For lambda calculus, the motto is "when everything is a function".
The boolean true is the function λx.λy.x, while false is λx.λy.y.
If b then x else y then simply becomes b x y.
In a functional language like Haskell that is basically a typed lambda calculus with lots of syntactic sugar, we can replicate this with:
type MyBool a = a -> a -> a
myTrue :: MyBool a
myTrue = \x y -> x
myFalse :: MyBool a
myFalse = \x y -> y
myIf :: MyBool a -> a -> a -> a
myIf b myThen myElse = b myThen myElse
main = print $ myIf myTrue "true" "false"
This is both the Church encoding and the Scott encoding of the abstract data type
data Bool = True | False
making it pretty much the only encoding you find in the literature.
This is quite different from the case of the natural numbers, where not only do the Church and Scott encoding differ, but there are several other reasonable representations fitting particular purposes.
