Re: Refutation of Bertrand Russell's Barber Paradox
From: Paul Holbach (paulholbachSPAMBAN_at_freenet.de)
Date: 06/11/04
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Date: 11 Jun 2004 09:21:26 -0700
> G. Frege <no_spam@aol.com> wrote in message news:<
> umfhc0tf01fg3kbdkcmirskq87390u5qbg@4ax.com>...
> > On 10 Jun 2004 12:09:12 -0700, paulholbachSPAMBAN@freenet.de (Paul
> > Holbach) wrote:
> > Ay(y = ixFx -> (E!y -> Fy)) (*)
> Right. This Formula is valid in Free Logic.
>
> Lambert mentions that
>
> " t = iyA -> A(t/y)
>
> holds, in positive free logics, only on the condition that
>
> E!t"
>
> I.e.
>
> E!t -> (t = iyA -> A(t/y))
>
> holds in PFL. And hence
>
> t = iyA -> (E!t -> A(t/y))
>
> (which is equivalent).
In NFL, which, as you know, is my favourite system of free logic, the
mere truth of
t = iyA
already implies
E!t, and E!iyA !
> But:
> This doesn't make sense h e r e.
>
> i.e. in Frege's classical framework.
>
>
> After all 'the so-and-so' is assigned, say, the number 0 if there
> _is no_ so-and so:
>
> ~ExFx -> 0 = ixFx (!)
>
> That's actually a consequence of the convention Frege's adopted for
> unfulfilled descriptions.
>
>
> "In accordance with what was said [earlier], an expression of
> the kind in question [descriptive descriptions --GF] must
> actually always be assured of reference, by means of a special
> stipulation, e.g. by the convention that the number 0 shall
> count as its reference, when the concept applies to no object
> or to more than one." ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
>
> (Gottlob Frege, "On sense and reference")
> > Well, you seem to be right, even though this is an offbeat theorem,
> > because it itself has the rather unpalatable consequence that definite
> > descriptions are no longer descriptive [...]
> They are IF there is something that i s actually described.
In Fregeīs system, if ~Ex(x is planet between the sun and Mercury),
ix(x is planet between the sun and Mercury) = 0 and E!0, then E!ix(x
is planet between the sun and Mercury).
That is, 0 is the planet between the sun and Mercury but it is no
planet between the sun and Mercury.
This means the planet between the sun and Mercury does not have the
property of being a planet between the sun and Mercury, which
circumstance shows that now definite descriptions are no longer
descriptive!
Normally, definite descriptions both n a m e and d e s c r i b e the
objects which are their referents.
But in Fregeīs system, if ~ExFx, "ixFx" merely names its referent!
> > (~ExFx & E!ixFx) -> ~FixFx !
> Right. Even simpler:
>
> ~ExFx -> ~FixFx
>
> "If there is no F, ixFx is (certainly) no F."
Do you see now that itīs pretty outlandish that for Frege the planet
between the sun and Mercury exists but is no planet between the sun
and Mercury?!
> Proof:
>
> 1 (1) ~ExFx A
> 2 (2) FixFx A
> 2 (3) ExFx 2 EI
> 1,2 (4) ExFx & ~ExFx 1,2 &I
> 1 (5) ~FixFx 2,4 RAA
> (6) ~ExFx -> ~FixFx 1,5 CP
> > If E!ixFx doesn't imply FixFx, then we'd better call definite
> > descriptions 'definite pseudo-descriptions'!
> Nonsense. If ~ExFx holds there is simply nothing to describe
> (relative to F).
Thatīs plain non-nonsense!
If ~ExFx and ixFx = 0, then there is 0 whose properties can be well
described!
What is even more, 0 doesnīt inherit the F-property but ixFx, which is
the same as 0, necessarily inherits all properties of 0:
(~ExFx & ixFx = 0) -> [~F0 & AP(P0 -> PixFx)]
Accordingly, even though 0 and planet between the sun and Mercury are
identical, 0 isnīt a planet between the sun and Mercury but the planet
between the sun and Mercury is a natural number!
Isnīt Fregeīs theory bizarre ...!?!
It could hardly be more u n n a t u r a l and more counter-intuitive!
Regards
PH
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