Re: Formulas & Sentences of PL...

"mikejde@xxxxxxxxx" wrote:
>
> Jim Spriggs wrote:
> > "mikejde@xxxxxxxxx" wrote:
> > >
> > > Jim Spriggs wrote:
> > > > "mikejde@xxxxxxxxx" wrote:
> > > > >
> > > > > .... Why would you want a duplicate
> > > > > quantification (x)(x)P to be well-formed?
> > > >
> > > > Just because ruling it out is a nuisance (which I think was William's
> > > > point).
> > >
> > > Oh, for a minute I thought being weird and being a nuisance were
> > > different. But why is it a nuisance?
> >
> > See William's post.
> >
> > > I can think of some cases where it
> > > would lead to problems.
> >
> > Example?
> >
>
> If we allow vacuous quantification then if x does not occur free or
> bound in G, then (x)F -> G is equivalent to (x)F -> (x)G (i.e. (x)G is
> a vacuous quantification). But in an "inclusive" (or free) logic whose
> theorems are valid in every, including the empty, domain, the two
> formulas above are not equivalent since any universal quantification is
> true in the empty domain. The same is not true for open formulas. E.g.
> (x)F -> G might be false under some interpretation in an empty domain
> while (x)F -> (x)G would be true under the same interpretation.

Ok, I was not thinking of free logic.
--
I don't know who you are Sir, or where you come from,
but you've done me a power of good.
.