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The Liar Paradox: My recent dialog with ChatGPT on my proposed resolution
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Dan Christensen
Sep 17, 2023, 6:46:45 PM9/17/23
to
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
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
bassam karzeddin
Sep 17, 2023, 10:14:53 PM9/17/23
to
BKK
Mild Shock
Sep 18, 2023, 4:41:05 AM9/18/23
to
I told you already so, that there are multiple solutions! And
what you presented is only sufficient, but not necessary.
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!
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
> 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.
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!
Mild Shock
Sep 18, 2023, 4:51:43 AM9/18/23
to
"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?
"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?
FromTheRafters
Sep 18, 2023, 4:54:36 AM9/18/23
to
Mild Shock has brought this to us :
not sufficient".
> 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
> what you presented is only sufficient, but not necessary.
not sufficient".
Mild Shock
Sep 18, 2023, 5:00:57 AM9/18/23
to
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.
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.
Mild Shock
Sep 18, 2023, 5:06:02 AM9/18/23
to
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.
concerning tidying up things. Which is synonymous to every mathematician
and logician, dealing with his structures from their mental petting zoo.
Dan Christensen
Sep 18, 2023, 9:20:40 AM9/18/23
to
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.)
[snip]
> 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),
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.)
Mild Shock
Sep 18, 2023, 10:50:08 AM9/18/23
to
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),
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?"
>
>
Mild Shock
Sep 18, 2023, 10:50:59 AM9/18/23
to
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.
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
Sep 18, 2023, 10:58:00 AM9/18/23
to
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
Dan Christensen
Sep 18, 2023, 11:05:05 AM9/18/23
to
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]
> 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.
>
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
Mild Shock
Sep 18, 2023, 11:29:33 AM9/18/23
to
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.
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.
Mild Shock
Sep 18, 2023, 11:41:22 AM9/18/23
to
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!
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)
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!
Dan Christensen
Sep 18, 2023, 11:59:39 AM9/18/23
to
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.
> 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
[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.
Mild Shock
Sep 18, 2023, 2:31:36 PM9/18/23
to
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.
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.
Mild Shock
Sep 18, 2023, 2:37:43 PM9/18/23
to
Another word for indeterminancy is truth gap, since
in indeterminacy when a e m, we also have ~a e f and ~a e t,
according to your Trichotomy. So there is a truth gap,
neither ~a e f nor ~a e t. The other side is a truth glut,
when we have a e f and a e t. This is for example the case
for an inconsistency. Even in your DC Proof, can you prove thus glut:
/* The Ex-Falso Quodlibet Glut */
A & ~A => B
A & ~A => ~B
For your resolution you counted to 3:
gap
true
false
But you can also count to 3:
true
false
glut
Or you can count to 4:
gap
true
false
glut
Its not rocket science. They all resolve the Liar Paradox.
in indeterminacy when a e m, we also have ~a e f and ~a e t,
according to your Trichotomy. So there is a truth gap,
neither ~a e f nor ~a e t. The other side is a truth glut,
when we have a e f and a e t. This is for example the case
for an inconsistency. Even in your DC Proof, can you prove thus glut:
/* The Ex-Falso Quodlibet Glut */
A & ~A => B
A & ~A => ~B
For your resolution you counted to 3:
gap
true
false
But you can also count to 3:
true
false
glut
Or you can count to 4:
gap
true
false
glut
Its not rocket science. They all resolve the Liar Paradox.
Mild Shock
Sep 18, 2023, 2:40:47 PM9/18/23
to
You find the notion "truth value gaps" also in the ChatGPT Protokoll:
> The notion of indeterminacy or TRUTH VALUE GAPS has
Have Fun!
> 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
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.
You can also google or ask ChatGPT about "truth value gluts".
paradoxes like the Liar Paradox.
Have Fun!
Dan Christensen
Sep 18, 2023, 4:05:02 PM9/18/23
to
> > 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]
> > > 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.
> You claim that the 4-valued logic cannot be modelled with
> the 2-valued logic
Mild Shock
Sep 18, 2023, 7:48:19 PM9/18/23
to
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]]
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]]
Mild Shock
Sep 18, 2023, 7:49:12 PM9/18/23
to
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.
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.
sci.math
Sep 18, 2023, 8:04:48 PM9/18/23
to
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.
> 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 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???
> 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.
