Graham Thirdness is also a partition of a set:
https://en.wikipedia.org/wiki/Partition_of_a_set

If you look at the cross product T x F, then Dan Trichotomy
does the following partition, they are mutually disjoint,
and Dan Trichotomy does label the first partition as m:

~a e f & ~a e t
a e f & ~a e t
~a e f & a e t

Now in Graham Thirdness, the partition reads as follows,
again all mutually disjoint. But now m labels the
last partation, and not the first partition:

a e f & ~a e t
~a e f & a e t
a e f & a e t

So both Dan Trichotomy and Graham Thirdness classify
sentences into 3 categories. Whereas Dan Trichotomy
does classify via these 3 categories:

indeterminate
false and only false
true and only true

Does Graham Thirdness classify via these 3 categories:

false and only false
true and only true
paradoxical

Mild Shock schrieb am Dienstag, 19. September 2023 um 08:30:33 UTC+2:
> This is not the case for the Liar Sentence. The Liar Sentence
> can also be in m and it would mean true and false.
>
> Just use Graham Thirdness:
> > ALL(a):[a ε s => [a ε t | a ε f] (Graham Thirdness Rules)
> > & [~a ε t | ~a ε f | a ε m]
> > & [~a ε m | a ε f]
> > & [~a ε m | a ε t]]
> And you can prove:
> ALL(b):[b ε s => [[b ε t <=> b ε f] => b ε m]]]]
> Dan Christensen schrieb am Dienstag, 19. September 2023 um 03:23:16 UTC+2:
> > Wrong. A sentence is either true (in t), false (in f), or of
> > indeterminate truth value (in m). Do you deny it?

What happened to your user name?
Too much Trichotomy? LMAO!

sci.math schrieb:

On Tuesday, September 19, 2023 at 4:38:23 PM UTC-4, Mild Shock wrote:
> What happened to your user name?
> Too much Trichotomy? LMAO!
>
> sci.math schrieb:

Testing

Mild Shock formulated the question :
> What happened to your user name?
> Too much Trichotomy? LMAO!
>
> sci.math schrieb:

It got googleized.

Dan Christensen used his keyboard to write :
> On Tuesday, September 19, 2023 at 4:38:23 PM UTC-4, Mild Shock wrote:
>> What happened to your user name?
>> Too much Trichotomy? LMAO!
>>
>> sci.math schrieb:
>
> Testing

Intermittently. It happened in another group I'm in too.

Either its standard or its not standard.
What should "more" standard mean?

The problem with your solution is that you reject LEM.
Whereas the solution by Graham Priest still accepts LEM.

In Graham Priest you can prove:

/* Provable in Graham Thirdness */
a e f v a e t

In your solution you cannot prove:

/* Not Provable in Dan Trichotomy */
a e f v a e t

Just ask ChatGPT, it amazes me again and again:

[...] Priest is associated with a solution to the Liar Paradox
known as "dialetheism." [...] it offers a solution to the Liar
Paradox that doesn't involve rejecting the law of excluded
middle (the principle that a statement is either true or false) [...]

I already almost forgot amout LEM.

sci.math schrieb am Dienstag, 19. September 2023 um 15:40:44 UTC+2:
> You can prove this using the more standard definition of a trichotomy:

On Tuesday, September 19, 2023 at 6:39:08 PM UTC-4, Mild Shock wrote:

> > You can prove this using the more standard definition of a trichotomy:

> Either its standard or its not standard.
> What should "more" standard mean?
>

More standard than your wonky notion of a trichotomy.

> The problem with your solution is that you reject LEM.

