Re: reductio ad falsum versus reductio ad absurdum


Torkel Franzen wrote:
> "futurist" <adamgolding@xxxxxxxxxxxxxxx> writes:
>
> > Conditional Proof:
> >
> > P|-Q => P->Q
> >
>
> You can use any notation you like as long as you define a precise
> syntax and semantics.

i want to use standard notation whenever possible for my purposes.



>
> > just the form you told me was called constructive reductio:
> >...
> > is one with no premises, and is presumably construed as a 'variety' of
> > RAA.
>
> There can be any number of premises in both constructive reductio
> and indirect proof - I just left them out in the notation.
>
> > 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
>
> Your "reductio ad falsum" is constructively valid, and we don't need
> non-constructive reductio to prove it as a derived rule.


ok, the main distinction i'm trying (hoping) to get at is this:

in ND, when you complete an argument, you either had premises or you
didn't. if you had (true) premises your conclusion is true. if you
had NO premises your conclusion is not only true but is a tautology.

in some applications of 'proof by contradiction' the contradiction is
between one of the formal (as opposed to material) consequences of your
assumption and one of the premises--that is to say, your assumption
contradicts the facts and is contingently false (reductio ad falsum,
i'd like to say.)

in the other case, the contradiction is independent of any premises,
and is a contradiction between two formal consequences of your
assumption, showing that your assumption is a *self* contradiction and
is thus *necessarily* false. (reductio ad absurdum, i'd like to say)

both are a 'proof by contradiction' or a 'reductio' in that in both
cases we are 'reducing' an assumption to a 'contradiction', but in the
first case it is a 'contradiction' of the *facts* (the premises), and
in the second case it is a *self-contradiction*--the first yields a
contingent conclusion, the second yields a necessary one.

these two modes of reasoning seem fundamentally different to me, and
thus seem to deserve separate names in ND, which is supposed to model
natural reasoning, after all. however i certainly don't want to assign
names to them without any heed of standard usage or etymology--hence my
attempt to logicize the terminology in the link in my first post.

PS however, i'm a little confused as to what distinguishes a
constructive from a nonconstructuve reductio (combining your
distinction and mine, we might even end up with a fourfold
distinction.. or not, please try to re-explain your distinction in
light of what i've written above)

.

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