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]]
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]]
Mild Shock
Sep 18, 2023, 8:11:34 PM9/18/23
to
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]]]]
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]]
> ALL(b):[b ε s => [[b ε t <=> b ε f] => b ε m]]]]
Mild Shock
Sep 18, 2023, 8:13:53 PM9/18/23
to
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/
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
F F T T
T F F T
T T T T
https://web.stanford.edu/class/cs103/tools/truth-table-tool/
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
Sep 18, 2023, 8:30:21 PM9/18/23
to
There is an art journal called contralytic:
https://www.contralytic.co.uk/
The perfect Christmas Present.
sci.math
Sep 18, 2023, 9:23:16 PM9/18/23
to
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]
> > > 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]]
> Your Trichotomy is not linguistically justified.
>
Mild Shock
Sep 19, 2023, 2:30:33 AM9/19/23
to
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)
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:
> & [~a ε m | a ε f]
> & [~a ε m | a ε t]]
ALL(b):[b ε s => [[b ε t <=> b ε f] => b ε m]]]]
Mild Shock
Sep 19, 2023, 3:08:09 AM9/19/23
to
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
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
sci.math
Sep 19, 2023, 9:40:44 AM9/19/23
to
On Tuesday, September 19, 2023 at 2:26:44 AM UTC-4, Mild Shock wrote:
> Dan Christensen schrieb am Dienstag, 19. September 2023 um 03:22:17 UTC+2:
> > Wrong. A sentence is either true (in t), false (in f), or of
> > indeterminate truth value (in m). Do you deny it?
> 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]]]]
You can prove this using the more standard definition of a trichotomy:
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]]
In words: t, f and m are subsets of s. Every element of the set s is an element of one and only one of those subsets.
Proof: https://dcproof.com/LiarParadoxV2.htm
> Dan Christensen schrieb am Dienstag, 19. September 2023 um 03:22:17 UTC+2:
> > Wrong. A sentence is either true (in t), false (in f), or of
> > indeterminate truth value (in m). Do you deny it?
> This is not the case for the Liar Sentence. The Liar Sentence
> can also be in m and it would mean true and false.
>
Still wrong.
> 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]]]]
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]]
Proof: https://dcproof.com/LiarParadoxV2.htm
Mild Shock
Sep 19, 2023, 4:38:23 PM9/19/23
to
What happened to your user name?
Too much Trichotomy? LMAO!
sci.math schrieb:
Too much Trichotomy? LMAO!
sci.math schrieb:
Dan Christensen
Sep 19, 2023, 5:31:15 PM9/19/23
to
FromTheRafters
Sep 19, 2023, 5:51:35 PM9/19/23
to
Mild Shock formulated the question :
> What happened to your user name?
> Too much Trichotomy? LMAO!
>
> sci.math schrieb:
It got googleized.
> Too much Trichotomy? LMAO!
>
> sci.math schrieb:
FromTheRafters
Sep 19, 2023, 5:52:54 PM9/19/23
to
Dan Christensen used his keyboard to write :
Intermittently. It happened in another group I'm in too.
Mild Shock
Sep 19, 2023, 6:39:08 PM9/19/23
to
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.
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.
Dan Christensen
Sep 19, 2023, 10:50:21 PM9/19/23
to
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:
> > 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.
> What should "more" standard mean?
>
> The problem with your solution is that you reject LEM.
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
>
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.")
Volney
Sep 20, 2023, 12:09:41 AM9/20/23
to
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.
Chris M. Thomasson
Sep 20, 2023, 12:10:53 AM9/20/23
to
Volney
Sep 20, 2023, 1:27:37 AM9/20/23
to
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.
Mild Shock
Sep 20, 2023, 3:06:52 AM9/20/23
to
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:
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:
Mild Shock
Sep 20, 2023, 4:15:44 AM9/20/23
to
Dan Christensen, our youngling wrote:
> - declarative sentence (statement)
> - interrogative sentence (question)
> - imperative sentence (command)
> - exclamative sentence (exclamation)
> https://www.englishclub.com/grammar/sentence/type.php
Well I guess some time in history the question mark
was invented, so that younglings like Dan Christensen
don't have to use their spare brain cells, and analyze the
sentence itself, just look at the end of the sentence,
if it has a QUESTION mark, its a INTERROGATIVE sentence.
What is the history of the question mark? Lets ask ChatGPT:
History of the question mark:
==========================
- The Point of Interrogation: In the 8th century,
Alcuin of York, an English scholar, introduced a
symbol called the "punctus interrogativus" (point
of interrogation), which was a dot placed at the
end of a sentence to signal a question.
