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 <[email protected]> wrote:
> On Aug 15, 2012 3:21 AM, "wren ng thornton" <[email protected]> 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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>
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