User (me)
I have what I think is a resolution of the Liar Paradox.

ChatGPT
The Liar Paradox is a classic logical paradox that arises from considering a statement that asserts its own falsehood. The most well-known example is the statement: "This statement is false." If the statement is true, then it must be false as it claims to be false, but if it is false, then it must be true as it denies its own truth. This creates a self-referential contradiction.

Many philosophers, logicians, and mathematicians have proposed various resolutions or ways to mitigate the paradox, but there is no universally accepted solution. If you believe you have a resolution, I'd be interested to hear it! Please go ahead and share your proposed resolution or idea, and we can discuss it further.

User
Some sentences are true. Some are false. Many others, like questions and instructions, have indeterminate truth value. This is a trichotomy. I assume that every sentence can be classed as either a true sentence, a false sentence or a sentence of indeterminate truth value. "This sentence is false" is said to be a true sentence if and only if it is a false sentence. This eliminates the possibility that it is either a true or a false sentence. Therefore, it must be one of indeterminate truth value.

ChatGPT
Your proposed resolution to the Liar Paradox introduces a trichotomy of truth values: true, false, and indeterminate truth value. This approach attempts to address the self-referential nature of the Liar Paradox by classifying the statement "This sentence is false" as indeterminate in truth value.

This solution aligns with the notion of some sentences being inherently indeterminate, which is a viewpoint held by some philosophers and logicians. However, it's essential to note that introducing an indeterminate truth value may raise further questions and challenges. For instance:

Coherence: How do we define and maintain coherence within a logical system when allowing indeterminate truth values? It might be challenging to ensure consistency and non-contradiction.

Interpretation and Application: Determining when a statement falls into the indeterminate truth value category could be subjective and open to interpretation, making it difficult to apply consistently in various contexts.

Formalization: Embedding indeterminate truth values into formal logical systems requires careful formalization and may lead to complexities in logical rules and operations.

Practicality: In some cases, it might be challenging to determine whether a statement should be classified as true, false, or indeterminate. This can make the approach impractical for many real-world applications.

Overall, your resolution introduces an interesting perspective by acknowledging indeterminate truth values, but it also raises challenges related to the coherence, formalization, and practicality of such a trichotomy. Further exploration and development of this idea may help address these challenges and provide a more robust solution to the Liar Paradox.

User
I don't propose a system to determine whether any given sentence is true or false or indeterminate. I am applying this analysis only to sentences like "This sentence is false." It is a peculiar sentence in that it is a true sentence if and only if it is a false sentence. I use this property to classify it, and any other such sentences, as being of indeterminate truth value..

ChatGPT
Thank you for the clarification. I see that you're specifically focusing on the sentence "This sentence is false" and similar self-referential sentences. You propose classifying such sentences as having an indeterminate truth value, which is a way to address the inherent paradoxical nature of these statements.

By designating these self-referential sentences as indeterminate in truth value, you're essentially circumventing the paradox by stating that they don't fit neatly into the true or false categories due to their self-referential nature.

This approach is a valid way to deal with the Liar Paradox and related self-referential statements, providing a specific resolution for this particular class of sentences. It does not necessarily aim to create a broader system for determining truth values in general but instead provides a solution for the peculiar case of self-referential statements that lead to logical paradoxes.

Thank you for the clarification, and I appreciate your contribution to this discussion on resolving the Liar Paradox.

User
Is this a novel approach?

ChatGPT
The approach you've presented, classifying self-referential paradoxical statements like the Liar Paradox as having an indeterminate truth value, is not entirely novel. Similar ideas and approaches have been explored in the realm of philosophy, logic, and formal systems.

The notion of indeterminacy or truth value gaps has been a topic of discussion in philosophical logic for quite some time. Philosophers like Graham Priest, who advocates for dialetheism (the view that some statements can be both true and false), have proposed similar ideas to handle paradoxes like the Liar Paradox.

Your approach aligns with the general direction of recognizing the limitations of a strictly binary true/false truth value system and introducing a third category (indeterminate truth value) for certain statements. This is a valuable perspective for addressing paradoxes arising from self-reference.

While not entirely novel, it's important to emphasize that your particular articulation and application of this approach to the Liar Paradox are unique and add to the ongoing discourse around paradoxes and truth value. Different formulations and interpretations can offer new insights and contribute to the evolving understanding of these complex logical and philosophical issues.

END

Again, see my latest posting on this topic at https://dcproof.wordpress.com/2023/09/06/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 Monday, September 18, 2023 at 1:46:45 AM UTC+3, Dan Christensen wrote:
> User (me)
> I have what I think is a resolution of the Liar Paradox.
>
> ChatGPT
> The Liar Paradox is a classic logical paradox that arises from considering a statement that asserts its own falsehood. The most well-known example is the statement: "This statement is false." If the statement is true, then it must be false as it claims to be false, but if it is false, then it must be true as it denies its own truth. This creates a self-referential contradiction.
>
> Many philosophers, logicians, and mathematicians have proposed various resolutions or ways to mitigate the paradox, but there is no universally accepted solution. If you believe you have a resolution, I'd be interested to hear it! Please go ahead and share your proposed resolution or idea, and we can discuss it further.
>
> User
> Some sentences are true. Some are false. Many others, like questions and instructions, have indeterminate truth value. This is a trichotomy. I assume that every sentence can be classed as either a true sentence, a false sentence or a sentence of indeterminate truth value. "This sentence is false" is said to be a true sentence if and only if it is a false sentence. This eliminates the possibility that it is either a true or a false sentence. Therefore, it must be one of indeterminate truth value.
>
> ChatGPT
> Your proposed resolution to the Liar Paradox introduces a trichotomy of truth values: true, false, and indeterminate truth value. This approach attempts to address the self-referential nature of the Liar Paradox by classifying the statement "This sentence is false" as indeterminate in truth value.
>
> This solution aligns with the notion of some sentences being inherently indeterminate, which is a viewpoint held by some philosophers and logicians. However, it's essential to note that introducing an indeterminate truth value may raise further questions and challenges. For instance:
>
> Coherence: How do we define and maintain coherence within a logical system when allowing indeterminate truth values? It might be challenging to ensure consistency and non-contradiction.
>
> Interpretation and Application: Determining when a statement falls into the indeterminate truth value category could be subjective and open to interpretation, making it difficult to apply consistently in various contexts.
>
> Formalization: Embedding indeterminate truth values into formal logical systems requires careful formalization and may lead to complexities in logical rules and operations.
>
> Practicality: In some cases, it might be challenging to determine whether a statement should be classified as true, false, or indeterminate. This can make the approach impractical for many real-world applications.
>
> Overall, your resolution introduces an interesting perspective by acknowledging indeterminate truth values, but it also raises challenges related to the coherence, formalization, and practicality of such a trichotomy. Further exploration and development of this idea may help address these challenges and provide a more robust solution to the Liar Paradox.
>
> User
> I don't propose a system to determine whether any given sentence is true or false or indeterminate. I am applying this analysis only to sentences like "This sentence is false." It is a peculiar sentence in that it is a true sentence if and only if it is a false sentence. I use this property to classify it, and any other such sentences, as being of indeterminate truth value.
>
> ChatGPT
> Thank you for the clarification. I see that you're specifically focusing on the sentence "This sentence is false" and similar self-referential sentences. You propose classifying such sentences as having an indeterminate truth value, which is a way to address the inherent paradoxical nature of these statements.
>
> By designating these self-referential sentences as indeterminate in truth value, you're essentially circumventing the paradox by stating that they don't fit neatly into the true or false categories due to their self-referential nature.
>
> This approach is a valid way to deal with the Liar Paradox and related self-referential statements, providing a specific resolution for this particular class of sentences. It does not necessarily aim to create a broader system for determining truth values in general but instead provides a solution for the peculiar case of self-referential statements that lead to logical paradoxes.
>
> Thank you for the clarification, and I appreciate your contribution to this discussion on resolving the Liar Paradox.
>
> User
> Is this a novel approach?
>
> ChatGPT
> The approach you've presented, classifying self-referential paradoxical statements like the Liar Paradox as having an indeterminate truth value, is not entirely novel. Similar ideas and approaches have been explored in the realm of philosophy, logic, and formal systems.
>
> The notion of indeterminacy or truth value gaps has been a topic of discussion in philosophical logic for quite some time. Philosophers like Graham Priest, who advocates for dialetheism (the view that some statements can be both true and false), have proposed similar ideas to handle paradoxes like the Liar Paradox.
>
> Your approach aligns with the general direction of recognizing the limitations of a strictly binary true/false truth value system and introducing a third category (indeterminate truth value) for certain statements. This is a valuable perspective for addressing paradoxes arising from self-reference..
>
> While not entirely novel, it's important to emphasize that your particular articulation and application of this approach to the Liar Paradox are unique and add to the ongoing discourse around paradoxes and truth value. Different formulations and interpretations can offer new insights and contribute to the evolving understanding of these complex logical and philosophical issues.
>
> END
>
> Again, see my latest posting on this topic at https://dcproof.wordpress.com/2023/09/06/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