- The Development of the Modern Question Mark:
Over time, the punctus interrogativus evolved into
the modern question mark. Its form gradually changed,
and it became a distinctive symbol resembling
the one we use today.
- Standardization: The standardization of punctuation,
including the question mark, occurred with the advent
of the printing press in the 15th century. Printers like Aldus
Manutius and Johannes Gutenberg played a role in codifying
punctuation rules, which included the question mark as we know it today.
Variants of the question mark:
==========================
- Spanish and Catalan: In Spanish and Catalan, a question
is indicated by placing an inverted question mark ("¿") at the
beginning of the sentence in addition to the regular question
mark at the end ("?"). For example: "¿Cómo estás?" (How are you?)
- Greek: In Greek, a question is often indicated by using a semicolon-
like symbol called the "Greek question mark" (;) at the end of a
sentence. However, the Latin-style question mark ("?") is also
commonly used, especially in contemporary writing
influenced by Western conventions.
You could make the same investigation for the exclamation mark.
https://chat.openai.com/share/bc695629-cbeb-46d6-acad-421353d8daf7
> - declarative sentence (statement)
> - interrogative sentence (question)
> - imperative sentence (command)
> - exclamative sentence (exclamation)
> https://www.englishclub.com/grammar/sentence/type.php
Well I guess some time in history the question mark
was invented, so that younglings like Dan Christensen
don't have to use their spare brain cells, and analyze the
sentence itself, just look at the end of the sentence,
if it has a QUESTION mark, its a INTERROGATIVE sentence.
What is the history of the question mark? Lets ask ChatGPT:
History of the question mark:
==========================
- The Point of Interrogation: In the 8th century,
Alcuin of York, an English scholar, introduced a
symbol called the "punctus interrogativus" (point
of interrogation), which was a dot placed at the
end of a sentence to signal a question.
- The Development of the Modern Question Mark:
Over time, the punctus interrogativus evolved into
the modern question mark. Its form gradually changed,
and it became a distinctive symbol resembling
the one we use today.
- Standardization: The standardization of punctuation,
including the question mark, occurred with the advent
of the printing press in the 15th century. Printers like Aldus
Manutius and Johannes Gutenberg played a role in codifying
punctuation rules, which included the question mark as we know it today.
Variants of the question mark:
==========================
- Spanish and Catalan: In Spanish and Catalan, a question
is indicated by placing an inverted question mark ("¿") at the
beginning of the sentence in addition to the regular question
mark at the end ("?"). For example: "¿Cómo estás?" (How are you?)
- Greek: In Greek, a question is often indicated by using a semicolon-
like symbol called the "Greek question mark" (;) at the end of a
sentence. However, the Latin-style question mark ("?") is also
commonly used, especially in contemporary writing
influenced by Western conventions.
You could make the same investigation for the exclamation mark.
https://chat.openai.com/share/bc695629-cbeb-46d6-acad-421353d8daf7
Mild Shock
Sep 20, 2023, 4:33:00 AM9/20/23
to
You find the question mark also in Prolog in the form of a "?-" prompt.
So what does it do? Do you have any clue? Here is an example:
/* GNU Prolog 1.5.0 (64 bits) */
| ?- [user].
compiling user for byte code...
men(aristoteles).
mortal(X) :- men(X).
user compiled, 2 lines read - 306 bytes written, 16005 ms
yes
| ?- mortal(aristoteles).
yes
You can download GNU Prolog here:
http://www.gprolog.org/
So what does it do? Do you have any clue? Here is an example:
/* GNU Prolog 1.5.0 (64 bits) */
| ?- [user].
compiling user for byte code...
men(aristoteles).
mortal(X) :- men(X).
user compiled, 2 lines read - 306 bytes written, 16005 ms
yes
| ?- mortal(aristoteles).
yes
You can download GNU Prolog here:
http://www.gprolog.org/
Dan Christensen
Sep 20, 2023, 10:43:40 AM9/20/23
to
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:
> >
> >> 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]
> 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.
>
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.
Mild Shock
Sep 20, 2023, 10:49:00 AM9/20/23
to
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
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
Mild Shock
Sep 20, 2023, 11:00:31 AM9/20/23
to
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
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
Dan Christensen
Sep 20, 2023, 11:31:33 AM9/20/23
to
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!"
> 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!"
>
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.
Mild Shock
Sep 20, 2023, 2:51:06 PM9/20/23
to
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.
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.
