## Lambda Calculus Boolean

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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} ```

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 {\fB Define Boolean From Scratch. \fP}

Also see [Ints] and [Lists].

 Coding Ockham's Razor, L. Allison, Springer A Practical Introduction to Denotational Semantics, L. Allison, CUP

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 :: list cons nil the [ ] list null predicate hd head (1st) tl tail (rest)

 © L. Allison   http://www.allisons.org/ll/   (or as otherwise indicated), Faculty of Information Technology (Clayton), Monash University, Australia 3800 (6/'05 was School of Computer Science and Software Engineering, Fac. Info. Tech., Monash University, was Department of Computer Science, Fac. Comp. & Info. Tech., '89 was Department of Computer Science, Fac. Sci., '68-'71 was Department of Information Science, Fac. Sci.) Created with "vi (Linux + Solaris)",  charset=iso-8859-1,  fetched Thursday, 02-Feb-2023 06:41:41 AEDT.