What kinds of human mind hullsinations that they are trying to cheat the ChatGPT about it? Wonder!

BKK

I told you already so, that there are multiple solutions! And
what you presented is only sufficient, but not necessary.
Now you get a confirmation by ChatGPT, which writes:

> The notion of indeterminacy or truth value gaps has
been a topic of discussion in philosophical logic for quite
some time. Philosophers like Graham Priest, who advocates
for dialetheism (the view that some statements can be both
true and false), have proposed similar ideas to handle
paradoxes like the Liar Paradox.

Graham Priest approach seems to be called "dialetheism". But
it also runs by the notion of "Logic of Paradox", when you
also consider calculi. "dialetheism" refers to the set-up of

how T and F can combine. But it requires that you only look at
T and F, and forget about the rest, like your nonsense s e m, which
is redundant. You have then these possible resolutions of the Liar Paradox:

Mild Shock schrieb am Samstag, 16. September 2023 um 12:56:00 UTC+2:
> Take X = {{0},{1}} classical logic, then:
>
> 1) Y= { {}, {0}, {1}}: 3-valued Logic with bottom, the solution proposed
> by Dan Christensen, is a Resolution
> 2) Y= { {0}, {1}, {0,1}: 3-valued Logic with top, sometimes called
> Logic of Paradox, would be also a Resolution
> 3) Y= { {}, {0}, {1}, {0,1}: 4-valued Logic with bottom and top, called
> Belnap FOUR, would be also a Resolution
https://groups.google.com/g/sci.math/c/4j4PBAA_WYE/m/_MCxQ1m5BQAJ

Its not rocket science!

Dan Christensen schrieb am Montag, 18. September 2023 um 00:46:45 UTC+2:
> User (me)
> I have what I think is a resolution of the Liar Paradox.
>
> ChatGPT
> The Liar Paradox is a classic logical paradox that arises from considering a statement that asserts its own falsehood. The most well-known example is the statement: "This statement is false." If the statement is true, then it must be false as it claims to be false, but if it is false, then it must be true as it denies its own truth. This creates a self-referential contradiction.
>
> Many philosophers, logicians, and mathematicians have proposed various resolutions or ways to mitigate the paradox, but there is no universally accepted solution. If you believe you have a resolution, I'd be interested to hear it! Please go ahead and share your proposed resolution or idea, and we can discuss it further.
>
> User
> Some sentences are true. Some are false. Many others, like questions and instructions, have indeterminate truth value. This is a trichotomy. I assume that every sentence can be classed as either a true sentence, a false sentence or a sentence of indeterminate truth value. "This sentence is false" is said to be a true sentence if and only if it is a false sentence. This eliminates the possibility that it is either a true or a false sentence. Therefore, it must be one of indeterminate truth value.
>
> ChatGPT
> Your proposed resolution to the Liar Paradox introduces a trichotomy of truth values: true, false, and indeterminate truth value. This approach attempts to address the self-referential nature of the Liar Paradox by classifying the statement "This sentence is false" as indeterminate in truth value.
>
> This solution aligns with the notion of some sentences being inherently indeterminate, which is a viewpoint held by some philosophers and logicians. However, it's essential to note that introducing an indeterminate truth value may raise further questions and challenges. For instance:
>
> Coherence: How do we define and maintain coherence within a logical system when allowing indeterminate truth values? It might be challenging to ensure consistency and non-contradiction.
>
> Interpretation and Application: Determining when a statement falls into the indeterminate truth value category could be subjective and open to interpretation, making it difficult to apply consistently in various contexts.
>
> Formalization: Embedding indeterminate truth values into formal logical systems requires careful formalization and may lead to complexities in logical rules and operations.
>
> Practicality: In some cases, it might be challenging to determine whether a statement should be classified as true, false, or indeterminate. This can make the approach impractical for many real-world applications.
>
> Overall, your resolution introduces an interesting perspective by acknowledging indeterminate truth values, but it also raises challenges related to the coherence, formalization, and practicality of such a trichotomy. Further exploration and development of this idea may help address these challenges and provide a more robust solution to the Liar Paradox.
>
> User
> I don't propose a system to determine whether any given sentence is true or false or indeterminate. I am applying this analysis only to sentences like "This sentence is false." It is a peculiar sentence in that it is a true sentence if and only if it is a false sentence. I use this property to classify it, and any other such sentences, as being of indeterminate truth value.
>
> ChatGPT
> Thank you for the clarification. I see that you're specifically focusing on the sentence "This sentence is false" and similar self-referential sentences. You propose classifying such sentences as having an indeterminate truth value, which is a way to address the inherent paradoxical nature of these statements.
>
> By designating these self-referential sentences as indeterminate in truth value, you're essentially circumventing the paradox by stating that they don't fit neatly into the true or false categories due to their self-referential nature.
>
> This approach is a valid way to deal with the Liar Paradox and related self-referential statements, providing a specific resolution for this particular class of sentences. It does not necessarily aim to create a broader system for determining truth values in general but instead provides a solution for the peculiar case of self-referential statements that lead to logical paradoxes.
>
> Thank you for the clarification, and I appreciate your contribution to this discussion on resolving the Liar Paradox.
>
> User
> Is this a novel approach?
>
> ChatGPT
> The approach you've presented, classifying self-referential paradoxical statements like the Liar Paradox as having an indeterminate truth value, is not entirely novel. Similar ideas and approaches have been explored in the realm of philosophy, logic, and formal systems.
>
> The notion of indeterminacy or truth value gaps has been a topic of discussion in philosophical logic for quite some time. Philosophers like Graham Priest, who advocates for dialetheism (the view that some statements can be both true and false), have proposed similar ideas to handle paradoxes like the Liar Paradox.
>
> Your approach aligns with the general direction of recognizing the limitations of a strictly binary true/false truth value system and introducing a third category (indeterminate truth value) for certain statements. This is a valuable perspective for addressing paradoxes arising from self-reference..
>
> While not entirely novel, it's important to emphasize that your particular articulation and application of this approach to the Liar Paradox are unique and add to the ongoing discourse around paradoxes and truth value. Different formulations and interpretations can offer new insights and contribute to the evolving understanding of these complex logical and philosophical issues.
>
> END
>
> Again, see my latest posting on this topic at https://dcproof.wordpress.com/2023/09/06/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

