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A relation between sets X and Y is a function between the power set of X and the power set of Y.
This function is order-preserving and preserves unions.
NOT could never be such a function because with NOT, you expect if you negate something small you get something big and if you negate something big you get something small, violating order preservation.
![](https://i.imgur.com/3327Y4u.jpg)

How to construct a relation btwn two order-reversing maps?
Galois adjunctions: the adjoint of a union-preserving map is a meet-preserving map.
In category theory terms: if limit-preserving, the opposite is colimit-preserving
Is a free way to introduce a negation in order theory.  by increasing the dimension, map can be order-preserving
![](https://i.imgur.com/QqICXZm.jpg)
![](https://i.imgur.com/mU9IZ4L.jpg)
![](https://i.imgur.com/pAXzxhH.jpg)


R: X -> Y
R': X \otimes X -> Y \otimes Y
f_R: P(X) -> P(Y)
f_R': P(X) -> P(Y)

"My sister is not a pet."
pet := 1/2 (cat + dog)
cat entails pet with error kc
dog entails pet with error kd
not("My sister is a pet.") = 1/2("My sister is not a cat." + "My sister is not a dog.")

"I am not going to the dentist"
"It is not me going to the dentist"
"I am not walking to the dentist"
"I am not going to that dentist"
