[FOM] 23 syllables

Hartley Slater slaterbh at cyllene.uwa.edu.au
Thu Dec 14 03:36:20 EST 2006

Bill Greenberg objects:

>On 12/12/06, Hartley Slater <slaterbh at cyllene.uwa.edu.au> wrote:
>>At 3:20 PM -0800 12/12/06, Bill Greenberg wrote:
>>>The standard definition of heterlogicality entails.
>>>2) -(G)(Des('Het',G) <-> Het=G)
>>>Does the plausibility of this definition provide more warrant for (2)
>>>than the 'plausibility' of unrestricted comprehension provides for the
>>>existence of the Russell Set?
>>>Or do you hold that "{x: x in x and -x in x}" is not univocal?
>>The cases are exactly parallel
>The cases can't be exactly parallel. Otherwise you would have to
>maintain the following.
>Ez(z = {x: -x in x} & -Ay(Ax(x in y <-> -x in x) <-> y = z))

Clearly one must parallel like with like in the two cases.  First, 
parallel to the definition of Heterologicality given before, namely
Het'x' iff (EF)((G)(Des('x', G) <-> G=F).-F'x'),
there is a definition of the Russell Set:
Rx iff (EF)((G)((Gx  <-> x isin x) <-> G=F).-Fx),
with 'Rx' having the form 'isin r', and with the uniqueness clause 
usually being taken to be redundant, since seemingly guaranteed in 
other ways.

But just as the former definition entails -(G)(Des('Het', G) <-> 
G=Het) (since assuming the reverse of this leads to a contradiction 
in combination with the definition), so the latter definition entails
-(G)((Gr  <-> r isin r) <-> G=R).
For if
(G)((Gr  <-> r isin r) <-> G=R),
then, from the second definition,
Rr iff -Rr.

This shows, as before, when I discussed direct proofs of 
non-univocality, that for no value of 'x' does  the sentence 'x isin 
x' define a *property* of x.  In fact it defines what is called a 
*trope*.  What might define a property, for a given 'x', is simply 
the predicate in the sentence 'x isin x', i.e. 'isin x'.  I have a 
short paper, which I will be reading at the LMPS conference in 
Beijing next year, on the difference between tropes and properties in 
this connection, and also in connection with supposed proofs of the 
Fixed Point Theorem.  Copies are available on request.
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, M207 School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 6488 1246 (W), 9386 4812 (H)
Fax: (08) 6488 1057
Url: http://www.philosophy.uwa.edu.au/staff/slater

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