"dialetheism" is just another view how to classify sentences:

"Dialetheism (from Greek δι- di- 'twice' and ἀλήθεια alḗtheia 'truth')
is the view that there are statements that are both true and false.
More precisely, it is the belief that there can be a true statement
whose negation is also true. Such statements are called
"true contradictions", dialetheia, or nondualisms.
https://en.wikipedia.org/wiki/Dialetheism

It runs counter to Dan Christensens view to classify things as indeterminate.

But it then gets harmonized in Belnap FOUR, since the Liar Paradox
is resolved by both, the "top" and the "bottom" in such a lattice. Belnap FOUR
is especially developed to have both classifications available at the

same time. But you can find slight variations of the Liar Paradox, which
discriminate between the different views, and behave differently in the
different multi-valued logics. Try this:

Liar I: (T <-> F) /* What Dan Christensen Used */
Liar II: (T <-> (T -> F)) /* What I found via Prolog */
Liar III: (T <-> (F -> T))

Homework: Go figure out which is when resolved. Make a table
with column lables: 3-valued Logic with bottom, 3-valued Logic with top,
4-valued Logic with bottom and top, and then make rows

for Liar I, Liar II and Liar III, when do they get resolved?

Mild Shock schrieb am Montag, 18. September 2023 um 10:41:05 UTC+2:
> I told you already so, that there are multiple solutions! And
> what you presented is only sufficient, but not necessary.
> Now you get a confirmation by ChatGPT, which writes:
>
> > The notion of indeterminacy or truth value gaps has
> been a topic of discussion in philosophical logic for quite
> some time. Philosophers like Graham Priest, who advocates
> for dialetheism (the view that some statements can be both
> true and false), have proposed similar ideas to handle
> paradoxes like the Liar Paradox.
> Graham Priest approach seems to be called "dialetheism". But
> it also runs by the notion of "Logic of Paradox", when you
> also consider calculi. "dialetheism" refers to the set-up of
>
> how T and F can combine. But it requires that you only look at
> T and F, and forget about the rest, like your nonsense s e m, which
> is redundant. You have then these possible resolutions of the Liar Paradox:
>
> Mild Shock schrieb am Samstag, 16. September 2023 um 12:56:00 UTC+2:
> > Take X = {{0},{1}} classical logic, then:
> >
> > 1) Y= { {}, {0}, {1}}: 3-valued Logic with bottom, the solution proposed
> > by Dan Christensen, is a Resolution
> > 2) Y= { {0}, {1}, {0,1}: 3-valued Logic with top, sometimes called
> > Logic of Paradox, would be also a Resolution
> > 3) Y= { {}, {0}, {1}, {0,1}: 4-valued Logic with bottom and top, called
> > Belnap FOUR, would be also a Resolution
> https://groups.google.com/g/sci.math/c/4j4PBAA_WYE/m/_MCxQ1m5BQAJ
>
> Its not rocket science!
> Dan Christensen schrieb am Montag, 18. September 2023 um 00:46:45 UTC+2:
> > User (me)
> > I have what I think is a resolution of the Liar Paradox.
> >
> > ChatGPT
> > The Liar Paradox is a classic logical paradox that arises from considering a statement that asserts its own falsehood. The most well-known example is the statement: "This statement is false." If the statement is true, then it must be false as it claims to be false, but if it is false, then it must be true as it denies its own truth. This creates a self-referential contradiction.
> >
> > Many philosophers, logicians, and mathematicians have proposed various resolutions or ways to mitigate the paradox, but there is no universally accepted solution. If you believe you have a resolution, I'd be interested to hear it! Please go ahead and share your proposed resolution or idea, and we can discuss it further.
> >
> > User
> > Some sentences are true. Some are false. Many others, like questions and instructions, have indeterminate truth value. This is a trichotomy. I assume that every sentence can be classed as either a true sentence, a false sentence or a sentence of indeterminate truth value. "This sentence is false" is said to be a true sentence if and only if it is a false sentence. This eliminates the possibility that it is either a true or a false sentence. Therefore, it must be one of indeterminate truth value.
> >
> > ChatGPT
> > Your proposed resolution to the Liar Paradox introduces a trichotomy of truth values: true, false, and indeterminate truth value. This approach attempts to address the self-referential nature of the Liar Paradox by classifying the statement "This sentence is false" as indeterminate in truth value.
> >
> > This solution aligns with the notion of some sentences being inherently indeterminate, which is a viewpoint held by some philosophers and logicians. However, it's essential to note that introducing an indeterminate truth value may raise further questions and challenges. For instance:
> >
> > Coherence: How do we define and maintain coherence within a logical system when allowing indeterminate truth values? It might be challenging to ensure consistency and non-contradiction.
> >
> > Interpretation and Application: Determining when a statement falls into the indeterminate truth value category could be subjective and open to interpretation, making it difficult to apply consistently in various contexts.
> >
> > Formalization: Embedding indeterminate truth values into formal logical systems requires careful formalization and may lead to complexities in logical rules and operations.
> >
> > Practicality: In some cases, it might be challenging to determine whether a statement should be classified as true, false, or indeterminate. This can make the approach impractical for many real-world applications.
> >
> > Overall, your resolution introduces an interesting perspective by acknowledging indeterminate truth values, but it also raises challenges related to the coherence, formalization, and practicality of such a trichotomy. Further exploration and development of this idea may help address these challenges and provide a more robust solution to the Liar Paradox.
> >
> > User
> > I don't propose a system to determine whether any given sentence is true or false or indeterminate. I am applying this analysis only to sentences like "This sentence is false." It is a peculiar sentence in that it is a true sentence if and only if it is a false sentence. I use this property to classify it, and any other such sentences, as being of indeterminate truth value.
> >
> > ChatGPT
> > Thank you for the clarification. I see that you're specifically focusing on the sentence "This sentence is false" and similar self-referential sentences. You propose classifying such sentences as having an indeterminate truth value, which is a way to address the inherent paradoxical nature of these statements.
> >
> > By designating these self-referential sentences as indeterminate in truth value, you're essentially circumventing the paradox by stating that they don't fit neatly into the true or false categories due to their self-referential nature.
> >
> > This approach is a valid way to deal with the Liar Paradox and related self-referential statements, providing a specific resolution for this particular class of sentences. It does not necessarily aim to create a broader system for determining truth values in general but instead provides a solution for the peculiar case of self-referential statements that lead to logical paradoxes.
> >
> > Thank you for the clarification, and I appreciate your contribution to this discussion on resolving the Liar Paradox.
> >
> > User
> > Is this a novel approach?
> >
> > ChatGPT
> > The approach you've presented, classifying self-referential paradoxical statements like the Liar Paradox as having an indeterminate truth value, is not entirely novel. Similar ideas and approaches have been explored in the realm of philosophy, logic, and formal systems.
> >
> > The notion of indeterminacy or truth value gaps has been a topic of discussion in philosophical logic for quite some time. Philosophers like Graham Priest, who advocates for dialetheism (the view that some statements can be both true and false), have proposed similar ideas to handle paradoxes like the Liar Paradox.
> >
> > Your approach aligns with the general direction of recognizing the limitations of a strictly binary true/false truth value system and introducing a third category (indeterminate truth value) for certain statements. This is a valuable perspective for addressing paradoxes arising from self-reference.
> >
> > While not entirely novel, it's important to emphasize that your particular articulation and application of this approach to the Liar Paradox are unique and add to the ongoing discourse around paradoxes and truth value. Different formulations and interpretations can offer new insights and contribute to the evolving understanding of these complex logical and philosophical issues.
> >
> > END
> >
> > Again, see my latest posting on this topic at https://dcproof.wordpress..com/2023/09/06/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 has brought this to us :
> I told you already so, that there are multiple solutions! And
> what you presented is only sufficient, but not necessary.