On the contrary, I make use of it in my proof (e.g. by eliminating '~~' and proof by contraction)

Maybe you didn't know, but introducing a trichotomy on a set does not necessarily require the rejection of LEM. As I show in the lemma attached to my proof, a trichotomy can be introduced on ANY set, even the empty set. Proof: http://www.dcproof.com/LiarParadoxLemma.htm

> Whereas the solution by Graham Priest still accepts LEM.
>
> In Graham Priest you can prove:
>
> /* Provable in Graham Thirdness */
> a e f v a e t
>
> In your solution you cannot prove:
>
> /* Not Provable in Dan Trichotomy */
> a e f v a e t
>
[snip]

Maybe you didn't know, but a sentence in natural language, for the purposes of resolving the the Liar Paradox, can be classified as being one of either: (1) a true sentence, (2) a false sentence, or (3) a sentence of indeterminate truth value (e.g. "Wash your hands" and "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

On 9/19/2023 4:38 PM, Mild Shock wrote:
> What happened to your user name?
> Too much Trichotomy? LMAO!
>
> sci.math schrieb:

He uses Giggle Groups, that's why.

Google doesn't fire poor programmers it accidentally hires now and then.
Instead, they assign them to work in a dungeon, deep below their
headquarters, where their new job is to make Google Groups worse and
worse. And they succeed doing so.

On 9/19/2023 9:09 PM, Volney wrote:
> On 9/19/2023 4:38 PM, Mild Shock wrote:
>> What happened to your user name?
>> Too much Trichotomy? LMAO!
>>
>> sci.math schrieb:
>
> He uses Giggle Groups, that's why.
>
> Google doesn't fire poor programmers it accidentally hires now and then.
> Instead, they assign them to work in a dungeon, deep below their
> headquarters, where their new job is to make Google Groups worse and
> worse. And they succeed doing so.

ROFL!!!! Its most likely very, oh so very true. Yikes!

On 9/20/2023 12:10 AM, Chris M. Thomasson wrote:
> On 9/19/2023 9:09 PM, Volney wrote:
>> On 9/19/2023 4:38 PM, Mild Shock wrote:
>>> What happened to your user name?
>>> Too much Trichotomy? LMAO!
>>>
>>> sci.math schrieb:
>>
>> He uses Giggle Groups, that's why.
>>
>> Google doesn't fire poor programmers it accidentally hires now and
>> then. Instead, they assign them to work in a dungeon, deep below their
>> headquarters, where their new job is to make Google Groups worse and
>> worse. And they succeed doing so.
>
> ROFL!!!! Its most likely very, oh so very true. Yikes!
>
It's funny because it's true! I've been saying this for years!

Google, whose biggest fame is its search engine, broke Usenet searching
around 2008 and didn't get it fixed for a year. Advanced group searches
got broken in 2015 and have never been fixed. Many other "improvements"
have worsened GG since buying DejaNews.

You don't make any sense. We told you already
that "Wash your hands!" and "Did you wash your hands?"
are irrelevant to the Liar Paradox.

They have an exclaimation mark "!" respectively
a question mark "?" at the end. So they are
anyway in a separate category.

Whats wrong with you?

The Liar Paradox has a so called full stop "."
at the end.

Dan Christensen schrieb:
> On Tuesday, September 19, 2023 at 6:39:08 PM UTC-4, Mild Shock wrote:
>
>>> You can prove this using the more standard definition of a trichotomy:
>
>> Either its standard or its not standard.
>> What should "more" standard mean?
>>
>
> More standard than your wonky notion of a trichotomy.
>
>> The problem with your solution is that you reject LEM.
>
> On the contrary, I make use of it in my proof (e.g. by eliminating '~~' and proof by contraction)
>
> Maybe you didn't know, but introducing a trichotomy on a set does not necessarily require the rejection of LEM. As I show in the lemma attached to my proof, a trichotomy can be introduced on ANY set, even the empty set. Proof: http://www.dcproof.com/LiarParadoxLemma.htm" rel="nofollow" target="_blank">http://www.dcproof.com/LiarParadoxLemma.htm
>
>> Whereas the solution by Graham Priest still accepts LEM.
>>
>> In Graham Priest you can prove:
>>
>> /* Provable in Graham Thirdness */
>> a e f v a e t
>>
>> In your solution you cannot prove:
>>
>> /* Not Provable in Dan Trichotomy */
>> a e f v a e t
>>
> [snip]
>
> Maybe you didn't know, but a sentence in natural language, for the purposes of resolving the the Liar Paradox, can be classified as being one of either: (1) a true sentence, (2) a false sentence, or (3) a sentence of indeterminate truth value (e.g. "Wash your hands" and "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
>

