On Jun 26, 2010, at 11:21 AM, Andrew Coppin wrote:
A first order logic quantifies over values, and a second order logic quantifies over values and sets of values (i.e., types, predicates, etc). The latter lets you express things like "For every property P, P x". Notice that this expression "is equivalent" to Haskell's bottom type "a". Indeed, Haskell is a weak second-order language. Haskell's language of values, functions, and function application is a first-order language.
I have literally no idea what you just said.
It's like, I have C. J. Date on the shelf, and the whole chapter on the Relational Calculus just made absolutely no sense to me because I don't understand the vocabulary.
Let's break it down then. A language is a set of "terms" or "expressions", together with rules for putting terms together (resulting in "sentences", in the logic vocabulary). A "logic" is a language with "rules of inference" that let you transform sets of sentences into new sentences.
Quantification is a tricky thing to describe. In short, if a language can "quantify over" something, it means that you can have variables "of that kind". So, Haskell can quantify over integers, since we can have variables like "x :: Integer". More generally, Haskell's run-time language quantifies over "values".
From this point of view, Haskell is TWO languages that interact nicely (or rather, a second-order language). First, there is the "term-level" run-time language. This is the stuff that gets evaluated when you actually run a program. It can quantify over values and functions. And we can express function application (a rule of inference to combine a function and a value, resulting in a new value).
Second, there is the type language, which relies on specific keywords:
data, class, instance, newtype, type, (::)
Using these keywords, we can quantify over types. That is, the constructs they enable take types as variables.
However, notice that a type is "really" a collection of values. For example, as the Gentle Introduction to Haskell says, we should think of the type Integer as being:
data Integer = 0 | 1 | -1 | 2 | -2 | ...
despite the fact that this isn't how it's really implemented. The Integer type is "just" an enumeration of the integers.
Putting this all together and generalizing a bit, a second-order language is a language with two distinct, unmixable kinds variables, where one kind of variable represents a collection of things that can fill in the other kind of variable.