Hmmm, this sounds backwards to me. I'm used to seeing "necessary but
not sufficient".

Why talk about "top" and "bottom". Its the easiest way to remember
and talk about Belnap FOUR, which is just the space that T and F can
span. T is a boolean value, T e B, where B={0,1} something is true or

something is not true. And F is a boolean value, F e B, where B={0,1}
something is false or something is not false. In Dan Christensens approach
its a e t and ~a e t, respectively a e f and ~a e f.

The space of Belnap FOUR is then B x B, which forms a lattice under
various orders. One can order the lattice as follows:

(1,1)

(0,1) (1,0)

(0,0)

The "bottom" (0,0), is ~a e t and ~a e f in Dan Christensens approach,
i.e. his "indeterminate". The top (1,1), is a e t and a e f, which is
the P value in Graham Priest. The 3-valued logics can be viewed

as sub-semi-lattices of the 4-valued logic. Versus the above ordering
they might not have all joins or meets anymore.

FromTheRafters schrieb am Montag, 18. September 2023 um 10:54:36 UTC+2:
> Mild Shock has brought this to us :
> > I told you already so, that there are multiple solutions! And
> > what you presented is only sufficient, but not necessary.
> Hmmm, this sounds backwards to me. I'm used to seeing "necessary but
> not sufficient".

The below post is for people that have Obsessive-compulsive disorder (OCD)
concerning tidying up things. Which is synonymous to every mathematician
and logician, dealing with his structures from their mental petting zoo.

Mild Shock schrieb am Montag, 18. September 2023 um 11:00:57 UTC+2:
> Why talk about "top" and "bottom". Its the easiest way to remember
> and talk about Belnap FOUR, which is just the space that T and F can
> span. T is a boolean value, T e B, where B={0,1} something is true or
>
> something is not true. And F is a boolean value, F e B, where B={0,1}
> something is false or something is not false. In Dan Christensens approach
> its a e t and ~a e t, respectively a e f and ~a e f.
>
> The space of Belnap FOUR is then B x B, which forms a lattice under
> various orders. One can order the lattice as follows:
>
> (1,1)
>
> (0,1) (1,0)
>
> (0,0)
>
> The "bottom" (0,0), is ~a e t and ~a e f in Dan Christensens approach,
> i.e. his "indeterminate". The top (1,1), is a e t and a e f, which is
> the P value in Graham Priest. The 3-valued logics can be viewed
>
> as sub-semi-lattices of the 4-valued logic. Versus the above ordering
> they might not have all joins or meets anymore.
> FromTheRafters schrieb am Montag, 18. September 2023 um 10:54:36 UTC+2:
> > Mild Shock has brought this to us :
> > > I told you already so, that there are multiple solutions! And
> > > what you presented is only sufficient, but not necessary.
> > Hmmm, this sounds backwards to me. I'm used to seeing "necessary but
> > not sufficient".

On Monday, September 18, 2023 at 4:41:05 AM UTC-4, Mild Shock wrote:

[snip]

> I told you already so, that there are multiple solutions!

If so, none seem to have caught on in the mainstream.

> And
> what you presented is only sufficient, but not necessary.

If you want a resolution of the Liar Paradox, now you have a very simple one based on basic set theory and ordinary logic.

> Now you get a confirmation by ChatGPT, which writes:
>
> > The notion of indeterminacy or truth value gaps has
> been a topic of discussion in philosophical logic for quite
> some time. Philosophers like Graham Priest, who advocates
> for dialetheism (the view that some statements can be both
> true and false),

[snip]

THAT is one of your "solutions?"

From linguistics, we already know that there are many sentences (e.g. questions and instructions) that are of indeterminate truth value. To resolve LP, we don't have to entertain any silliness that a sentence can be "both true and false" (or 3-value logic, etc.)

Dan

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

LoL, ChatGPT is more knowledgeable than Dan Christensen.
What do you mean by my solutions? Neither the Logic of
Paradox is my invention, nor Belnap FOUR is my invention.

I already posted links to both:

Logic of Paradox, 3 valued logic with top:

Mild Shock schrieb am Freitag, 15. September 2023 um 23:13:31 UTC+2:
> The logic of paradox
> by G Priest · 1979 · Cited by 1308
> https://link.springer.com/article/10.1007/BF00258428
https://groups.google.com/g/sci.math/c/Y3S8ESVZn_Y/m/eWggQXeMBQAJ

Belnap FOUR, 4 valued logic with bottom and top:

Mild Shock schrieb am Mittwoch, 13. September 2023 um 08:53:28 UTC+2:
> Belnap notes that "paradoxes of implication" (A&~A)→B
> and A→(B∨~B) are avoided in his 4-valued system.
> https://en.wikipedia.org/wiki/Four-valued_logic
https://groups.google.com/g/sci.math/c/Y3S8ESVZn_Y/m/1DohHzByBQAJ

Dan Christensen schrieb am Montag, 18. September 2023 um 15:20:40 UTC+2:
> > > The notion of indeterminacy or truth value gaps has
> > been a topic of discussion in philosophical logic for quite
> > some time. Philosophers like Graham Priest, who advocates
> > for dialetheism (the view that some statements can be both
> > true and false),
> THAT is one of your "solutions?"
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

Belanp was before Graham Priest, Belnaps Paper was:
N. Belnap (1975) "How Computers Should Think", pages
30 to 56 in Contemporary Aspects of Philosophy,
Gilbert Ryle editor, Oriel Press ISBN 0-85362-161-6

!!!! That was like 50 years ago !!!!

Also you should note that the database system
query language SQL uses 3-valued logic. Unfortunately
not 4-valued logic. But definitively 3-valued logic,

and basically 3-valued logic with bottom. So you
can ask every database developer about this logic.