Mild Shock
Sep 20, 2023, 2:51:38 PM9/20/23
to
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?
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?
Dan Christensen
Sep 20, 2023, 3:08:42 PM9/20/23
to
On Wednesday, September 20, 2023 at 2:51:38 PM UTC-4, Mild Shock wrote:
>
> Dan Christensen schrieb:
>
> 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]
> 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.
> When its about mathematical structures, its pretty easy
> to see that 3-, 3+ and 4 are solutions to the Liar Paradox.
Message has been deleted
Mild Shock
Sep 20, 2023, 3:32:07 PM9/20/23
to
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
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
Mild Shock
Sep 20, 2023, 4:42:42 PM9/20/23
to
Thats a funny find that these two are logically equivalent:
a e t <=> a e f
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!
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!
Dan Christensen
Sep 20, 2023, 4:44:44 PM9/20/23
to
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]
> 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
>
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.
Mild Shock
Sep 20, 2023, 4:46:53 PM9/20/23
to
Dan Christensen was loosing his marbles:
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
> 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]]
------------------------------------ end proof --------------------------------------
https://groups.google.com/g/sci.math/c/JQTvBJvYxDA/m/lrDgf9VjCAAJ
Dan Christensen
Sep 20, 2023, 5:03:13 PM9/20/23
to
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 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.
Dan
Mild Shock
Sep 21, 2023, 4:20:57 AM9/21/23
to
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:.
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
Mild Shock
Sep 21, 2023, 4:27:59 AM9/21/23
to
The concept of equivalence class is basic math, its for
example used in rational numbers construction since 2/3 ~ 4/6.
https://en.wikipedia.org/wiki/Equivalence_class#Properties
Equivalence classes are usually written as [a]. So under
the ~ equivalence relation, instead of dealing with 3 subsets:
+---------------+----------------+----------------+
| f | t | m |
+---------------+----------------+----------------+
You can take 3 values, that represent the 3 subsets,
with F = [a] where a e f, T = [a] where a e t, U = [a] where a e m.
F, T and U
But in DC Proof, I cannot write uppercase for a value,
so I had to use lower case:
49 ALL(x):[Value(x) => [[x=t <=> x=f] => x=u]]
Rem DNeg, 48
Thats a technical problem of DC Poop.
example used in rational numbers construction since 2/3 ~ 4/6.
https://en.wikipedia.org/wiki/Equivalence_class#Properties
Equivalence classes are usually written as [a]. So under
the ~ equivalence relation, instead of dealing with 3 subsets:
+---------------+----------------+----------------+
| f | t | m |
+---------------+----------------+----------------+
You can take 3 values, that represent the 3 subsets,
with F = [a] where a e f, T = [a] where a e t, U = [a] where a e m.
F, T and U
But in DC Proof, I cannot write uppercase for a value,
so I had to use lower case:
49 ALL(x):[Value(x) => [[x=t <=> x=f] => x=u]]
Rem DNeg, 48
Mild Shock
Sep 21, 2023, 4:32:03 AM9/21/23
to
Ultimately I am using slightly different equivalence classes.
On the left hand side is the set condition, on the right hand
side the equivalence class:
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
Quite impressive Zoo of resolutions, already.
On the left hand side is the set condition, on the right hand
side the equivalence class:
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 = t
~a e t & a e f ~~> x = f
a e t & a e f ~~> x = u
~a e t & a e f ~~> x = f
Quite impressive Zoo of resolutions, already.
Dan Christensen
Sep 21, 2023, 6:16:08 AM9/21/23
to
On Thursday, September 21, 2023 at 4:20:57 AM UTC-4, Mild Shock wrote:
> 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]]
>
You are making this needlessly complicated. Why would you say, for example, that the sentences "What time is it?" and "Wash your hands" are somehow equivalent? They are simply sentences of indeterminate truth value.
> 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]]
>
There is no need for an equivalence relation here. The only structure you need here is the simple trichotomy: Every sentence is either (1) a true sentence, (2) a false sentence, or (3) one of indeterminate truth value. (See line 6 here: https://dcproof.com/LiarParadoxV2.htm )
Mild Shock
Sep 21, 2023, 1:53:15 PM9/21/23
to
There is no constant for the truth value "T" in DC Proof.
In classical logic you could defined the truth values
"T" and "F" as follows:
a ~ "T" :<=> (a <=> p & ~p)
a ~ "F" :<=> (a <=> p v ~p)
Now concerning your non-classical logic that resolves the
Liar Paradox, how do you define the involved truth values?
