User (me)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.
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
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 andshare your proposed resolution or idea, and we can discuss it further.
Userindeterminate 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.
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
ChatGPTsentence is false" as indeterminate in truth value.
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
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 furtherquestions 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.development of this idea may help address these challenges and provide a more robust solution to the Liar Paradox.
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
Useris a false sentence. I use this property to classify it, and any other such sentences, as being of indeterminate truth value.
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
ChatGPTaddress the inherent paradoxical nature of these statements.
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
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.values in general but instead provides a solution for the peculiar case of self-referential statements that lead to logical paradoxes.
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
Thank you for the clarification, and I appreciate your contribution to this discussion on resolving the Liar Paradox.and formal systems.
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,
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), haveproposed 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 foraddressing 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 andinterpretations 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
The notion of indeterminacy or truth value gaps hasbeen a topic of discussion in philosophical logic for quite
Take X = {{0},{1}} classical logic, then:https://groups.google.com/g/sci.math/c/4j4PBAA_WYE/m/_MCxQ1m5BQAJ
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
User (me)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.
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
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 andshare your proposed resolution or idea, and we can discuss it further.
Userindeterminate 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.
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
ChatGPTsentence is false" as indeterminate in truth value.
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
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 furtherquestions 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.development of this idea may help address these challenges and provide a more robust solution to the Liar Paradox.
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
Useris a false sentence. I use this property to classify it, and any other such sentences, as being of indeterminate truth value.
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
ChatGPTaddress the inherent paradoxical nature of these statements.
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
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.values in general but instead provides a solution for the peculiar case of self-referential statements that lead to logical paradoxes.
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
Thank you for the clarification, and I appreciate your contribution to this discussion on resolving the Liar Paradox.and formal systems.
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,
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), haveproposed 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 foraddressing 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 andinterpretations 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 told you already so, that there are multiple solutions! Andclaims 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.
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 hasbeen 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 Resolutionhttps://groups.google.com/g/sci.math/c/4j4PBAA_WYE/m/_MCxQ1m5BQAJ
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
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
share your proposed resolution or idea, and we can discuss it further.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
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.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
sentence is false" as indeterminate in 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
questions and challenges. For instance: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
development of this idea may help address these challenges and provide a more robust solution to the Liar Paradox.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
is a false sentence. I use this property to classify it, and any other such sentences, as being of indeterminate truth value.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
address the inherent paradoxical nature of these statements.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
values in general but instead provides a solution for the peculiar case of self-referential statements that lead to logical paradoxes.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
and formal systems.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,
proposed similar ideas to handle paradoxes like the Liar Paradox.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
addressing paradoxes arising from self-reference.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
interpretations can offer new insights and contribute to the evolving understanding of these complex logical and philosophical issues.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
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! AndHmmm, this sounds backwards to me. I'm used to seeing "necessary but
what you presented is only sufficient, but not necessary.
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! AndHmmm, this sounds backwards to me. I'm used to seeing "necessary but
what you presented is only sufficient, but not necessary.
not sufficient".
I told you already so, that there are multiple solutions! And
what you presented is only sufficient, but not necessary.
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 hasbeen 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),
The logic of paradoxhttps://groups.google.com/g/sci.math/c/Y3S8ESVZn_Y/m/eWggQXeMBQAJ
by G Priest · 1979 · Cited by 1308 https://link.springer.com/article/10.1007/BF00258428
Belnap notes that "paradoxes of implication" (A&~A)→Bhttps://groups.google.com/g/sci.math/c/Y3S8ESVZn_Y/m/1DohHzByBQAJ
and A→(B∨~B) are avoided in his 4-valued system. https://en.wikipedia.org/wiki/Four-valued_logic
THAT is one of your "solutions?"The notion of indeterminacy or truth value gaps hasbeen 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),
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 paradoxhttps://groups.google.com/g/sci.math/c/Y3S8ESVZn_Y/m/eWggQXeMBQAJ
by G Priest · 1979 · Cited by 1308 https://link.springer.com/article/10.1007/BF00258428
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)→Bhttps://groups.google.com/g/sci.math/c/Y3S8ESVZn_Y/m/1DohHzByBQAJ
and A→(B∨~B) are avoided in his 4-valued system. https://en.wikipedia.org/wiki/Four-valued_logic
Dan Christensen schrieb am Montag, 18. September 2023 um 15:20:40 UTC+2:
THAT is one of your "solutions?"The notion of indeterminacy or truth value gaps hasbeen 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),
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.