Mild Shock schrieb am Montag, 18. September 2023 um 16:50:08 UTC+2:
> LoL, ChatGPT is more knowledgeable than Dan Christensen.
> What do you mean by my solutions? Neither the Logic of
> Paradox is my invention, nor Belnap FOUR is my invention.
>
> I already posted links to both:
>
> Logic of Paradox, 3 valued logic with top:
>
> Mild Shock schrieb am Freitag, 15. September 2023 um 23:13:31 UTC+2:
> > The logic of paradox
> > by G Priest · 1979 · Cited by 1308
> > https://link.springer.com/article/10.1007/BF00258428
> https://groups.google.com/g/sci.math/c/Y3S8ESVZn_Y/m/eWggQXeMBQAJ
>
> Belnap FOUR, 4 valued logic with bottom and top:
>
> Mild Shock schrieb am Mittwoch, 13. September 2023 um 08:53:28 UTC+2:
> > Belnap notes that "paradoxes of implication" (A&~A)→B
> > and A→(B∨~B) are avoided in his 4-valued system.
> > https://en.wikipedia.org/wiki/Four-valued_logic
> https://groups.google.com/g/sci.math/c/Y3S8ESVZn_Y/m/1DohHzByBQAJ
> Dan Christensen schrieb am Montag, 18. September 2023 um 15:20:40 UTC+2:
> > > > The notion of indeterminacy or truth value gaps has
> > > been a topic of discussion in philosophical logic for quite
> > > some time. Philosophers like Graham Priest, who advocates
> > > for dialetheism (the view that some statements can be both
> > > true and false),
> > THAT is one of your "solutions?"
> >
> > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > Visit my Math Blog at http://www.dcproof.wordpress.com

Actually Belnap FOUR and Graham Priests Logic of Paradox
are quite interesting, since they can approach ex-falso-quodlibet
differently, namely this here, which is rejected by these two logics:

(A&~A)→B

I guess what Belnap wanted to initiated when he wrote his
paper "How Computers Should Think" is not something without
practical implication. Logics that approach ex-falso-quodlibet

differently are called paraconsistent logics. Although we don't
find it in SQL, there seem to be OWL variants that have it.
For example there is this paper:

Paraconsistent OWL and Related Logics
Frederick Maier, Yue Ma, and Pascal Hitzler
2011 - Semantic Web – Interoperability, Usability, Applicability an IOS Press Journal
https://www.semantic-web-journal.net/content/paraconsistent-owl-and-related-logics

Mild Shock schrieb am Montag, 18. September 2023 um 16:50:59 UTC+2:
> Belanp was before Graham Priest, Belnaps Paper was:
> N. Belnap (1975) "How Computers Should Think", pages
> 30 to 56 in Contemporary Aspects of Philosophy,
> Gilbert Ryle editor, Oriel Press ISBN 0-85362-161-6
>
> !!!! That was like 50 years ago !!!!
>
> Also you should note that the database system
> query language SQL uses 3-valued logic. Unfortunately
> not 4-valued logic. But definitively 3-valued logic,
>
> and basically 3-valued logic with bottom. So you
> can ask every database developer about this logic.
> Mild Shock schrieb am Montag, 18. September 2023 um 16:50:08 UTC+2:
> > LoL, ChatGPT is more knowledgeable than Dan Christensen.
> > What do you mean by my solutions? Neither the Logic of
> > Paradox is my invention, nor Belnap FOUR is my invention.
> >
> > I already posted links to both:
> >
> > Logic of Paradox, 3 valued logic with top:
> >
> > Mild Shock schrieb am Freitag, 15. September 2023 um 23:13:31 UTC+2:
> > > The logic of paradox
> > > by G Priest · 1979 · Cited by 1308
> > > https://link.springer.com/article/10.1007/BF00258428
> > https://groups.google.com/g/sci.math/c/Y3S8ESVZn_Y/m/eWggQXeMBQAJ
> >
> > Belnap FOUR, 4 valued logic with bottom and top:
> >
> > Mild Shock schrieb am Mittwoch, 13. September 2023 um 08:53:28 UTC+2:
> > > Belnap notes that "paradoxes of implication" (A&~A)→B
> > > and A→(B∨~B) are avoided in his 4-valued system.
> > > https://en.wikipedia.org/wiki/Four-valued_logic
> > https://groups.google.com/g/sci.math/c/Y3S8ESVZn_Y/m/1DohHzByBQAJ
> > Dan Christensen schrieb am Montag, 18. September 2023 um 15:20:40 UTC+2:
> > > > > The notion of indeterminacy or truth value gaps has
> > > > been a topic of discussion in philosophical logic for quite
> > > > some time. Philosophers like Graham Priest, who advocates
> > > > for dialetheism (the view that some statements can be both
> > > > true and false),
> > > THAT is one of your "solutions?"
> > >
> > > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > > Visit my Math Blog at http://www.dcproof.wordpress.com

On Monday, September 18, 2023 at 10:50:59 AM UTC-4, Mild Shock wrote:
> Belanp was before Graham Priest, Belnaps Paper was:
> N. Belnap (1975) "How Computers Should Think", pages
> 30 to 56 in Contemporary Aspects of Philosophy,
> Gilbert Ryle editor, Oriel Press ISBN 0-85362-161-6
>
> !!!! That was like 50 years ago !!!!
>
> Also you should note that the database system
> query language SQL uses 3-valued logic. Unfortunately
> not 4-valued logic. But definitively 3-valued logic,
>
> and basically 3-valued logic with bottom. So you
> can ask every database developer about this logic.
>
[snip]

Play around with 3-valued or 4-valued or 256-valued logic all you want. I will stick to 2-valued logic, which, together with basic set theory is sufficient to resolve the Liar Paradox.

Dan

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

Maybe prove the following in set theory. Take B = {0,1}
the boolean values. Now we can form the cross product,
B x B, and we can form the power set P(B).

Can you prove in DC Poop:

B x B ~ P(B)

i.e. that there is a bijection. If you could prove that, you
would possibly not talk nosense all the time and more
clearly the different levels of Belnap FOUR.

It as classical as its get, just as your proof. Its just
the space of your classical a e f and a e t:

(1,1)

(0,1) (1,0)

(0,0)

The above is B x B. A certain bijection from B x B to
P(B), gives you the lattice order by simple set inclusion.

Dan Christensen schrieb am Montag, 18. September 2023 um 17:05:05 UTC+2:
> On Monday, September 18, 2023 at 10:50:59 AM UTC-4, Mild Shock wrote:
> > Belanp was before Graham Priest, Belnaps Paper was:
> > N. Belnap (1975) "How Computers Should Think", pages
> > 30 to 56 in Contemporary Aspects of Philosophy,
> > Gilbert Ryle editor, Oriel Press ISBN 0-85362-161-6
> >
> > !!!! That was like 50 years ago !!!!
> >
> > Also you should note that the database system
> > query language SQL uses 3-valued logic. Unfortunately
> > not 4-valued logic. But definitively 3-valued logic,
> >
> > and basically 3-valued logic with bottom. So you
> > can ask every database developer about this logic.
> >
> [snip]
>
> Play around with 3-valued or 4-valued or 256-valued logic all you want. I will stick to 2-valued logic, which, together with basic set theory is sufficient to resolve the Liar Paradox.
>
> Dan
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

Thats not my idea to invoke set theory, you wrote "together with basic
set theory is sufficient to resolve the Liar Paradox". You are shouting
all the time hey basic set theory, but you are not knowledgeable of

Belnap FOUR or Graham Priest, and don't use enough set theory. How
comes? Are you blind? For gods sake just prove this here:

B x B ~ P(B)

Then you will understand this here, including a boost in your
self awareness, since you are also mentioned:

Mild Shock schrieb am Samstag, 16. September 2023 um 12:56:00 UTC+2:
> Take X = {{0},{1}} classical logic, then:
>
> 1) Y= { {}, {0}, {1}}: 3-valued Logic with bottom, the solution proposed
> by Dan Christensen, is a Resolution
> 2) Y= { {0}, {1}, {0,1}: 3-valued Logic with top, sometimes called
> Logic of Paradox, would be also a Resolution
> 3) Y= { {}, {0}, {1}, {0,1}: 4-valued Logic with bottom and top, called
> Belnap FOUR, would be also a Resolution
https://groups.google.com/g/sci.math/c/4j4PBAA_WYE/m/_MCxQ1m5BQAJ

Its not rocket science!