On Wednesday, September 20, 2023 at 3:06:52 AM UTC-4, Mild Shock wrote:
> Dan Christensen schrieb:
> > On Tuesday, September 19, 2023 at 8:47:07 PM UTC-4, Mild Shock (aka Mr. Collapse) wrote:
> >
> >> Dan Christensen schrieb:
> >>> Maybe you didn't know, but a sentence in natural language, [for the purpose of resolving the Liar Paradox] can be classified as being one of either: (1) a true sentence, (2) a false sentence, or (3) a sentence of indeterminate truth value (e.g. "Wash your hands" and "This sentence is false.")
> >
> >> Who cares about *can*, thats not science and/or
> >> mathematics or logic.
> >
> > Maybe you also didn't know, but there are other ways to classify sentences in natural language. Here are some other ways:
> >
> > - declarative sentence (statement) <--------------- true or false
> > - interrogative sentence (question) <--------------- indeterminate
> > - imperative sentence (command) <----------------- indeterminate
> > - exclamative sentence (exclamation) <----------- could also be declarative or imperative

> >
> > https://www.englishclub.com/grammar/sentence/type.php
> >

> You don't make any sense. We told you already
> that "Wash your hands!" and "Did you wash your hands?"
> are irrelevant to the Liar Paradox.
>
[snip]

Wrong. Each is an example of a sentence of indeterminate truth value.

FYI in English (unlike German?) imperative sentences need not end in an exclamation point.

Dan

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

Its MORE STANDARD that they end with exclamation mark:

Q: To ChatGPT
Does an imperative sentence end with an exclamation mark?

A: From ChatGPT
Yes, an imperative sentence typically ends with an exclamation
mark. Examples include "Run!" "Stop!" and "Do your homework!"

https://www.quora.com/Does-an-imperative-sentence-end-with-an-exclamation-mark

Dan Christensen schrieb am Mittwoch, 20. September 2023 um 16:43:40 UTC+2:
> FYI in English (unlike German?) imperative sentences need not end in an exclamation point.

There is an intersting historic bit. Spanish language uses again
two exclamation marks. Here is an example:

¡Ve a la India y vuelve!

Without the second exclamation mark, Christopher Columbus
would not have come back to spain in his first voyage 1493.

https://en.wikipedia.org/wiki/Voyages_of_Christopher_Columbus

Mild Shock schrieb am Mittwoch, 20. September 2023 um 16:49:00 UTC+2:
> Its MORE STANDARD that they end with exclamation mark:
>
> Q: To ChatGPT
> Does an imperative sentence end with an exclamation mark?
>
> A: From ChatGPT
> Yes, an imperative sentence typically ends with an exclamation
> mark. Examples include "Run!" "Stop!" and "Do your homework!"
>
> https://www.quora.com/Does-an-imperative-sentence-end-with-an-exclamation-mark
> Dan Christensen schrieb am Mittwoch, 20. September 2023 um 16:43:40 UTC+2:
> > FYI in English (unlike German?) imperative sentences need not end in an exclamation point.

On Wednesday, September 20, 2023 at 10:49:00 AM UTC-4, Mild Shock wrote:

> Its MORE STANDARD that they end with exclamation mark:
>
> Q: To ChatGPT
> Does an imperative sentence end with an exclamation mark?
>
> A: From ChatGPT
> Yes, an imperative sentence typically ends with an exclamation
> mark. Examples include "Run!" "Stop!" and "Do your homework!"
>

Not my experience. To Canadians anyway, an exclamation point is used only for emphasis or to express surprise, elation or disgust. It's like shouting. It can also be used on declarative sentences as well, e.g. "They ran away!" "We won the game!"

