik, are you good in polynomials?
Is it a problem for you to juggle with polynomials, even
more so for multi-variant polynomials? Anyway

here is a lead, maybe, for a proof, informal or formal?

Formal Proofs of Transcendence for e and π as an
Application of Multivariate and Symmetric Polynomials
Sophie Bernard et al. - 2015
https://arxiv.org/abs/1512.02791

Mild Shock schrieb:

[Mild Shock]

Why not prove something nice in DC Proof related
to constructive mathematics. Can you prove that

π is transcendental

Or maybe start with e, the Euler number:

e is transcendental

in DC Proof. What does that even mean? Do you
have some clue. Whats the history of this question?

Can we bring in some Pascal Triangle? See also:

Math Bite: Finding e in Pascal’s Triangle - Harlan J. Brothers
http://www.brotherstechnology.com/docs/Finding_e_in_Pascals_Triangle.pdf

[Mild Shock]

So lets walk the path of the philosopher stone, and
resolve the Liar Paradox. The adivse is here, very modern
with a touch of LGBT:

"In like manner the Philosophers would have the quadrangle
reduced into a triangle, that is, into body, Spirit, and Soul,
which three do appear in three previous colors before redness,
for example, the body or earth in the blackness of Saturn,
the Spirit in a lunar whiteness, as water, the Soul or air in a solar
citrinity: then will the triangle be perfect, but this likewise must
be changed into a circle, that is, into an invariable redness:
By which operation the woman is converted into the man,
and made one with him, and the senary the first number of
the perfect completed by one, two, having returned again to an
unit, in which is eternal rest and peace.
— Michael Maier, Atalanta Fugiens, Emblem XXI.

So lets turn "woman" into "man" and assume there are not
two truth values {T, F} but only a single truth value {*}. The
truth table for "is" is very simple:

A B A <-> B
* * *

When we now stipulate negation as:

A ~A
* *

We can resolve the Liar Paradox by p <-> ~p:

"This sentences is false."

Dan Christensen schrieb am Dienstag, 25. Juli 2023 um 20:06:15 UTC+2:
> Some sentences are neither true nor false. We can say that their truth values are _indeterminate_ , not unlike the numerical value of 1/0. Some examples:
>
> - What time is it?
>
> - Wash the dishes.
>
> - This sentence is false.
>
> The latter will take some explaining: Suppose we have a set of sentences s. Define subsets t, f and m of set s such that
>
> t = the subset of "true" sentences
> f = the subset of "false" sentences
> m = the subset of sentences of indeterminate truth value
>
> Each element of s will be an element of precisely one of these subsets.
>
> Now, "This sentence is false" is problematic. It is assumed to be true if and only if it is false. Therefore, its true value must be indeterminate.
>
> Formal proof: https://dcproof.com/LiarParadox2.htm (only 44 lines in DC Proof format)
>
> Your comments?
>
> Dan

[Mild Shock]

This is the proper resolution, like the Russell Paradox, it sacrifices
something. The Russell Paradox resolution in ZF set theory sacrifices
for example that the universal class would be a set,

here we sacrificy the idea that there are "men" and "women", we
go from two truth values {T, F} to only a single truth value {*}. The
advantage is that we can stay in the framework of sentences,

we don't need to call in expressions such that:

sentences ⊆ expressions

So its a proper resolution of the Liar Paradox, not a fake resolution
like in Dan Christensens proof, which assumes a broader class
of phrases that can have a value outside of truth values.

A terminological error in Dan Christensens proof is that he
calls this broader class of phrases "sentences" whereas in
mathematical logic and computer science one would call

them "expressions". Just like 2+2 is a expression and 2+2=4
is a sentence. But the single truth value resolution doesn't need
"expressions", a phrase which valuates to {*} is still a sentence.

[Mild Shock]

Ha Ha, a story when cranks asses hit bottom.

Dan Christensens logic and math stamina is very weak.
A few days ago he was all fire for his Trichotomy nonsense.
Now all his steam is gone, not a single attempt anymore

to show the benefit of his Liar Paradox resolution with
Trichotomy? Does this imply that the Grelling Antinomy
is also a sentence of the third kind? Or the Russell Paradox,

is it also a sentence of the third kind? After all we can
reduce the the Antinomy and the Paradox also to p <=> ~p.

