Boolean values can be defined in the λ calculus, although they are often "built into" programming languages based on λ calculus.

let
  TRUE  = lambda a. lambda b. a,
  FALSE = lambda a. lambda b. b

in let
  AND   = lambda p. lambda q. p q FALSE,
  OR    = lambda p. lambda q. p TRUE q,
  NOT   = lambda p. lambda a. lambda b. p b a,
  IF    = lambda p. lambda a. lambda b. p a b,
  EQ    = lambda x. lambda y.
            if x = y then TRUE else FALSE

in {simple test:}
  IF TRUE                   1  (-1) ::
  IF FALSE                (-2)   2  ::
  IF (OR  FALSE TRUE)       3  (-3) ::
  IF (AND FALSE TRUE)     (-4)   4  ::
  IF (NOT FALSE)            5  (-5) ::
  IF (EQ 1 1)               6  (-6) ::
  IF (OR (EQ 1 2) (EQ 2 2)) 7  (-7) ::
  nil

{ Define Boolean From Scratch. }

The example defines 'TRUE', 'FALSE', 'AND', 'OR', etc. from first principles but defines 'EQ' using the built in '=' which is of course a cheat (to keep the example small). However, the section on integers shows how to define 'ISZERO' which could be used to define 'EQ' from first principles.

let TRUE = lambda a. lambda b. a, FALSE = lambda a. lambda b. b in let AND = lambda p. lambda q. p q FALSE, OR = lambda p. lambda q. p TRUE q, NOT = lambda p. lambda a. lambda b. p b a, IF = lambda p. lambda a. lambda b. p a b, EQ = lambda x. lambda y. if x = y then TRUE else FALSE in {simple test:} IF TRUE 1 (-1) :: IF FALSE (-2) 2 :: IF (OR FALSE TRUE) 3 (-3) :: IF (AND FALSE TRUE) (-4) 4 :: IF (NOT FALSE) 5 (-5) :: IF (EQ 1 1) 6 (-6) :: IF (OR (EQ 1 2) (EQ 2 2)) 7 (-7) :: nil {<B> Define Boolean From Scratch. </B>}

output area

Also see [Ints] and [Lists].