Back to the main point: A sentence, for the purposes of resolving the Liar Paradox, can be classified as being one of either: (1) a true sentence, (2) a false sentence, or (3) a sentence of indeterminate truth value.

Dan

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

Thats only LESS STANDARD, we can say SUB STANDARD.
At least this every mathematicians and logicians standard.
Because they don't care what philosophers feel or halucinate.

When its about mathematical structures, its pretty easy
to see that 3-, 3+ and 4 are solutions to the Liar Paradox.
You don't need to invoke experience or taste.

Just do your mathematical reasoning.

Dan Christensen schrieb am Mittwoch, 20. September 2023 um 17:31:33 UTC+2:
> Back to the main point: A sentence, for the purposes of resolving the Liar Paradox,
can be classified as being one of either: (1) a true sentence, (2) a false sentence, or
(3) a sentence of indeterminate truth value.

As a first step you need to be able to really
talk about truth values. Currently where do you
model truth values? You don't have yet truth values!

Hint, you need an equivalence relation over s,
the set of sentences. And then you can form equivalence
classes. For example if the mathematical structure

is 3-, as modelled via your Trichotomy here:

[a e t | a e f | a e m]
& ~[a e t & a e f]
& ~[a e t & a e m]
& ~[a e f & a e m]
https://www.dcproof.com/LiarParadoxResolution.htm

Then you can define the following equivalence
relation over sentences:

a ~ b :<=> [[a e t & b e t] |
[a e f & b e f] | [a e m & b e m]]

How many equivalence classes are there? What would
happen if we call the equivalence class 'F', 'T'
and 'U', 3 different constants.

How would we then model the solution really with
truth values, and not with sets of sentences that
share the same truth value?

Do you know how to do it?

Mild Shock schrieb am Mittwoch, 20. September 2023 um 20:51:06 UTC+2:
> Thats only LESS STANDARD, we can say SUB STANDARD.
> At least this every mathematicians and logicians standard.
> Because they don't care what philosophers feel or halucinate.
>
> When its about mathematical structures, its pretty easy
> to see that 3-, 3+ and 4 are solutions to the Liar Paradox.
> You don't need to invoke experience or taste.
>
> Just do your mathematical reasoning.
> Dan Christensen schrieb am Mittwoch, 20. September 2023 um 17:31:33 UTC+2:
> > Back to the main point: A sentence, for the purposes of resolving the Liar Paradox,
> can be classified as being one of either: (1) a true sentence, (2) a false sentence, or
> (3) a sentence of indeterminate truth value.

On Wednesday, September 20, 2023 at 2:51:38 PM UTC-4, Mild Shock wrote:
>
> Dan Christensen schrieb:
> > Back to the main point: A sentence, for the purposes
> of resolving the Liar Paradox, can be classified as being one of either:
> (1) a true sentence, (2) a false sentence, or (3) a
> sentence of indeterminate truth value.

[snip]

> When its about mathematical structures, its pretty easy
> to see that 3-, 3+ and 4 are solutions to the Liar Paradox.

Huh??? The Liar Paradox is about a sentence in natural language. I have applied basic set theory to a longstanding problem in linguistics. Deal with, Mr. Collapse.

Dan

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

On Wednesday, September 20, 2023 at 2:51:06 PM UTC-4, Mild Shock wrote:
>>
> Dan Christensen schrieb:
> > Back to the main point: A sentence, for the purposes
> of resolving the Liar Paradox, can be classified as being one of either:
> (1) a true sentence, (2) a false sentence, or (3) a
> sentence of indeterminate truth value.

[snip]

> When its about mathematical structures, its pretty easy
> to see that 3-, 3+ and 4 are solutions to the Liar Paradox.

Huh??? The Liar Paradox is about a sentence in natural language. I have applied basic set theory to a longstanding problem in linguistics. Deal with it, Mr. Collapse.