[Mild Shock]

If we have Trichotomy, how is this proof valid?

The set of all things cannot exist.
https://www.dcproof.com/UniversalSet.htm

In the end phase the proof uses reduction ad absurdum,
via r e r & ~r e r, but essentially we can also derive r & r <=> ~r & r.
So in the end phase the following classical inference is used:

G, A |- f
---------------- (Reductio Ad Absurdum)
G |- ~A

But what if r & r <=> ~r & r is of the third kind. Is this
a valid inference then? What logic is that would have,
where m = the subset of sentences of indeterminate truth value:

G, A |- m
---------------- (Rabbit out of the hat)
G |- ~A

LoL

[Dan Christensen]

On Friday, August 25, 2023 at 3:46:58 AM UTC-4, Mild Shock wrote:
> If we have Trichotomy, how is this proof valid?
>
> The set of all things cannot exist.
> https://www.dcproof.com/UniversalSet.htm
>

That proof has nothing to do with any trichotomy.

> In the end phase the proof uses reduction ad absurdum,
> via r e r & ~r e r,

As such the universal set u cannot exist. (Line 23)

> but essentially we can also derive r & r <=> ~r & r.

[snip]

r is not a logical proposition. Try again.

[olcott]

On 7/25/2023 1:06 PM, Dan Christensen wrote:
> Some sentences are neither true nor false. We can say that their truth values are _indeterminate_ , not unlike the numerical value of 1/0. Some examples:
>
> - What time is it?

Indeterminate means cannot be determined a more accurate term is
non-existent. We we are trying to determine N such that N > 5 and N < 3
the issue at hand is not that we are incapable of determining the
correct N, the issue is that such an N does not exist.

[Dan Christensen]

On Friday, August 25, 2023 at 2:21:58 PM UTC-4, olcott wrote:
> On 7/25/2023 1:06 PM, Dan Christensen wrote:
> > Some sentences are neither true nor false. We can say that their truth values are _indeterminate_ , not unlike the numerical value of 1/0. Some examples:
> >
> > - What time is it?
> Indeterminate means cannot be determined a more accurate term is
> non-existent. We we are trying to determine N such that N > 5 and N < 3
> the issue at hand is not that we are incapable of determining the
> correct N, the issue is that such an N does not exist.

I have presented a reasonable definition of "indeterminate truth value" in this context. Deal with it, Ollie.

Dan

[olcott]

It is misleading in the same way that the mathematical notion of
incompleteness is misleading.

Every self-contradictory expression is neither provable nor refutable
thus the mathematical notion of incompleteness would blame the
formal system and not the self-contradictory expression.

If one cannot determine a truth value because such a truth value
does not exist it is more accurate to call this an [non-existent]
rather than than [indeterminate] truth value.

When we define [lazy horse] as any horse that will not run 100 feet
then [dead horse] perfectly meets the definition of [lazy horse] yet
we all know that a dead horse is not lazy.

[Dan Christensen]

On Friday, August 25, 2023 at 3:13:14 PM UTC-4, olcott wrote:
> On 8/25/2023 1:49 PM, Dan Christensen wrote:
> > On Friday, August 25, 2023 at 2:21:58 PM UTC-4, olcott wrote:
> >> On 7/25/2023 1:06 PM, Dan Christensen wrote:
> >>> Some sentences are neither true nor false. We can say that their truth values are _indeterminate_ , not unlike the numerical value of 1/0. Some examples:
> >>>
> >>> - What time is it?
> >> Indeterminate means cannot be determined a more accurate term is
> >> non-existent. We we are trying to determine N such that N > 5 and N < 3
> >> the issue at hand is not that we are incapable of determining the
> >> correct N, the issue is that such an N does not exist.
> >
> > I have presented a reasonable definition of "indeterminate truth value" in this context. Deal with it, Ollie.
> >

> It is misleading in the same way that the mathematical notion of
> incompleteness is misleading.
>

"indeterminate: adjective, not measured, counted, or clearly known"
https://dictionary.cambridge.org/us/dictionary/english/indeterminate

So, there is nothing "misleading" about saying that the the truth value of the of the sentence "What time is it?" is indeterminate.

Is your quibbling here about terminology your own only criticism of my proposed resolution The Liar Paradox?