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
Take X = {{0},{1}} classical logic, then:https://groups.google.com/g/sci.math/c/4j4PBAA_WYE/m/_MCxQ1m5BQAJ
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
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[snip]
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.
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[snip]
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.
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
The notion of indeterminacy or TRUTH VALUE GAPS hasbeen a topic of discussion in philosophical logic for quite
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.
Mild Shock schrieb am Montag, 18. September 2023 um 20:31:36 UTC+2:
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[snip]
all the time hey basic set theory, but you are not knowledgeable of Belnap FOUR or Graham Priest.
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
[snip]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.
You claim that the 4-valued logic cannot be modelled with
the 2-valued logic
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[snip]
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 withDo pay attention, Mr. Collapse. As I have shown here, resolving LP does NOT require modelling 4-value logic. Deal with it.
the 2-valued logic
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[snip]
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 withDo pay attention, Mr. Collapse. As I have shown here, resolving LP does NOT require modelling 4-value logic. Deal with it.
the 2-valued logic
Dan
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
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]]]]
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 LPVice versa. It does not require your Trichotomy to get resolved.
does NOT require modelling 4-value logic. Deal with 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]]]]
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)An actual trichotomy would work better:
& [~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]]
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)And then prove this here:
& [~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]]]]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)An actual trichotomy would work better:
& [~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]]
Dan
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
Dan Christensen 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]]]]
There is no conclusion possible, that the above implies your Trichotomy.[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]]
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)And then prove this here:
& [~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]]]]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)An actual trichotomy would work better:
& [~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]]
Dan
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
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.
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]]
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.
Just use Graham Thirdness:
ALL(a):[a ε s => [a ε t | a ε f] (Graham Thirdness Rules)And you can prove:
& [~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]]]]
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?
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.
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]]]]
What happened to your user name?
Too much Trichotomy? LMAO!
sci.math schrieb:
What happened to your user name?
Too much Trichotomy? LMAO!
sci.math schrieb:
On Tuesday, September 19, 2023 at 4:38:23 PM UTC-4, Mild Shock wrote:
What happened to your user name?
Too much Trichotomy? LMAO!
sci.math schrieb:
Testing
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?
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
What happened to your user name?
Too much Trichotomy? LMAO!
sci.math schrieb:
On 9/19/2023 4:38 PM, Mild Shock wrote:
What happened to your user name?
Too much Trichotomy? LMAO!
sci.math schrieb:
He uses Giggle Groups, that's why.
Google doesn't fire poor programmers it accidentally hires now and then. Instead, they assign them to work in a dungeon, deep below their headquarters, where their new job is to make Google Groups worse and
worse. And they succeed doing so.
On 9/19/2023 9:09 PM, Volney wrote:
On 9/19/2023 4:38 PM, Mild Shock wrote:
What happened to your user name?
Too much Trichotomy? LMAO!
sci.math schrieb:
He uses Giggle Groups, that's why.
Google doesn't fire poor programmers it accidentally hires now and
then. Instead, they assign them to work in a dungeon, deep below their
headquarters, where their new job is to make Google Groups worse and
worse. And they succeed doing so.
ROFL!!!! Its most likely very, oh so very true. Yikes!
On Tuesday, September 19, 2023 at 6:39:08 PM UTC-4, Mild Shock wrote:LiarParadoxLemma.htm
You can prove this using the more standard definition of a trichotomy:
Either its standard or its not standard.
What should "more" standard mean?
More standard than your wonky notion of a trichotomy.
The problem with your solution is that you reject LEM.
On the contrary, I make use of it in my proof (e.g. by eliminating '~~' and proof by contraction)
Maybe you didn't know, but introducing a trichotomy on a set does not necessarily require the rejection of LEM. As I show in the lemma attached to my proof, a trichotomy can be introduced on ANY set, even the empty set. Proof: http://www.dcproof.com/
Wash your hands" and "This sentence is false.")Whereas the solution by Graham Priest still accepts LEM.[snip]
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. "
Dan
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
Dan Christensen, our youngling wrote:com/LiarParadoxLemma.htm
- 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 schrieb am Mittwoch, 20. September 2023 um 09:06:52 UTC+2:
You don't make any sense. We told you already
that "Wash your hands!" and "Did you wash your hands?"
are irrelevant to the Liar Paradox.
They have an exclaimation mark "!" respectively
a question mark "?" at the end. So they are
anyway in a separate category.
Whats wrong with you?
The Liar Paradox has a so called full stop "."
at the end.