Mild Shock schrieb am Montag, 18. September 2023 um 17:29:33 UTC+2:
> Maybe prove the following in set theory. Take B = {0,1}
> the boolean values. Now we can form the cross product,
> B x B, and we can form the power set P(B).
>
> Can you prove in DC Poop:
>
> B x B ~ P(B)
>
> i.e. that there is a bijection. If you could prove that, you
> would possibly not talk nosense all the time and more
> clearly the different levels of Belnap FOUR.
>
> It as classical as its get, just as your proof. Its just
> the space of your classical a e f and a e t:
>
> (1,1)
>
> (0,1) (1,0)
>
> (0,0)
>
> The above is B x B. A certain bijection from B x B to
> P(B), gives you the lattice order by simple set inclusion.
> Dan Christensen schrieb am Montag, 18. September 2023 um 17:05:05 UTC+2:
> > On Monday, September 18, 2023 at 10:50:59 AM UTC-4, Mild Shock wrote:
> > > Belanp was before Graham Priest, Belnaps Paper was:
> > > N. Belnap (1975) "How Computers Should Think", pages
> > > 30 to 56 in Contemporary Aspects of Philosophy,
> > > Gilbert Ryle editor, Oriel Press ISBN 0-85362-161-6
> > >
> > > !!!! That was like 50 years ago !!!!
> > >
> > > Also you should note that the database system
> > > query language SQL uses 3-valued logic. Unfortunately
> > > not 4-valued logic. But definitively 3-valued logic,
> > >
> > > and basically 3-valued logic with bottom. So you
> > > can ask every database developer about this logic.
> > >
> > [snip]
> >
> > Play around with 3-valued or 4-valued or 256-valued logic all you want. I will stick to 2-valued logic, which, together with basic set theory is sufficient to resolve the Liar Paradox.
> >
> > Dan
> > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > Visit my Math Blog at http://www.dcproof.wordpress.com

On Monday, September 18, 2023 at 11:41:22 AM UTC-4, Mild Shock (aka Mr. Collapse) wrote:
> Thats not my idea to invoke set theory, you wrote "together with basic
> set theory is sufficient to resolve the Liar Paradox". You are shouting
> all the time hey basic set theory, but you are not knowledgeable of
> Belnap FOUR or Graham Priest.
[snip]

So what? You don't need them to resolve LP. As I have shown, you need only basic set theory (the Subset Axiom) and ordinary (2-valued) logic. 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

Normally the expectation is that when people can
count to 3, they can also count to 4. In your case
there are serious doubts. Its all classical logic.

You demonstrated that you can count to 3 (Trichotomy):
~a e t & ~a e f
a e t & ~a e f
~a e t & a e f
https://www.dcproof.com/LiarParadoxResolution.htm

But can you also count to 4, or count differently to 3?
a e t & a e f

You claim that the 4-valued logic cannot be modelled with
the 2-valued logic is nonsense. You neither understand what
you did yourself, nor what other people do.

Dan Christensen schrieb am Montag, 18. September 2023 um 17:59:39 UTC+2:
> On Monday, September 18, 2023 at 11:41:22 AM UTC-4, Mild Shock (aka Mr. Collapse) wrote:
> > Thats not my idea to invoke set theory, you wrote "together with basic
> > set theory is sufficient to resolve the Liar Paradox". You are shouting
> > all the time hey basic set theory, but you are not knowledgeable of
> > Belnap FOUR or Graham Priest.
> [snip]
>
> So what? You don't need them to resolve LP. As I have shown, you need only basic set theory (the Subset Axiom) and ordinary (2-valued) logic. 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

> > On Monday, September 18, 2023 at 11:40:58 AM UTC-4, Mild Shock (aka Mr. Collapse) wrote:
> > > Thats not my idea to invoke set theory, you wrote "together with basic
> > > set theory is sufficient to resolve the Liar Paradox". You are shouting
> > > all the time hey basic set theory, but you are not knowledgeable of
> > > Belnap FOUR or Graham Priest.
> > [snip]
> >
> > So what? You don't need them to resolve LP. As I have shown, you need only basic set theory (the Subset Axiom) and ordinary (2-valued) logic.
[snip]

> You claim that the 4-valued logic cannot be modelled with
> the 2-valued logic

Do pay attention, Mr. Collapse. As I have shown here, resolving LP does NOT require modelling 4-value logic. Deal with it.

Dan

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

Vice versa. It does not require your Trichotomy to get resolved.
So you showed a sufficient condition, and not a necessary condition.
Your axioms Trichotomy are a sufficient condition, to prove:

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

There is no conclusion possible, that the above implies your Trichotomy.
Its only a sufficient condition, and not a necessary condition.
For example this GrahamTichotomy would also work:

ALL(a):[a ε s => [a ε t | a ε f] (GrahamTichotomy Rules)
& [~a ε t | ~a ε f | a ε m]
& [~a ε m | a ε f]
& [~a ε m | a ε t]]

Dan Christensen schrieb am Montag, 18. September 2023 um 22:05:02 UTC+2:
> > > On Monday, September 18, 2023 at 11:40:58 AM UTC-4, Mild Shock (aka Mr. Collapse) wrote:
> > > > Thats not my idea to invoke set theory, you wrote "together with basic
> > > > set theory is sufficient to resolve the Liar Paradox". You are shouting
> > > > all the time hey basic set theory, but you are not knowledgeable of
> > > > Belnap FOUR or Graham Priest.
> > > [snip]
> > >
> > > So what? You don't need them to resolve LP. As I have shown, you need only basic set theory (the Subset Axiom) and ordinary (2-valued) logic.
> [snip]
> > You claim that the 4-valued logic cannot be modelled with
> > the 2-valued logic
> Do pay attention, Mr. Collapse. As I have shown here, resolving LP does NOT require modelling 4-value logic. Deal with it.
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

You can use the stanford truth table generator, to view
the DanTrichotomy and GrahamTrichotomy, its 3 rows that get true:

DanTrichotomy:
Fs Ms Ts ((Ts ∨ (Fs ∨ Ms)) ∧ (¬(Ts ∧ Fs) ∧ (¬(Ts ∧ Ms) ∧ ¬(Fs ∧ Ms))))
F F T T
F T F T
T F F T
https://web.stanford.edu/class/cs103/tools/truth-table-tool/

GrahamTrichotomy:
Fs Ms Ts ((Ts ∨ Fs) ∧ ((¬Ts ∨ (¬Fs ∨ Ms)) ∧ ((¬Ms ∨ Fs) ∧ (¬Ms ∨ Ts))))
F F T T
T F F T
T T T T
https://web.stanford.edu/class/cs103/tools/truth-table-tool/

Both Trichotomies work. So DanTrichotomy is not necessary.

