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Topic: A formal counter-example of Ax Ey P(x,y) -> Ey Ax P(x,y)
Replies: 2   Last Post: Dec 12, 2012 11:53 AM

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 Dan Christensen Posts: 6,595 Registered: 7/9/08
Re: A formal counter-example of Ax Ey P(x,y) -> Ey Ax P(x,y)
Posted: Dec 12, 2012 11:53 AM

On Dec 12, 11:22 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
> Dan Christensen <Dan_Christen...@sympatico.ca> writes:
> > This problem is central to predicate calculus.
>
>   What problem is that? There is no apparent problem in the observation
> that (x)(Ey)P(x,y) does not imply (Ey)(x)P(x,y).
>

It's nice to have a fairly concrete and formalizable illustration of
the point, don't you think?

In developing my program, I found this issue to be the most difficult
one in formal logic -- how to ensure that we cannot derive Ey Ax
P(x,y) from Ax Ey P(x,y) in the most natural way possible. My thinking
on this evolved over the years. Further insights into this problem
were the basis for my version 2.0. My program no longer had to track
dependencies among variables resulting from existential specification
-- no more Skolem functions or anything like them, and no weird,
counter-intuitive axioms. My new approach greatly simplified my
program and solved many nagging theoretical and practical problems. It
wasn't a new system of logic -- more like a set of guidelines
(restrictions) for the use of FOL, guidelines that, it seemed to me,
most mathematicians use instinctively.

Dan