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Boolean values can be defined in the
Lambda Calculus, although they are often "built into"
programming languages based on Lambda 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
{\fB Define Boolean From Scratch. \fP}
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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.
Also see
[Ints]
and
[Lists].
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window on the wide world:
λ ...
| :: | list cons |
| nil | the [ ] list |
| null | predicate |
| hd | head (1st) |
| tl | tail (rest) |
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