Re: reductio ad falsum versus reductio ad absurdum


Torkel Franzen wrote:
> adamgolding@xxxxxxxxx writes:
>
> > the => means |-, here?
>
> We don't need to take => to refer to provability in any formal calculus.
> Just read G=>A as "A follows from G". The rule
>
> G,A => B
> --------
> G => A->B
>
> then allows us to conclude, given that B follows from G together with
> A, that A->B follows from G.
>
> > so, by this line of thinking, saying that CP is
> >
> > (P |- Q) |- (P -> Q)
> >
> > makes a lot of sense to me--although i gather from the responses that
> > there is something wrong with this--although i don't quite gather
> > what--is there something formally/techincally wrong? is it false?
>
> To make good sense of the above, you need to specify the language
> you use and how |- is to be understood.

ok, and since |- seems to have a rather precise meaning attatched to
it, which prohibits such nesting, perhaps i should use the more
versatile "=>" (which i have seen used with nothing sort of five
different meanings) like so:

Conditional Proof:

P|-Q => P->Q


perhaps?



>
> > >
> > > ~A => Q ~A => ~Q
> > > -------
> > > => A
> >
> > Allen & Hand call the above rule "Impossible Antecedent", and they lump
> > all the rest under RAA...
>
> "Impossible Antecedent" is not standard terminology. But the
> important thing is that indirect proof is not constructively
> valid, but constructive reductio is.

hrm, i'm guessing you're referring to intuitionistic logic, which i'm
not exactly familiar with


> > ok, so with RAA often being used as an umbrella term, 'constructive
> > reductio' makes it clear that one means the one with no premises.
>
> Here I don't see what you have in mind.

just the form you told me was called constructive reductio:

---
> A (assumption)
> ...
> Q
> ...
> ~Q
> ~A

is one with no premises, and is presumably construed as a 'variety' of
RAA.


>
> > there another specifying term in somewhat common parlance to specify
> > the other kind, i.e. not constructive reduction, not impossible
> > antecedent, but the one i called 'reductio ad falsum' ??
>
> Not that I know of.

i suppose we could simply call it the 'nonconstructive reductio' but
'reductio ad falsum' seems a rather good term to me--relating it to the
link i gave before

.