Re: [Haskell-cafe] Data structure containing elements which are instances of the same type class

In classical logic A -> B is the equivalent to ~A v B (with ~ = not and v = or) So (forall a. P(a)) -> Q {implication = not-or} ~(forall a. P(a)) v Q {forall a. X is equivalent to there does not exist a such that X doesn't hold} ~(~exists a. ~P(a)) v Q {double negation elimination} (exists a. ~P(a)) v Q {a is not free in Q} exists a. (~P(a) v Q) {implication = not-or} exists a. (P(a) -> Q) These steps are all equivalencies, valid in both directions. On Wed, Aug 15, 2012 at 9:32 AM, David Feuer wrote:

On Aug 15, 2012 3:21 AM, "wren ng thornton" wrote:

...

It's even easier than that.

(forall a. P(a)) -> Q <=> exists a. (P(a) -> Q)

Where P and Q are metatheoretic/schematic variables. This is just the usual thing about antecedents being in a "negative" position, and thus flipping as you move into/out of that position.

Most of this conversation is going over my head. I can certainly see how exists a. (P(a)->Q) implies that (forall a. P(a))->Q. The opposite certainly doesn't hold in classical logic. What sort of logic are you folks working in?

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