Query on Booleans in Lambda Calculus

Question

This is the lambda calculus representation for the AND operator:

lambda(m).lambda(n).lambda (a).lambda (b). m(n a b) b

Can anyone help me i

Answer 1

Actually it's a little more than just the AND operator. It's the lambda calculus' version of if m and n then a else b. Here's the explanation:

In lambda calculus true is represented as a function taking two arguments* and returning the first. False is represented as function taking two arguments* and returning the second.

The function you showed above takes four arguments*. From the looks of it m and n are supposed to be booleans and a and b some other values. If m is true, it will evaluate to its first argument which is n a b. This in turn will evaluate to a if n is true and b if n is false. If m is false, it will evaluate to its second argument b.

So basically the function returns a if both m and n are true and b otherwise.

(*) Where "taking two arguments" means "taking an argument and returning a function taking another argument".

Edit in response to your comment:

and true false as seen on the wiki page works like this:

The first step is simply to replace each identifier with its definition, i.e. (λm.λn. m n m) (λa.λb. a) (λa.λb. b). Now the function (λm.λn. m n m) is applied. This means that every occurrence of m in m n m is replaced with the first argument ((λa.λb. a)) and every occurrence of n is replaced with the second argument ((λa.λb. b)). So we get (λa.λb. a) (λa.λb. b) (λa.λb. a). Now we simply apply the function (λa.λb. a). Since the body of this function is simply a, i.e. the first argument, this evaluates to (λa.λb. b), i.e. false (as λx.λy. y means false).