Dan Christensen schrieb:
On Tuesday, September 19, 2023 at 6:39:08 PM UTC-4, Mild Shock wrote:
You can prove this using the more standard definition of a trichotomy:
Either its standard or its not standard.
What should "more" standard mean?
More standard than your wonky notion of a trichotomy.
The problem with your solution is that you reject LEM.
On the contrary, I make use of it in my proof (e.g. by eliminating '~~' and proof by contraction)
Maybe you didn't know, but introducing a trichotomy on a set does not necessarily require the rejection of LEM. As I show in the lemma attached to my proof, a trichotomy can be introduced on ANY set, even the empty set. Proof: http://www.dcproof.
"Wash your hands" and "This sentence is false.")Whereas the solution by Graham Priest still accepts LEM.[snip]
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.
Dan
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
- declarative sentence (statement)
- interrogative sentence (question)
- imperative sentence (command)
- exclamative sentence (exclamation) https://www.englishclub.com/grammar/sentence/type.php
You don't make any sense. We told you alreadyLiarParadoxLemma.htm
that "Wash your hands!" and "Did you wash your hands?"
are irrelevant to the Liar Paradox.
They have an exclaimation mark "!" respectively
a question mark "?" at the end. So they are
anyway in a separate category.
Whats wrong with you?
The Liar Paradox has a so called full stop "."
at the end.
Dan Christensen schrieb:
On Tuesday, September 19, 2023 at 6:39:08 PM UTC-4, Mild Shock wrote:
You can prove this using the more standard definition of a trichotomy:
Either its standard or its not standard.
What should "more" standard mean?
More standard than your wonky notion of a trichotomy.
The problem with your solution is that you reject LEM.
On the contrary, I make use of it in my proof (e.g. by eliminating '~~' and proof by contraction)
Maybe you didn't know, but introducing a trichotomy on a set does not necessarily require the rejection of LEM. As I show in the lemma attached to my proof, a trichotomy can be introduced on ANY set, even the empty set. Proof: http://www.dcproof.com/
Wash your hands" and "This sentence is false.")Whereas the solution by Graham Priest still accepts LEM.[snip]
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. "
Dan
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
FYI in English (unlike German?) imperative sentences need not end in an exclamation point.
Dan Christensen schrieb:Wash your hands" and "This sentence is false.")
On Tuesday, September 19, 2023 at 8:47:07 PM UTC-4, Mild Shock (aka Mr. Collapse) wrote:
Dan Christensen schrieb:
Maybe you didn't know, but a sentence in natural language, [for the purpose of resolving the Liar Paradox] can be classified as being one of either: (1) a true sentence, (2) a false sentence, or (3) a sentence of indeterminate truth value (e.g. "
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.
Its MORE STANDARD that they end with exclamation mark:
Q: To ChatGPT
Does an imperative sentence end with an exclamation mark?
A: From ChatGPT
Yes, an imperative sentence typically ends with an exclamation
mark. Examples include "Run!" "Stop!" and "Do your homework!"
https://www.quora.com/Does-an-imperative-sentence-end-with-an-exclamation-mark
Dan Christensen schrieb am Mittwoch, 20. September 2023 um 16:43:40 UTC+2:
FYI in English (unlike German?) imperative sentences need not end in an exclamation point.
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
Dan Christensen schrieb:
Back to the main point: A sentence, for the purposesof 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.
Dan Christensen schrieb:
Back to the main point: A sentence, for the purposesof 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.
Thats only LESS STANDARD, we can say SUB STANDARD.
At least this every mathematicians and logicians standard.
Because they don't care what philosophers feel or halucinate.
When its about mathematical structures, its pretty easy
to see that 3-, 3+ and 4 are solutions to the Liar Paradox.
You don't need to invoke experience or taste.
Just do your mathematical reasoning.
Dan Christensen schrieb am Mittwoch, 20. September 2023 um 17:31:33 UTC+2:
Back to the main point: A sentence, for the purposes of resolving the Liar Paradox,can be classified as being one of either: (1) a true sentence, (2) a false sentence, or
(3) a sentence of indeterminate truth value.