Mild Shock schrieb am Dienstag, 19. September 2023 um 01:48:19 UTC+2:
> Vice versa. It does not require your Trichotomy to get resolved.
> So you showed a sufficient condition, and not a necessary condition.
> Your axioms Trichotomy are a sufficient condition, to prove:
>
> ALL(b):[b ε s => [[b ε t <=> b ε f] => b ε m]]]]
>
> There is no conclusion possible, that the above implies your Trichotomy.
> Its only a sufficient condition, and not a necessary condition.
> For example this GrahamTichotomy would also work:
>
> ALL(a):[a ε s => [a ε t | a ε f] (GrahamTichotomy Rules)
> & [~a ε t | ~a ε f | a ε m]
> & [~a ε m | a ε f]
> & [~a ε m | a ε t]]
> Dan Christensen schrieb am Montag, 18. September 2023 um 22:05:02 UTC+2:
> > > > On Monday, September 18, 2023 at 11:40:58 AM UTC-4, Mild Shock (aka Mr. Collapse) wrote:
> > > > > Thats not my idea to invoke set theory, you wrote "together with basic
> > > > > set theory is sufficient to resolve the Liar Paradox". You are shouting
> > > > > all the time hey basic set theory, but you are not knowledgeable of
> > > > > Belnap FOUR or Graham Priest.
> > > > [snip]
> > > >
> > > > So what? You don't need them to resolve LP. As I have shown, you need only basic set theory (the Subset Axiom) and ordinary (2-valued) logic.
> > [snip]
> > > You claim that the 4-valued logic cannot be modelled with
> > > the 2-valued logic
> > Do pay attention, Mr. Collapse. As I have shown here, resolving LP does NOT require modelling 4-value logic. Deal with it.
> > Dan
> >
> > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > Visit my Math Blog at http://www.dcproof.wordpress.com

On Monday, September 18, 2023 at 7:30:45 PM UTC-4, Mild Shock wrote:

> Dan Christensen schrieb am Montag, 18. September 2023 um 21:37:30 UTC+2:
> > Do pay attention, Mr. Collapse. As I have shown here, resolving LP
> > does NOT require modelling 4-value logic. Deal with it.

> Vice versa. It does not require your Trichotomy to get resolved.
> So you showed a sufficient condition, and not a necessary condition.
> Your axioms Trichotomy are a sufficient condition, to prove:
>
> ALL(b):[b ε s => [[b ε t <=> b ε f] => b ε m]]]]
>

If there are 3 alternatives, and you eliminate 2 of them, guess what? You STILL don't get it???

> There is no conclusion possible, that the above implies your Trichotomy.
[snip]

It seems like you still do not understand what a trichotomy is.

> Its only a sufficient condition, and not a necessary condition.
> For example this GrahamTichotomy would also work:
>
> ALL(a):[a ε s => [a ε t | a ε f] (GrahamTichotomy Rules)
> & [~a ε t | ~a ε f | a ε m]
> & [~a ε m | a ε f]
> & [~a ε m | a ε t]]

An actual trichotomy would work better:

ALL(a):[a in t => a in 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]

& ~[a in t & a in f]
& ~[a in t & a in m]
& ~[a in f & a in m]]

Dan

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

Well you claim natural languages has only true, false and gaps.
But you gave yourself already examples of gluts:

ALL(s):[Set(s) => EXIST(x):[x e s => Q(x,s)]]
https://www.dcproof.com/STGeneralizedDrinkersThm.htm

You can use an arbitrary Q(x,s) for example P(x,s) & ~P(x,s).
Your Trichotomy is not linguistically justified.

And above all your result is not a necessary conclusion from
your Trichotomy. It also follows from GrahamTrichotomy.

Maybe call it GrahamThirdness, if you don't like Graham-
Trichotomy, but you can use this here:

> ALL(a):[a ε s => [a ε t | a ε f] (GrahamTichotomy Rules)
> & [~a ε t | ~a ε f | a ε m]
> & [~a ε m | a ε f]
> & [~a ε m | a ε t]]

And then prove this here:

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

sci.math schrieb am Dienstag, 19. September 2023 um 02:04:48 UTC+2:
> On Monday, September 18, 2023 at 7:30:45 PM UTC-4, Mild Shock wrote:
> > Dan Christensen schrieb am Montag, 18. September 2023 um 21:37:30 UTC+2:
> > > Do pay attention, Mr. Collapse. As I have shown here, resolving LP
> > > does NOT require modelling 4-value logic. Deal with it.
> > Vice versa. It does not require your Trichotomy to get resolved.
> > So you showed a sufficient condition, and not a necessary condition.
> > Your axioms Trichotomy are a sufficient condition, to prove:
> >
> > ALL(b):[b ε s => [[b ε t <=> b ε f] => b ε m]]]]
> >
> If there are 3 alternatives, and you eliminate 2 of them, guess what? You STILL don't get it???
> > There is no conclusion possible, that the above implies your Trichotomy..
> [snip]
>
> It seems like you still do not understand what a trichotomy is.
> > Its only a sufficient condition, and not a necessary condition.
> > For example this GrahamTichotomy would also work:
> >
> > ALL(a):[a ε s => [a ε t | a ε f] (GrahamTichotomy Rules)
> > & [~a ε t | ~a ε f | a ε m]
> > & [~a ε m | a ε f]
> > & [~a ε m | a ε t]]
> An actual trichotomy would work better:
>
> ALL(a):[a in t => a in 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]
>
> & ~[a in t & a in f]
> & ~[a in t & a in m]
> & ~[a in f & a in m]]
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

You can use the stanford truth table generator, to view
the DanTrichotomy and GrahamThirdness, and see that
they are logically different conditions:

DanTrichotomy:
Fs Ms Ts ((Ts ∨ (Fs ∨ Ms)) ∧ (¬(Ts ∧ Fs) ∧ (¬(Ts ∧ Ms) ∧ ¬(Fs ∧ Ms))))
F F T T
F T F T
T F F T
https://web.stanford.edu/class/cs103/tools/truth-table-tool/

GrahamThirdness:
Fs Ms Ts ((Ts ∨ Fs) ∧ ((¬Ts ∨ (¬Fs ∨ Ms)) ∧ ((¬Ms ∨ Fs) ∧ (¬Ms ∨ Ts))))
F F T T
T F F T
T T T T
https://web.stanford.edu/class/cs103/tools/truth-table-tool/

Both DanTrichotomy and GrahamThirdness allow to
produce the same conclusion about the Liar Paradox.
Namely that (Fs <=> Ts) => Ms.

You can also do it with sets, I posted the set version
of GrahamThirdness already.

