[FOM] Hodges' and Simmons' paradoxes

Hartley Slater slaterbh at cyllene.uwa.edu.au
Wed Feb 12 19:23:19 EST 2003

Sandy Hodges discussed a short while ago a denotation paradox about 
Abelard and Heloise.  Like Simmons before him, however, he forgets 
about the possibility of non-attributive (and also anaphoric) 
reference.  One can formalise non-attributive reference using epsilon 
terms, as I have shown now in a number of places (see, for example, 
my entry in the Internet Encyclopedia of Philosophy, 'Epsilon 
Calculi', section 2).  Briefly, if a referential phrase correctly 
describes something, a user of it has no choice about what it refers 
to, but if it does not correctly describe anything a user has a 
choice, sometimes arbitrary, about its referent.  I copy below from a 
paper of mine the consequent solution of one of Simmons' puzzles, 
using 'e' for epsilon.

Suppose one puts three referential expressions on the board, 'six', 
'pi', and 'the sum of the numbers denoted by expressions on this 
board'.  One might be tempted to conclude, first, that, since 
-(En)(n=6+pi+n) the third phrase did not denote anything.  But then, 
secondly, that if it does not denote anything, then the sum of the 
numbers denoted by expressions on the board is 6+pi, which means that 
the third phrase does denote something after all.

If one is tempted to argue in this kind of way, I think it is merely 
because one has been schooled to choose interpretations of such cases 
which lead to paradox, when there are other interpretations which 
generate no trouble.  For even if -(En)(n=6+pi+n), still the third 
phrase may have a reference - simply a non-attributive one.  If 
a=en(n=6+pi+n) then since, by the the definition of epsilon terms, 
(Ex)Fx iff FexFx, all that follows is that  -(a=6+pi+a), and so the 
referent of 'a' can be chosen at will.  The second part of the above 
case also has an alternative reading, since it involves the anaphor 
'then': if the third phrase indeed did not refer to anything, the sum 
of the numbers *then* denoted by expressions on the board would be 
6+pi.  But the referring phrase just used is different from the third 
above, because of the presence of the anaphor.  So it is not that 
there is one expression which does and does not denote something, 
there are simply two different expressions.  If the original phrase 
is used to refer non-attributively, to n, the sum then is simply 
6+pi+n, which means there is a sum of the numbers but it is 
non-constant, since it is functional on another variable: 
(n)(Em)(m=6+pi+n).  The variability of the sum explains, in a 
different way, why the third expression on the board must be 
non-attributive in its application: there is no such thing as *the 
sum* merely such things as *the sum if the chosen denotation of 'a' 
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 9380 1246 (W), 9386 4812 (H)
Fax: (08) 9380 1057
Url: http://www.arts.uwa.edu.au/PhilosWWW/Staff/slater.html

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