Edit Question - KnowledgeHub

Title Body

Well, I am studying Haskell Monads. When I read the Wikibook Category theory article, I found that the signature of monad morphisms looks pretty like tautologies in logic, but you need to convert M a to ~~A , here ~ is the logic negation. return :: a -> M a -- Map to tautology A => ~~A, double negation introduction (>>=) :: M a -> (a -> M b) -> M b -- Map to tautology ~~A => (A => ~~B) => ~~B the other operations is also tautologies: fmap :: (a -> b) -> M a -> M b -- Map to (A => B) -> (~~A => ~~B) join :: M (M a) -> M a -- Map to ~~(~~A) => ~~A It's also understood that according to the fact that the Curry-Howard correspondence of normal functional languages is the intuitive logic, not classical logic, so we cannot expect a tautology like ~~A => A can have a correspondence. But I am thinking of something else. Why the Monad can only relate to a double negation? what is the correspondence of single negation? This lead me to the following class definition: class Nomad n where rfmap :: (a -> b) -> n b -> n a dneg :: a -> n (n a) return :: Nomad n => a -> n (n a) return = dneg (>>=) :: Nomad n => n (n a) -> (a -> n (n b)) -> n (n b) x >>= f = rfmap dneg $ rfmap (rfmap f) x Here I defined a concept called "Nomad", and it supports two operations (both related to a logic axiom in intuitive logic). Notice the name "rfmap" means the fact that its signature is similar to functor's fmap , but the order of a and b are reversed in result. Now I can re-define the Monad operations with them, with replace M a to n

Tags (comma-separated) Cancel