Boolean Logic In Lambda Calculus

Alonzo Church defined the boolean values ‘true’ and ‘false’ in lambda calculus as:

• true=λab.a
• false=λab.b

Given a predicate (a function that returns a boolean value) p, the statement that would usually be written ‘if p then E₁ else E₂’ becomes simply pE₁E₂.

The various boolean functions are also simple:

• not p=(p)(false)(true)
• a and b=(a)(b)(false)
• a or b=(a)(true)(b)
• a xor b=(a)((b)(false)(true))(b)