[olcott]

This may be beyond your technical capability to understand
yet the fact the LP := ~True(LP) has a cycle in the directed
graph of its evaluation sequence proves that it has no truth
value what-so-ever.

Every other formalization of the Liar Paradox makes sure
to represent self-reference incorrectly, thus ends up with
an incorrect assessment.

[Dan Christensen]

A lot of unnecessary machinery! I guess it might impress some.

Again, using nothing more than the basic rules of logic and elementary set theory I obtain the theorem:

ALL(s):ALL(t):ALL(f):ALL(m):[Set(s) & Set(t) & Set(f) & Set(m)

=> [ALL(a):[a in t => a in s] (Subsets of s)
& ALL(a):[a in f => a in s]
& ALL(a):[a in m => a in s]

& ALL(a):[a in s => [a in t | a in f | a in m] (Trichotomy)
& ~[a in t & a in f] & ~[a in t & a in m]
& ~[a in f & a in m]]

=> ALL(b):[b in s => [[b in t <=> b in f] => b in m]]]]

Where we can interpret:

s = a set of sentences, not necessarily every sentence
t = the subset of true sentences in s
f = the subset of false sentence in s
m = the set of sentences of indeterminate truth value in s

Formal proof: https://dcproof.com/LiarParadox2.htm

[olcott]

The seems to have nothing to do with the Liar Paradox or its
closely related set theory isomorphism Russell's Paradox.

*In both cases self-reference prevents a resolution to a truth value*

ZFC simply determined that RP is incorrect.
We need the same thing for the Liar Paradox.

[Dan Christensen]

[unsnip]

=> [ALL(a):[a in t => a in s] (Subsets of s)
& ALL(a):[a in f => a in s]
& ALL(a):[a in m => a in s]

& ALL(a):[a in s => [a in t | a in f | a in m] (Trichotomy)
& ~[a in t & a in f] & ~[a in t & a in m]
& ~[a in f & a in m]]

=> ALL(b):[b in s => [[b in t <=> b in f] => b in m]]]]

Where we can interpret:

s = a set of sentences, not necessarily every sentence
t = the subset of true sentences in s
f = the subset of false sentence in s
m = the set of sentences of indeterminate truth value in s

Formal proof: https://dcproof.com/LiarParadox2.htm

> The seems to have nothing to do with the Liar Paradox

Willful blindness???

> or its
> closely related set theory isomorphism Russell's Paradox.
>
> *In both cases self-reference prevents a resolution to a truth value*
>

RP cannot be blamed on self-reference. Self-reference is handled quite easily in FOL. In DC Proof:

1 ALL(b):[R(b,x) <=> ~R(b,b)] <--------------- Self-reference on RHS
Premise

2 R(x,x) <=> ~R(x,x)
U Spec, 1

3 ~EXIST(a):ALL(b):[R(b,a) <=> ~R(b,b)]
Conclusion, 1

> ZFC simply determined that RP is incorrect.

No, ZFC just makes it impossible to infer the existence of the set of all sets that are not elements of themselves. Self-reference (as above) is still allowed in FOL.

[olcott]

On 8/25/2023 7:51 PM, Dan Christensen wrote:

> [unsnip]

>
> => [ALL(a):[a in t => a in s] (Subsets of s)
> & ALL(a):[a in f => a in s]
> & ALL(a):[a in m => a in s]
>
> & ALL(a):[a in s => [a in t | a in f | a in m] (Trichotomy)
> & ~[a in t & a in f] & ~[a in t & a in m]
> & ~[a in f & a in m]]
>
> => ALL(b):[b in s => [[b in t <=> b in f] => b in m]]]]
>
> Where we can interpret:
>
> s = a set of sentences, not necessarily every sentence
> t = the subset of true sentences in s
> f = the subset of false sentence in s
> m = the set of sentences of indeterminate truth value in s
>
> Formal proof: https://dcproof.com/LiarParadox2.htm
>

>> The seems to have nothing to do with the Liar Paradox
>

> Willful blindness???

LP := ~True(LP)
Where a single sentence asserts that itself is untrue
is the actual liar paradox.

Prolog correctly determines that all sentences of the general
form of the Liar Paradox are semantically unsound because they
have a cycle in the directed graph of their evaluation sequence.

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

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