What is predicativity?

Question

I have pretty decent intuition about types Haskell prohibits as \"impredicative\": namely ones where a forall appears in an argument to a type constructor other

Answer 1

I'd like to make a comment on the etymology issue, since @sclv's answer isn't quite right (etymologically, not conceptually).

Go back in time, to the days of Russell when everything is set theory— including logic. One of the logical notions of particular import is the "principle of comprehension"; that is, given some logical predicate φ:A→2 we would like to have some principle to determine the set of all elements satisfying that predicate, written as "{x | φ(x) }" or some variation thereon. The key point to bear in mind is that "sets" and "predicates" are viewed as being fundamentally different things: predicates are mappings from objects to truth values, and sets are objects. Thus, for example, we may allow quantifying over sets but not quantifying over predicates.

Now, Russell was rather concerned by his eponymous paradox, and sought some way to get rid of it. There are numerous fixes, but the one of interest here is to restrict the principle of comprehension. But first, the formal definition of the principle: ∃S.∀x.S x ↔︎ φ(x); that is, for our particular φ there exists some object (i.e., set) S such that for every object (also a set, but thought of as an element) x, we have that S x (you can think of this as meaning "x∈S", though logicians of the time gave "∈" a different meaning than mere juxtaposition) is true just in case φ(x) is true. If we take the principle exactly as written then we end up with an impredicative theory. However, we can place restrictions on which φ we're allowed to take the comprehension of. (For example, if we say that φ must not contain any second-order quantifiers.) Thus, for any restriction R, if a set S is determined (i.e., generated via comprehension) by some R-predicate, then we say that S is "R-predicative". If every set in our language is R-predicative then we say that our language is "R-predicative". And then, as is often the case with hyphenated prefix things, the prefix gets dropped off and left implicit, whence "predicative" languages. And, naturally, languages which are not predicative are "impredicative".

That's the old school etymology. Since those days the terms have gone off and gotten lives of their own. The ways we use "predicative" and "impredicative" today are quite different, because the things we're concerned about have changed. So it can sometimes be a bit hard to see how the heck our modern usage ties back to this stuff. Honestly, I don't think knowing the etymology really helps any in terms of figuring out what the words are really about (these days).