Dan

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

Truth values help tremendously the thinking process!
We only have to prove once:

Mild Shock schrieb am Mittwoch, 20. September 2023 um 21:12:12 UTC+2:
> Here is a solution with truth values:
> 49 ALL(x):[Value(x) => [[x=t <=> x=f] => x=u]]
https://groups.google.com/g/sci.math/c/JQTvBJvYxDA/m/lrDgf9VjCAAJ

Now we can think about sentence classifications. We can even
make a stronger liar paradox, instead of only:
a e t <=> a e f

We can let it read, in 3- this is irrelevant, needed for the others:
a e t & ~a e f <=> ~a e t & a e f

Now there are many possible assignments to truth values:

Model 3-
~a e t & ~a e f ~~> x = u
a e t & ~a e f ~~> x = t
~a e t & a e f ~~> x = f

Model 3+
a e t & ~a e f ~~> x = t
~a e t & a e f ~~> x = f
a e t & a e f ~~> x = u

Model 4
~a e t & ~a e f ~~> x = u
a e t & ~a e f ~~> x = t
~a e t & a e f ~~> x = f
a e t & a e f ~~> x = u

Dan Christensen schrieb am Mittwoch, 20. September 2023 um 21:08:42 UTC+2:
> On Wednesday, September 20, 2023 at 2:51:38 PM UTC-4, Mild Shock wrote:
> >
> > Dan Christensen schrieb:
> > > Back to the main point: A sentence, for the purposes
> > of resolving the Liar Paradox, can be classified as being one of either:
> > (1) a true sentence, (2) a false sentence, or (3) a
> > sentence of indeterminate truth value.
> [snip]
> > When its about mathematical structures, its pretty easy
> > to see that 3-, 3+ and 4 are solutions to the Liar Paradox.
> Huh??? The Liar Paradox is about a sentence in natural language. I have applied basic set theory to a longstanding problem in linguistics. Deal with, Mr. Collapse.
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

Thats a funny find that these two are logically equivalent:

a e t <=> a e f
a e t & ~a e f <=> ~a e t & a e f

Wasn't that much aware of the fact. But it seems to be true.
Was aware of this fact, confirmed by Wolfgang Schwarz tree tool:

(A↔B) ↔ (¬B ↔ ¬A) is valid.
https://www.umsu.de/trees/#(A~4B)~4(~3B~4~3A)

Now if you overlay the two, you have also this logical equivalence:

(A↔B) ↔ ((A∧¬B) ↔ (¬A∧B)) is valid.
https://www.umsu.de/trees/#(A~4B)~4(A~1~3B~4~3A~1B)

Thats kind of cool!

Mild Shock schrieb am Mittwoch, 20. September 2023 um 21:32:07 UTC+2:
> Truth values help tremendously the thinking process!
> We only have to prove once:
>
> Mild Shock schrieb am Mittwoch, 20. September 2023 um 21:12:12 UTC+2:
> > Here is a solution with truth values:
> > 49 ALL(x):[Value(x) => [[x=t <=> x=f] => x=u]]
> https://groups.google.com/g/sci.math/c/JQTvBJvYxDA/m/lrDgf9VjCAAJ
>
> Now we can think about sentence classifications. We can even
> make a stronger liar paradox, instead of only:
> a e t <=> a e f
>
> We can let it read, in 3- this is irrelevant, needed for the others:
> a e t & ~a e f <=> ~a e t & a e f
>
> Now there are many possible assignments to truth values:
>
> Model 3-
> ~a e t & ~a e f ~~> x = u
> a e t & ~a e f ~~> x = t
> ~a e t & a e f ~~> x = f
>
> Model 3+
> a e t & ~a e f ~~> x = t
> ~a e t & a e f ~~> x = f
> a e t & a e f ~~> x = u
>
> Model 4
> ~a e t & ~a e f ~~> x = u
> a e t & ~a e f ~~> x = t
> ~a e t & a e f ~~> x = f
> a e t & a e f ~~> x = u
> Dan Christensen schrieb am Mittwoch, 20. September 2023 um 21:08:42 UTC+2:
> > On Wednesday, September 20, 2023 at 2:51:38 PM UTC-4, Mild Shock wrote:
> > >
> > > Dan Christensen schrieb:
> > > > Back to the main point: A sentence, for the purposes
> > > of resolving the Liar Paradox, can be classified as being one of either:
> > > (1) a true sentence, (2) a false sentence, or (3) a
> > > sentence of indeterminate truth value.
> > [snip]
> > > When its about mathematical structures, its pretty easy
> > > to see that 3-, 3+ and 4 are solutions to the Liar Paradox.
> > Huh??? The Liar Paradox is about a sentence in natural language. I have applied basic set theory to a longstanding problem in linguistics. Deal with, Mr. Collapse.
> > Dan
> >
> > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > Visit my Math Blog at http://www.dcproof.wordpress.com