Here is a solution with truth values:https://groups.google.com/g/sci.math/c/JQTvBJvYxDA/m/lrDgf9VjCAAJ
49 ALL(x):[Value(x) => [[x=t <=> x=f] => x=u]]
On Wednesday, September 20, 2023 at 2:51:38 PM UTC-4, Mild Shock wrote:
Dan Christensen schrieb:[snip]
Back to the main point: A sentence, for the purposesof 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 easyHuh??? The Liar Paradox is about a sentence in natural language. I have applied basic set theory to a longstanding problem in linguistics. Deal with, Mr. Collapse.
to see that 3-, 3+ and 4 are solutions to 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
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:https://groups.google.com/g/sci.math/c/JQTvBJvYxDA/m/lrDgf9VjCAAJ
49 ALL(x):[Value(x) => [[x=t <=> x=f] => x=u]]
Now we can think about sentence classifications. We can even
make a stronger liar paradox, instead of only:
a e t <=> a e f
We can let it read, in 3- this is irrelevant, needed for the others:
a e t & ~a e f <=> ~a e t & a e f
Now there are many possible assignments to truth values:
Model 3-
~a e t & ~a e f ~~> x = u
a e t & ~a e f ~~> x = t
~a e t & a e f ~~> x = f
Model 3+
a e t & ~a e f ~~> x = t
~a e t & a e f ~~> x = f
a e t & a e f ~~> x = u
Model 4
~a e t & ~a e f ~~> x = u
a e t & ~a e f ~~> x = t
~a e t & a e f ~~> x = f
a e t & a e f ~~> x = u
Dan Christensen schrieb am Mittwoch, 20. September 2023 um 21:08:42 UTC+2:
On Wednesday, September 20, 2023 at 2:51:38 PM UTC-4, Mild Shock wrote:
Dan Christensen schrieb:[snip]
Back to the main point: A sentence, for the purposesof 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 easyHuh??? The Liar Paradox is about a sentence in natural language. I have applied basic set theory to a longstanding problem in linguistics. Deal with, Mr. Collapse.
to see that 3-, 3+ and 4 are solutions to 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
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
We are talking here about 3 DISJOINT categories.
On Wednesday, September 20, 2023 at 3:32:07 PM UTC-4, Mild Shock wrote:
Now we can think about sentence classifications. We can even
make a stronger liar paradox, instead of only:
a e t <=> a e f
We can let it read, in 3- this is irrelevant, needed for the others:
a e t & ~a e f <=> ~a e t & a e f
[snip]
Looks much the same. The LHS is equivalent to a in t. The RHS to a in f.
If x is in one of 3 categories, then x is NOT in the other 2 categories. We are talking here about 3 DISJOINT categories.
Dan
Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com
Dan Christensen was loosing his marbles:
We are talking here about 3 DISJOINT categories.Well the 3 categories f, t and u are disjoin, just have a look.
The suggested mappings for the models 3-, 3+ and 4 are
also disjoint.
What ARE you talking about?
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.
Mild Shock schrieb am Donnerstag, 21. September 2023 um 10:20:57 UTC+2:
You can use this equivalence relation:
a ~ b :<=> [[a e t & b e t] v [a e f & b e f] v [a e m & b e m]]
To get equivalence classes:
https://en.wikipedia.org/wiki/Equivalence_class
The equivalence classes are the truth values.
Thats missing in your take. If you then further
abstract, you get this nifty result:
49 ALL(x):[Value(x) => [[x=t <=> x=f] => x=u]]
Rem DNeg, 48
Dan Christensen schrieb am Mittwoch, 20. September 2023 um 23:03:13 UTC+2:.
What ARE you talking about?
You can use this equivalence relation:
a ~ b :<=> [[a e t & b e t] v [a e f & b e f] v [a e m & b e m]]
To get equivalence classes:
https://en.wikipedia.org/wiki/Equivalence_class
The equivalence classes are the truth values.
Thats missing in your take. If you then further
abstract, you get this nifty result:
49 ALL(x):[Value(x) => [[x=t <=> x=f] => x=u]]
Rem DNeg, 48
Dan Christensen schrieb am Mittwoch, 20. September 2023 um 23:03:13 UTC+2:.
What ARE you talking about?
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]]
Every sentence is either (1) a true sentence,(2) a false sentence, or (3) one of indeterminate truth value.
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?
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?
[ALL(a):[a in t => a in s] <------------- 3 Subsets of s& ALL(a):[a in f => a in s]
ALL(b):[b in s => [[b in t <=> b in f] => b in m]]]]
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?
| Sysop: | Keyop |
|---|---|
| Location: | Huddersfield, West Yorkshire, UK |
| Users: | 307 |
| Nodes: | 16 (2 / 14) |
| Uptime: | 96:32:24 |
| Calls: | 6,850 |
| Calls today: | 1 |
| Files: | 12,352 |
| Messages: | 5,414,947 |