Why do we need a separate notation for П-types?



Main

I am confused about the motivation behind the need for a separate notation for П-types, that you can find in type systems from λ2 on. The answer usually goes like so – think about how one can represent a signature of identity function – it can be λa:type.λx:a.x or λb:type.λx:b.x. The subtle part, they say, is that these two signatures not only not equal, they are not alpha-equivalent as type variables a and b are free variables inside their correspondent abstractions. So to overcome this pesky syntactic issue, we present П binder that plays nicely with alpha-conversion.

So the question: why is that? Why not just fix the notion of alpha-equivalence?

UPDATE z:

Oh, silly of me, λa:type.λx:a.x and λb:type.λx:b.x are alpha equivalent. But why a:type -> a -> a and b:type -> b -> b arent then.

UPDATE suc z:

Aha, interesting, I guess this is a perfect example of selective blindness =D

I am reading the book Type Theory and Formal Proof, and in the chapter about lambda2 author motivates the existence of П using exactly that kind of argumentation – one cant say that \t:*.\v:t.v : * -> t -> t because this makes two alpha-equivalent terms\t:*.\v:t.v and \g:*.\v:g.v have different types, as corresponding types are not alpha-equivalent, where types like t:* -> t -> t are in fact alpha-invariant. Mind the difference between t:* -> t -> t and * -> t -> t. But, doesn’t it make this argument a bit trivial, and is it even something meaningful to talk about type a -> b where a and b are unbound by any quantifiers variables. Andrej Bauer pointed out in the comments that П is indeed resembles a lambda abstraction with a few additional bells and whistles.

All in all, I am done with that one, thank you guys.