Talk:De Morgan algebra

Dunn's algebra

"Another example is Dunn's 4-valued logic, in which false < neither-true-nor-false < true and false < both-true-and-false < true, while neither-true-nor-false and both-true-and-false are not comparable."

It is claimed that this algebra has four values but five appear. It is claimed that two of these values are not comparable but the four/five values appear totally ordered.

I think that this situation must be cleared and that some words must be spent to distinguish this De Morgan algebra from the 4-valued boolean algebra. 92.184.108.122 (talk) 07:22, 11 March 2023 (UTC)

You misquoted that. It's not false < neither-true-nor-false < true and false < both-true-and-false < true.

The correct quote is false < neither-true-nor-false < true and false < both-true-and-false < true (my emphasis). See the difference? Paradoctor (talk) 08:22, 11 March 2023 (UTC)

Ok for the "and". But I am still unable to distinguish Dunn's algebra from the 4-valued boolean algebra { 0, {a}, {b}, {a,b} } for which we have also 0 ⊂ {a} ⊂ {a,b} and 0 ⊂ {b} ⊂ {a,b} while {a} and {b} are not comparable. Some more words to distinguish them or to say they are the same ? 92.184.107.177 (talk) 11:41, 15 March 2023 (UTC)

De Morgan algebras are more general than Boolean algebras. The distinction is explained in the second sentence of the lead. ;) Paradoctor (talk) 12:52, 15 March 2023 (UTC)

Thank you for your quick answer. I think that I begin to understand : Dunn's algebra is obtained by taking the two intermediate N and B values equal to their negation. If it is the case I think that this could be added in the article to help the reader's comprehension. 92.184.104.255 (talk) 17:05, 15 March 2023 (UTC)

Can you prove that N≠¬B? Paradoctor (talk) 18:15, 15 March 2023 (UTC)

I can't give any answer if I don't know the theory in which I have to work. Please precise your question. 92.184.104.255 (talk) 19:27, 15 March 2023 (UTC)

And therein lies the answer. If you can't prove this for all theories, you cannot presume it is a theorem that Dunn's semantics implies ¬N=N and ¬B=B. Paradoctor (talk) 19:54, 15 March 2023 (UTC)

Ok Paradoxctor. 92.184.100.248 (talk) 07:18, 16 March 2023 (UTC)

For me it follows from our strange talk that the simplest De Morgan algebra which is nor a boolean algebra is 3-valued : T, M, B with ¬M = M. Mention it in the article ? 92.184.100.248 (talk) 11:55, 16 March 2023 (UTC)

This implies that all two-valued De Morgan algebras have the complementation laws as theorems. Can you WP:PROVEIT? Paradoctor (talk) 12:28, 16 March 2023 (UTC)

For me the question is not to prove anything (in which theory , with which logic ?), it is question to check that Dunn's 4-valued algebra and the 3-valued algebra just above are or not De Morgan algebras. I think they are. You ? 92.184.100.248 (talk) 13:25, 16 March 2023 (UTC)

Um, Dunn's semantics is explicitly mentioned as an example of a non-Boolean De Morgan algebra.

the question is not to prove anything (in which theory , with which logic ? If I link something, the idea is for you to at least to take a look at it. It's part of what I say. Had you done so, you wouldn't have had to ask. Paradoctor (talk) 14:46, 16 March 2023 (UTC)