Re: Help with a problem

On 25 Nov 2005 13:38:22 +0100, Torkel Franzen <torkel@xxxxxxxxxx> wrote:

> >
> > If one's aim is to use nd to formalize the inferences of informal
> > reasoning, then a system like Copi's would work best.
> >
> A bizarre notion. The reasoning from "If r is positive, r is in E"
> to "If r is positive and r is in S, r is in E" in the "elegant and
> simple" derivation given by an unfortunate expert on Copi's system
> goes
>
> If r is positive, r is in E. So either if r is positive, r is in E,
> or r is not in S. So either r is not in S
> or if r is positive, r is in E. So if r is in S, if
> r is positive, r is in E. So if r is in S and r is
> positive, r is in E. So if r is positive and r is in S,
> r is in E.
>

Background information:
---------------------------------------------

A fine example of the awful effects that a study of Copi's system
may have is given by the following derivation of N&P->O from N->O

1. N -> O [assumption (i.e., premise)]
2. (N -> O) v ~P [from 1, Add.]
3. ~P v (N -> O) [from 2, Com.]
4. P -> (N -> O) [from 3, Impl.]
5. (P & N) -> O [from 4, Exp.]
6. (N & P) -> O [from 5, Com.]

This is described as an elegant and simple derivation by Keith
Burgess-Jackson (co-author with Copi of _Informal Logic_).

---------------------------------------------

>
> This bears no resemblance to any informal reasoning from "If r is
> positive, r is in E" to "If r is positive and r is in S, r is in E"
> found in nature.
>

Informal proof:

If r is positive, r is in E.
Assume r is positive and r is in S.
Then r is positive. Hence r is in E.
With other words, if r is positive
and r is in S, r is in E.

The proof formalized in [one of] Gentzen's system[s] of ND:

1 (1) N -> O A
2 (2) N & P A
2 (3) N 2 &E
1,2 (4) O 1,3 MPP (->E)
1 (5) N & P -> O 2,4 CP (->I)

"[...] Gentzen's systems are also natural in the more
superficial sense of corresponding rather well to informal practices;
in other words, the structures of informal proofs are often preserved
rather well when formalized within the systems of natural deduction."

(Prawitz)


F.

--
"I do tend to feel Hughes & Cresswell is a more authoritative
source than you." (D. Ullrich)
.

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