On Wednesday, September 20, 2023 at 3:32:07 PM UTC-4, Mild Shock wrote:
> Now we can think about sentence classifications. We can even
> make a stronger liar paradox, instead of only:
> a e t <=> a e f
>
> We can let it read, in 3- this is irrelevant, needed for the others:
> a e t & ~a e f <=> ~a e t & a e f
>
[snip]

Looks much the same. The LHS is equivalent to a in t. The RHS to a in f.

If x is in one of 3 categories, then x is NOT in the other 2 categories. We are talking here about 3 DISJOINT categories.

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 was loosing his marbles:
> We are talking here about 3 DISJOINT categories.

Well the 3 categories f, t and u are disjoin, just have a look.
The suggested mappings for the models 3-, 3+ and 4 are
also disjoint. Here is the proof again, see axiom 2, 3 and 4:

------------------------------------ begin proof --------------------------------------

1 ALL(x):[Value(x) <=> x=f | x=t | x=u]
Axiom

2 ~f=t
Axiom

3 ~f=u
Axiom

4 ~t=u
Axiom

[...]

49 ALL(x):[Value(x) => [[x=t <=> x=f] => x=u]]
Rem DNeg, 48

------------------------------------ end proof --------------------------------------

https://groups.google.com/g/sci.math/c/JQTvBJvYxDA/m/lrDgf9VjCAAJ

Dan Christensen schrieb am Mittwoch, 20. September 2023 um 22:44:44 UTC+2:
> On Wednesday, September 20, 2023 at 3:32:07 PM UTC-4, Mild Shock wrote:
> > Now we can think about sentence classifications. We can even
> > make a stronger liar paradox, instead of only:
> > a e t <=> a e f
> >
> > We can let it read, in 3- this is irrelevant, needed for the others:
> > a e t & ~a e f <=> ~a e t & a e f
> >
> [snip]
>
> Looks much the same. The LHS is equivalent to a in t. The RHS to a in f.
>
> If x is in one of 3 categories, then x is NOT in the other 2 categories. We are talking here about 3 DISJOINT categories.
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

On Wednesday, September 20, 2023 at 4:46:53 PM UTC-4, Mild Shock wrote:
> Dan Christensen was loosing his marbles:
> > We are talking here about 3 DISJOINT categories.
> Well the 3 categories f, t and u are disjoin, just have a look.
> The suggested mappings for the models 3-, 3+ and 4 are
> also disjoint.

What ARE you talking about? Maybe you should get some sleep.

Dan

You can use this equivalence relation:

a ~ b :<=> [[a e t & b e t] v [a e f & b e f] v [a e m & b e m]]

To get equivalence classes:

https://en.wikipedia.org/wiki/Equivalence_class

The equivalence classes are the truth values.
Thats missing in your take. If you then further
abstract, you get this nifty result:

49 ALL(x):[Value(x) => [[x=t <=> x=f] => x=u]]
Rem DNeg, 48

Dan Christensen schrieb am Mittwoch, 20. September 2023 um 23:03:13 UTC+2:.
> What ARE you talking about?