From: holmes@garnet.idbsu.edu (Randall Holmes)
Subject: Re: Of apples and their skins
Organization: Boise State University
Date: Fri, 9 Oct 1992 15:59:35 GMT

In article <1992Oct8.151700.23766@netcom.com> steveth@netcom.com (Steve Thomas) writes:
>In article <TORKEL.92Oct7213511@bast.sics.se> torkel@sics.se (Torkel Franzen) writes:
>>In article <1992Oct7.190125.9953@netcom.com> steveth@netcom.com (Steve Thomas)
>> writes:
>>
>>   >I would liken this to Bertrand Russel's "The sentence and am now making is
>>   >false.", what does he mean?
>>
>>  Why is the Liar like the statement "This is an apple"?
>
>
>
>In the context of the post, I was comparing two philosopher's statements
>that I thought were similar in motive--obviously, they were probably talking
>about two different subjects.  (I'd better reread my post--I whipped it
>up in a big hurry: perhaps it's not very clear...).
>
>
>______________
>Steve Thomas
>steveth@rossinc.com

The Liar (This statement is false) was originally proposed in a spirit
of deliberate paradox (in ancient Greece, not by Russell).  Thomas's
criticism may apply to the original proposer.

It certainly does not apply to Russell.  Russell was interested in the
Liar because he had produced it unintentionally in the course of
reasoning which seemed perfectly sensible.  If reason produces the
Liar, reason is in deep trouble; thus Russell's genuine, legitimate
concern.

Here's how he did it.  It seems prefectly acceptable that for any
property P (say, redness) there is a set of all things with property P
(the set of all red things).  Now, if P is the property "is a set",
the set we get (the set of all sets) contains itself as a member; it
is a set.  If P is "is red", the set we get does not contain itself;
the set of all red things is not a red thing (it is an abstraction).
Let P be "is a set which does not contain itself as a member".  The
collection we get is the "Russell class", the set of all sets which
are not members of themselves.  If R is a member of itself, then it
fails to have its defining property and is not a member of itself.  If
R is not a member of itself, then it has its defining property and
should be a member of itself.  This is a contradiction.

The conclusion of this argument is not "there is something wrong with
reason"; it is that the "perfectly acceptable principle" above is
false; it is not the case that every property corresponds to a
collection in this way.  But notice that the sentence "R is a member
of itself" is basically the Liar; it asserts its own falsehood.

The definition of R is not artificial, either.  It results from
applying Cantor's proof that the collection of all subsets of a set A
is "larger" than the set A to the largest possible set, the universe.

Russell's motive was not to show off or to confuse anyone; it was
serious alarm about basic concepts being applied in mathematics at the
time.  When Russell discussed the Liar, his aim was to look for
correct principles of reasoning which would avoid the paradox above
and other similar problems.






-- 
The opinions expressed		|     --Sincerely,
above are not the "official"	|     M. Randall Holmes
opinions of any person		|     Math. Dept., Boise State Univ.
or institution.			|     holmes@opal.idbsu.edu
Newsgroups: sci.philosophy.meta