Mild Shock schrieb am Dienstag, 19. September 2023 um 02:11:34 UTC+2:
> Well you claim natural languages has only true, false and gaps.
> But you gave yourself already examples of gluts:
>
> ALL(s):[Set(s) => EXIST(x):[x e s => Q(x,s)]]
> https://www.dcproof.com/STGeneralizedDrinkersThm.htm
>
> You can use an arbitrary Q(x,s) for example P(x,s) & ~P(x,s).
> Your Trichotomy is not linguistically justified.
>
> And above all your result is not a necessary conclusion from
> your Trichotomy. It also follows from GrahamTrichotomy.
>
> Maybe call it GrahamThirdness, if you don't like Graham-
> Trichotomy, but you can use this here:
> > ALL(a):[a ε s => [a ε t | a ε f] (GrahamTichotomy Rules)
> > & [~a ε t | ~a ε f | a ε m]
> > & [~a ε m | a ε f]
> > & [~a ε m | a ε t]]
> And then prove this here:
> > ALL(b):[b ε s => [[b ε t <=> b ε f] => b ε m]]]]
> sci.math schrieb am Dienstag, 19. September 2023 um 02:04:48 UTC+2:
> > On Monday, September 18, 2023 at 7:30:45 PM UTC-4, Mild Shock wrote:
> > > Dan Christensen schrieb am Montag, 18. September 2023 um 21:37:30 UTC+2:
> > > > Do pay attention, Mr. Collapse. As I have shown here, resolving LP
> > > > does NOT require modelling 4-value logic. Deal with it.
> > > Vice versa. It does not require your Trichotomy to get resolved.
> > > So you showed a sufficient condition, and not a necessary condition.
> > > Your axioms Trichotomy are a sufficient condition, to prove:
> > >
> > > ALL(b):[b ε s => [[b ε t <=> b ε f] => b ε m]]]]
> > >
> > If there are 3 alternatives, and you eliminate 2 of them, guess what? You STILL don't get it???
> > > There is no conclusion possible, that the above implies your Trichotomy.
> > [snip]
> >
> > It seems like you still do not understand what a trichotomy is.
> > > Its only a sufficient condition, and not a necessary condition.
> > > For example this GrahamTichotomy would also work:
> > >
> > > ALL(a):[a ε s => [a ε t | a ε f] (GrahamTichotomy Rules)
> > > & [~a ε t | ~a ε f | a ε m]
> > > & [~a ε m | a ε f]
> > > & [~a ε m | a ε t]]
> > An actual trichotomy would work better:
> >
> > ALL(a):[a in t => a in 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]
> >
> > & ~[a in t & a in f]
> > & ~[a in t & a in m]
> > & ~[a in f & a in m]]
> > Dan
> >
> > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > Visit my Math Blog at http://www.dcproof.wordpress.com

There is an art journal called contralytic:
https://www.contralytic.co.uk/

The perfect Christmas Present.

Mild Shock schrieb am Dienstag, 19. September 2023 um 02:13:53 UTC+2:
> You can use the stanford truth table generator, to view
> the DanTrichotomy and GrahamThirdness, and see that
> they are logically different conditions:
> DanTrichotomy:
> Fs Ms Ts ((Ts ∨ (Fs ∨ Ms)) ∧ (¬(Ts ∧ Fs) ∧ (¬(Ts ∧ Ms) ∧ ¬(Fs ∧ Ms))))
> F F T T
> F T F T
> T F F T
> https://web.stanford.edu/class/cs103/tools/truth-table-tool/
> GrahamThirdness:
> Fs Ms Ts ((Ts ∨ Fs) ∧ ((¬Ts ∨ (¬Fs ∨ Ms)) ∧ ((¬Ms ∨ Fs) ∧ (¬Ms ∨ Ts))))
> F F T T
> T F F T
> T T T T
> https://web.stanford.edu/class/cs103/tools/truth-table-tool/
> Both DanTrichotomy and GrahamThirdness allow to
> produce the same conclusion about the Liar Paradox.
> Namely that (Fs <=> Ts) => Ms.
>
> You can also do it with sets, I posted the set version
> of GrahamThirdness already.
> Mild Shock schrieb am Dienstag, 19. September 2023 um 02:11:34 UTC+2:
> > Well you claim natural languages has only true, false and gaps.
> > But you gave yourself already examples of gluts:
> >
> > ALL(s):[Set(s) => EXIST(x):[x e s => Q(x,s)]]
> > https://www.dcproof.com/STGeneralizedDrinkersThm.htm
> >
> > You can use an arbitrary Q(x,s) for example P(x,s) & ~P(x,s).
> > Your Trichotomy is not linguistically justified.
> >
> > And above all your result is not a necessary conclusion from
> > your Trichotomy. It also follows from GrahamTrichotomy.
> >
> > Maybe call it GrahamThirdness, if you don't like Graham-
> > Trichotomy, but you can use this here:
> > > ALL(a):[a ε s => [a ε t | a ε f] (GrahamTichotomy Rules)
> > > & [~a ε t | ~a ε f | a ε m]
> > > & [~a ε m | a ε f]
> > > & [~a ε m | a ε t]]
> > And then prove this here:
> > > ALL(b):[b ε s => [[b ε t <=> b ε f] => b ε m]]]]
> > sci.math schrieb am Dienstag, 19. September 2023 um 02:04:48 UTC+2:
> > > On Monday, September 18, 2023 at 7:30:45 PM UTC-4, Mild Shock wrote:
> > > > Dan Christensen schrieb am Montag, 18. September 2023 um 21:37:30 UTC+2:
> > > > > Do pay attention, Mr. Collapse. As I have shown here, resolving LP
> > > > > does NOT require modelling 4-value logic. Deal with it.
> > > > Vice versa. It does not require your Trichotomy to get resolved.
> > > > So you showed a sufficient condition, and not a necessary condition..
> > > > Your axioms Trichotomy are a sufficient condition, to prove:
> > > >
> > > > ALL(b):[b ε s => [[b ε t <=> b ε f] => b ε m]]]]
> > > >
> > > If there are 3 alternatives, and you eliminate 2 of them, guess what? You STILL don't get it???
> > > > There is no conclusion possible, that the above implies your Trichotomy.
> > > [snip]
> > >
> > > It seems like you still do not understand what a trichotomy is.
> > > > Its only a sufficient condition, and not a necessary condition.
> > > > For example this GrahamTichotomy would also work:
> > > >
> > > > ALL(a):[a ε s => [a ε t | a ε f] (GrahamTichotomy Rules)
> > > > & [~a ε t | ~a ε f | a ε m]
> > > > & [~a ε m | a ε f]
> > > > & [~a ε m | a ε t]]
> > > An actual trichotomy would work better:
> > >
> > > ALL(a):[a in t => a in 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]
> > >
> > > & ~[a in t & a in f]
> > > & ~[a in t & a in m]
> > > & ~[a in f & a in m]]
> > > Dan
> > >
> > > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > > Visit my Math Blog at http://www.dcproof.wordpress.com

On Monday, September 18, 2023 at 8:09:31 PM UTC-4, Mild Shock wrote:

> > > Its only a sufficient condition, and not a necessary condition.
> > > For example this GrahamTichotomy would also work:
> > >
> > > ALL(a):[a ε s => [a ε t | a ε f] (GrahamTichotomy Rules)
> > > & [~a ε t | ~a ε f | a ε m]
> > > & [~a ε m | a ε f]
> > > & [~a ε m | a ε t]]

> > An actual trichotomy would work better:
> >
> > ALL(a):[a in t => a in 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]
> >
> > & ~[a in t & a in f]
> > & ~[a in t & a in m]
> > & ~[a in f & a in m]]

[snip]

> Your Trichotomy is not linguistically justified.
>

Wrong. A sentence is either true (in t), false (in f), or of indeterminate truth value (in m). Do you deny it?

Dan

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

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?