Corr.: Typo in the proof:
have x e sat(L) and x e Y \ X. But by definition ~(x e Y \ X),

Should read:
have x e sat(L) and x e X. But by definition ~(x e Y \ X),

Mild Shock schrieb am Samstag, 16. September 2023 um 12:33:04 UTC+2:
> In Belnap FOUR, the Liar Sentences is not diag(X).
> In Belnap FOUR we have X = { {}, {0}, {1}, {0,1} },
> the diag is way too set theoretic, it jumps out of
>
> X cannot be used to analyse multi-valued logic.
> diag(X) gives the set X itself. Namely we find
> quite trivially that the diagonal is:
> diag(X) = { {}, {0}, {1}, {0,1} } = X
>
> And X is of course element of the universal class
> V, so that we have X e V, and hence X e anti(X,V).
> And X is also element of the power set, i.e.
>
> X e P(X), and hence X e anti(X,P(X)). But for multi-valued
> logic, we need to define another type of diagonal.
> Or we go with the set anti(X,Y) directly.
>
> A propositional formula identifies a set of propositional
> variable assignments, the satisfaction set. So looking
> for a formula with satisfaction set anti(X,Y) works fine.
>
> Every formula L, where sat(L) has this property:
> sat(L) =\= {} and sat(L) ⊆ anti(X,Y) and .
>
> We can show two theorems:
> 1) L is an antinomy in X
> 2) L is resolved in Y
>
> Proof: 1) sat(L) ⊆ anti(X,Y) is the same as sat(L) ⊆ Y \ X.
> Now assume that there were a propositional variable assignment
> x that would make L true, and that were in X. Then we would
> have x e sat(L) and x e Y \ X. But by definition ~(x e Y \ X),
> and hence it should be also ~(x e sat(L)), hence our assumption is false.
> 2) sat(L) implies that it is resolved. sat(L) ⊆ anti(X,Y) implies
> that it is resolved in Y, since sat(L) ⊆ Y \ X also implies sat(L) ⊆ Y.
>
> Q.E.D.
> Mild Shock schrieb am Samstag, 16. September 2023 um 12:20:40 UTC+2:
> > Corr.: Made a little Typo here, should say =\= {} instead = {}.
> >
> > Better call it the anti set, i.e. we can define:
> > anti(X,Y) := { x e Y | ∀y(y e X => x Δ y =\= {}) }
> > Now if we have extensionality available, we find:
> > anti(X,Y) = { x e Y | ∀y(y e X => x =\= y) }
> > And by contraposition:
> > anti(X,Y) = { x e Y | ∀y(x = y => ~y e X) }
> > And by substitution:
> > anti(X,Y) = { x e Y | ~x e X }
> >
> > Hence the anti set is just the set difference:
> > anti(X, Y) = Y \ X.
> >
> > Diagonal then referes to a popular construction of a set,
> > namely this construction:
> > diag(X) := { x e X | ~(x e x) }
> >
> > It can be used for Russells Paradox, in the form of Dan Christensens
> > Generalized Drinker paradox, we can use the construction to show
> > that anti(X,V) is non-empty, since it is provable:
> > diag(X) e anti(X,V)
> >
> > Or it can be used for Cantors Theorem, to show:
> > diag(X) e anti(X,P(X))
> >
> > Have Fun!

Mild Shock wrote:

> Corr.: Typo in the proof:
> have x e sat(L) and x e Y \ X. But by definition ~(x e Y \ X),
>
> Should read:
> have x e sat(L) and x e X. But by definition ~(x e Y \ X),

how should I read that, when 𝘆𝗼𝘂 𝘄𝗿𝗶𝘁𝗲 𝘀𝗼𝗺𝗲𝘁𝗵𝗶𝗻𝗴 𝗲𝗹𝘀𝗲. Which one shall I read
then?? Your country 𝗺𝗮𝗸𝗲𝘀 𝗳𝘂𝗻𝗻𝘆 𝗺𝗼𝗻𝗲𝘆 𝗼𝘂𝘁 𝗼𝗳 𝘁𝗵𝗲 𝘁𝗵𝗶𝗻 𝗮𝗶𝗿, on a computer,
𝘄𝗶𝘁𝗵𝗼𝘂𝘁 𝘄𝗼𝗿𝗸. Which is illegal. It is stealing 𝗳𝗿𝗼𝗺 𝗺𝗮𝗱𝗲 𝗽𝗼𝗼𝗿 𝗽𝗲𝗼𝗽𝗹𝗲, going
hungry in bed. You disgusting thief. It is written in The Bible that 𝘆𝗼𝘂
𝘀𝘁𝗲𝗮𝗹 𝗴𝗼𝗼𝗱𝘀 𝗮𝗻𝗱 𝘄𝗲𝗮𝗹𝘁𝗵, with the fake money you make. You disgusting
satanists.

𝗭𝗲𝗹𝗲𝗻𝘀𝗸𝘆_𝗽𝗼𝗹𝗶𝘁𝗶𝗰𝗮𝗹_𝗿𝗶𝘃𝗮𝗹_𝗰𝗵𝗮𝗿𝗴𝗲𝗱_𝘄𝗶𝘁𝗵_𝘁𝗿𝗲𝗮𝘀𝗼𝗻
https://r%74.com/ru%73%73ia/583051-ukraine-zelensky-shufrych-treason/

they have military bases around the world, dictating "democracy", then
accuse the President of China of being "dictator".

𝗚𝗲𝗿𝗺𝗮𝗻_𝗙𝗠_𝗹𝗮𝗯𝗲𝗹𝘀_𝗫𝗶_‘𝗮_𝗱𝗶𝗰𝘁𝗮𝘁𝗼𝗿’
The West must help Ukraine win its conflict against Russia to deter people
like the Chinese leader, 𝗔𝗻𝗻𝗮𝗹-ena Baerbock has claimed
https://r%74.com/news/583050-germany-baerbock-xi-dictator/

That's an insult coming from a real dictator and a nazi. Germans are
genetically genocidal and that's a fact, not an insult. They've proven it,
not once, not twice, but 7-8 times and are now proving in Bosnia with what
their nazi dictator is doing to Republic of Srpska.

She complains of Dictator at the same time she Dictates ...
western bureaucrat ignorance never ceases to amaze

Seems like the braincells that had been left also jumped ship.

Incredible shame that damn fool (Gorbachev) returned East Germany after
the Nazis lost it in a fair fight.

Typical German bimbo repeating the dumb Americans warmongers talking
points.

These 𝘄𝗲𝘀𝘁𝗲𝗿𝗻_𝗿𝗮𝗰𝗶𝘀𝘁_𝗰𝗿𝗲𝗮𝘁𝘂𝗿𝗲𝘀 think they can do or talk whatever they like
from their filthy mouth

The US & EU label other leaders they cannot control as dictators. Just
more demonization campaign against China.

Dumb FOOL cannot recognize that the REAL Dictator is in the US, 𝘄𝗶𝘁𝗵 𝗺𝗶𝗹𝗶𝘁𝗮𝗿𝘆
𝗯𝗮𝘀𝗲𝘀 𝗶𝗻 𝗵𝗲𝗿 𝗼𝘄𝗻 𝗰𝗼𝘂𝗻𝘁𝗿𝘆, dictating 𝘄𝗵𝗮𝘁 𝘁𝗵𝗶𝘀 𝘂𝗴𝗹𝘆 𝗯𝗶𝘁𝗰𝗵 𝗵𝗮𝘀 𝘁𝗼 𝘀𝗮𝘆. How much more
STUPID can you get. What a disgrace to the German Population