Motivation behind Phantom Types?

Don Stewart's Haskell in the Large's presentation mentioned Phantom Types:

data Ratio n = Ratio Double
1.234 :: Ratio D3

data Ask ccy = Ask Double
Ask 1.5123 :: Ask GBP

I read over his bullet points about them, but I did not understand them. In addition, I read the Haskell Wiki on the topic. Yet I still am missing their point.

What's the motivation to use a phantom type?

To answer the "what's the motivation to use a phantom type". There is two points:

• to make invalid states unrepresentable, which is well explained in Aadit's answer
• Carry some of the information on the type level

For example you could have distances tagged by the length unit:

{-# LANGUAGE GeneralizedNewtypeDeriving #-}

newtype Distance a = Distance Double
  deriving (Num, Show)

data Kilometer
data Mile

marathonDistance :: Distance Kilometer
marathonDistance = Distance 42.195

distanceKmToMiles :: Distance Kilometer -> Distance Mile
distanceKmToMiles (Distance km) = Distance (0.621371 * km)

marathonDistanceInMiles :: Distance Mile
marathonDistanceInMiles = distanceKmToMiles marathonDistance

And you can avoid Mars Climate Orbiter disaster:

>>> marathonDistanceInMiles
Distance 26.218749345

>>> marathonDistanceInMiles + marathonDistance

<interactive>:10:27:
    Couldn't match type ‘Kilometer’ with ‘Mile’
    Expected type: Distance Mile
      Actual type: Distance Kilometer
    In the second argument of ‘(+)’, namely ‘marathonDistance’
    In the expression: marathonDistanceInMiles + marathonDistance

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There are slight varitions to this "pattern". You can use DataKinds to have closed set of units:

{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE DataKinds #-}

data LengthUnit = Kilometer | Mile

newtype Distance (a :: LengthUnit) = Distance Double
  deriving (Num, Show)

marathonDistance :: Distance 'Kilometer
marathonDistance = Distance 42.195

distanceKmToMiles :: Distance 'Kilometer -> Distance 'Mile
distanceKmToMiles (Distance km) = Distance (0.621371 * km)

marathonDistanceInMiles :: Distance 'Mile
marathonDistanceInMiles = distanceKmToMiles marathonDistance

And it will work similarly:

>>> marathonDistanceInMiles
Distance 26.218749345

>>> marathonDistance + marathonDistance
Distance 84.39

>>> marathonDistanceInMiles + marathonDistance

<interactive>:28:27:
    Couldn't match type ‘'Kilometer’ with ‘'Mile’
    Expected type: Distance 'Mile
      Actual type: Distance 'Kilometer
    In the second argument of ‘(+)’, namely ‘marathonDistance’
    In the expression: marathonDistanceInMiles + marathonDistance

But now the Distance can be only in kilometers or miles, we can't add more units later. That might be useful in some use cases.

---

We could also do:

data Distance = Distance { distanceUnit :: LengthUnit, distanceValue :: Double }
   deriving (Show)

In the distance case we can work out the addition, for example translate to kilometers if different units are involved. But this doesn't work well for currencies which ratio isn't constant over time etc.

---

And it's possible to use GADTs for that instead, which may be simpler approach in some situations:

{-# LANGUAGE GeneralizedNewtypeDeriving #-}
{-# LANGUAGE KindSignatures #-}
{-# LANGUAGE DataKinds #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE StandaloneDeriving #-}

data Kilometer
data Mile

data Distance a where
  KilometerDistance :: Double -> Distance Kilometer
  MileDistance :: Double -> Distance Mile

deriving instance Show (Distance a)

marathonDistance :: Distance Kilometer
marathonDistance = KilometerDistance 42.195

distanceKmToMiles :: Distance Kilometer -> Distance Mile
distanceKmToMiles (KilometerDistance km) = MileDistance (0.621371 * km)

marathonDistanceInMiles :: Distance Mile
marathonDistanceInMiles = distanceKmToMiles marathonDistance

Now we know the unit also on the value level:

>>> marathonDistanceInMiles 
MileDistance 26.218749345

---

This approach especially greately simplifies Expr a example from Aadit's answer:

{-# LANGUAGE GADTs #-}

data Expr a where
  Number     :: Int -> Expr Int
  Boolean    :: Bool -> Expr Bool
  Increment  :: Expr Int -> Expr Int
  Not        :: Expr Bool -> Expr Bool

---

It's worth pointing out that the latter variations require non-trivial language extensions (GADTs, DataKinds, KindSignatures), which might not be supported in your compiler. That's might be the case with Mu compiler Don mentions.