Assume we have already defined a ~ "T" and a ~ "F" in
your system. I guess you want to defined "U" as follows, right?
a ~ "U" :<=> ~a ~ "T" & ~a ~ "F"
But then there is a bug in DC Proof, I cannot use
capital letter for constant values. So I cannot
use the STANDARD constant values "F", "T" and "U" from here:
https://en.wikipedia.org/wiki/Three-valued_logic
Because DC Proof doesn't accept:
1 ALL(x):[Value(x) <=> x=F | x=T | x=U]
Axiom
It only accepts:
1 ALL(x):[Value(x) <=> x=f | x=t | x=u]
Axiom
:-( Is this a bug or feature?
In classical logic you could defined the truth values
"T" and "F" as follows:
a ~ "T" :<=> (a <=> p & ~p)
a ~ "F" :<=> (a <=> p v ~p)
Now concerning your non-classical logic that resolves the
Liar Paradox, how do you define the involved truth values?
Assume we have already defined a ~ "T" and a ~ "F" in
your system. I guess you want to defined "U" as follows, right?
a ~ "U" :<=> ~a ~ "T" & ~a ~ "F"
But then there is a bug in DC Proof, I cannot use
capital letter for constant values. So I cannot
use the STANDARD constant values "F", "T" and "U" from here:
https://en.wikipedia.org/wiki/Three-valued_logic
Because DC Proof doesn't accept:
1 ALL(x):[Value(x) <=> x=F | x=T | x=U]
Axiom
It only accepts:
1 ALL(x):[Value(x) <=> x=f | x=t | x=u]
Axiom
Mild Shock
Sep 21, 2023, 2:04:05 PM9/21/23
to
Dan Christensen halucinated complete bullshit:
a) p => ~p
b) ~p => p
They are not equivalent to these two propositional formulas:
1) p v ~p /* true sentence */
2) p & ~p /* false sentence */
Do you claim they are equivalent to the Liar Paradox?
> Every sentence is either (1) a true sentence,
(2) a false sentence, or (3) one of indeterminate truth value.
Take these two propositional formulas:
(2) a false sentence, or (3) one of indeterminate truth value.
a) p => ~p
b) ~p => p
They are not equivalent to these two propositional formulas:
1) p v ~p /* true sentence */
2) p & ~p /* false sentence */
Do you claim they are equivalent to the Liar Paradox?
Dan Christensen
Sep 21, 2023, 6:14:48 PM9/21/23
to
On Thursday, September 21, 2023 at 1:53:15 PM UTC-4, Mild Shock wrote:
> There is no constant for the truth value "T" in DC Proof.
Neither is there in most textbook math proofs. As I have shown here, it isn't necessary for a resolution of The Liar Paradox. Basic set theory and ordinary logic are enough.
> There is no constant for the truth value "T" in DC Proof.
> In classical logic you could defined the truth values
> "T" and "F" as follows:
>
> a ~ "T" :<=> (a <=> p & ~p)
>
> a ~ "F" :<=> (a <=> p v ~p)
>
> Now concerning your non-classical logic ...
There is nothing "non-classical" about it. It's the ordinary logic of most textbook math proofs.
> ... that resolves the
> Liar Paradox, how do you define the involved truth values?
I prove a theorem of basic set theory:
ALL(s):ALL(t):ALL(f):ALL(m):[Set(s) & Set(t) & Set(f) & Set(m) <--------- 4 Arbitrary sets
=> [ALL(a):[a in t => a in s] <------------- 3 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 Rules
& ~[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]]]]
Since "This sentence is false" is a true sentence if and only if it is false sentence, we can resolve LP by interpreting:
s = a set of sentences
t = the subset of true sentences in s
f = the subset of false sentence in s
m = the set sentences "of indeterminate truth value") in s
Thus, "This sentence is false" can be proven to be a sentence of indeterminate truth value using only basic set theory and ordinary logic with no need to postulate any other structures.
> But then there is a bug in DC Proof, I cannot use
> capital letter for constant values.
> So I cannot
> use the STANDARD constant values "F", "T" and "U" from here:
>
> https://en.wikipedia.org/wiki/Three-valued_logic
>
> Because DC Proof doesn't accept:
>
> 1 ALL(x):[Value(x) <=> x=F | x=T | x=U]
> Axiom
>
> It only accepts:
> 1 ALL(x):[Value(x) <=> x=f | x=t | x=u]
> Axiom
> :-( Is this a bug or feature?
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