At 09:02 +0100 2003/04/17, Simon Peyton-Jones wrote:
Allow 'exists' Deprecate 'forall' (for defining existentials, that is) Eventually allow only 'exists'
Does anyone have any opinions on this topic? It's a small point, but one that bites quite frequently.
I have to come to think about this type of questions, but in another context, namely, when I was implementing a version of Prolog in the form of a proof-verification system building on metamathematics (= metalogic) rather than the standard logic. See for example, Elliott Mendelson, "Introduction to Mathematical Logic". You might pick down a program called "Qu-Prolog" to get an input on similar metalogical ideas in the form of a generalized Prolog system for bound variables: Qu-Prolog: http://www.svrc.it.uq.edu.au/pages/QuProlog_release.html Earlier versions http://svrc.it.uq.edu.au/Bibliography/svrc-tr.html?00-20 http://svrc.it.uq.edu.au/Bibliography/svrc-tr.html?00-21 Ergo http://www.svrc.it.uq.edu.au/pages/Ergo_release.html In such a context, it looks as though the natural extension of the type theory is axiomatic set theory. So I am therefore inclined to think that one should do whatever is correct in math, in order to pave the way for a natural development. As for naming conventions, I favor a naming without prepositions and endings, if possible. For example, "+" might be called "addition", even though "a + b" is read "a added to b". So one might use the words "all" and "exist" simply. At 10:05 +0200 2003/04/17, Doaitse Swierstra wrote:
I think that (in line with the other type notation conventions where you do not have to be explicit) you should not be forced to write anything at all, but I am sure I will remain a minority ;-}
Actually, in metamathematics, there is a difference between quantified and unquantified formulas (variables with "all").
Many of you have grown to love existential data types, which we current write like this:
data T = forall a. Foo a (a -> Int)
Mark and I chose 'forall' rather than 'exists' to save grabbing another keyword from the programmer. And indeed, the type of the constructor Foo is Foo :: forall a. a -> (a->Int) -> T
But every single time I explain this to someone, I find I have to make excuses for using the term 'forall'. I say "it really means 'exists', but we didn't want to lose another keyword".
A mathematical formulation of T is perhaps: T := V_{a in I} a x Hom(a, Int) where V sands for the disjoint union (= coproduct in the category of sets). So one computer approximating analogue might then be "exist". Hans Aberg