Re: Simple yet Profound Metatheorem

G. Frege wrote:
> On 16 Dec 2005 01:36:20 -0800, "sradhakr" <sradhakr@xxxxxxxxxx> wrote:
>
> >
> > *Any* proposition can be proven in intuitionistic logic if you start
> > with the [assumption] P&~P.
> >
> No. That's false. But it is true that any proposition can be derived
> as a /conclusion/ from the /premiss/ P & ~P.
>
> Comment: Obviously you are not very familiar with systems
> of natural deduction. There assumptions can be "discharged".
> Hence the do not count as premisses of the final argument.
>
> >
> > The above "proof" is fundamentally flawed and *should not* be accepted
> > as a valid proof of ~(P&~P).
> >
> Nonsense. The proof is completely valid.
>
> >
> > [...] What you could do is to make a straightforward, bald assertion
> > of ~(P&~P) without claiming to prove it.
> >
> In logic we would call such a statement an /axiom/ then.
>
> B u t we do not have to do that, since we can actually prove the
> statement in the usual systems (calculi) of intuitionistic logic.
> (See the other posts where several proofs in several different
> systems [calculi] of intuitionistic logic have been posted.)
>
>
> F.

I wonder if anyone realizes what a waste of time this discussion is?
You are arguing about the syntax of propositional calculus. What could
be more trivial or less consequential? It's just case analysis.
Doesn't Resolution alone suffice as the rules of inference?
(That's what programmers often use, AFAIK.)

It's almost a variant of people attacking the syntax of a proposal
rather than addressing the substance of the idea. In that case, they
seem to want to do anything to (1) attack the proposal, and (2) not
have to do any work in the process. So they look for syntax errors and
all they talk about is the syntax and how anyone who doesn't know the
syntax shouldn't be trusted or taken seriously. Then they argue ad
nauseum about the "true meaning" of symbols.

There are plenty of mathematical results that have never been
formalized. Why not address something new rather than something so
mundane as propositional calculus?

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

Gad! That is hopeless!

.

Relevant Pages

  • Re: Simple yet Profound Metatheorem
    ... > You are arguing about the syntax of propositional calculus. ... This is a misunderstanding on your part. ... semantics of constructive logic. ...
    (sci.logic)
  • Re: Simple yet Profound Metatheorem
    ... >> You are arguing about the syntax of propositional calculus. ... ~is a propositional calculus wff, ... and all propositional calculus proofs are simply case analysis ...
    (sci.logic)
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