[FOM] Simmons' denotation paradoxes
Hartley Slater
slaterbh at cyllene.uwa.edu.au
Sun Feb 16 22:43:09 EST 2003
Sandy Hodges, in FOM Digest Vol 2 issue 16 asks me to analyse the
following case:
On a certain day Abelard says "17" and says "The sum of the numbers
referred to attributively by Heloise today", and Heloise says says "62"
and says "The sum of the numbers referred to attributively by Master
Abelard today".... Alberic says "The sum of the numbers referred to
attributively by Master Abelard today".
In particular he wants me to say which of the five utterances refer
attributively (i.e. correctly describe what they refer to). Let us
take it in stages. I showed in FOM Digest Vol 2 issue 14 that with
one of Simmons' paradoxes there was a ready solution using the
epsilon calculus. That puzzle arose from having certain denoting
phrases on a board - just 'six', 'pi' and 'the sum of the numbers
referred to on the board', say. Since -(En)(n=6+pi+n) it may easily
be thought that the referring phrase 'that number with is the sum of
6, pi and itself' cannot have a denotation, but the epsilon reduction
of the negative existential statement still allows 'en(n=6+pi+n)'
(where 'e' is epsilon) to have a reference, simply a non-attributive
one, since all that is known is that -(a=6+pi+a) (using 'a' to
abbreviate the epsilon term), which leaves 'a' with a quite arbitrary
reference.
Moving on to a case resembling Hodges one above more closely,
consider: On a certain day Abelard says "17" and says "The sum of the
numbers referred to by Heloise today", and Heloise says says "62" and
says "The sum of the numbers referred to by Master Abelard today".
Here we have -(Em)(En)(17+n=m & 62+m=n), i.e. there is no solution to
two simultaneous equations. So there is no way that both of the
'sum' descriptive phrases can be attributive. If the reference of
the second is arbitrarily chosen to be m', however, we can make the
first attributive: take it to refer to 62+m'. Of course we are not
obliged to do this; both 'sum' phrases might be non-attributive.
Similar points go if the case is re-phrased using 'referred to
attributively' rather than just 'referred to'. Assuming both 'sum'
phrases are attributive, the same two equations in two variables
result; they cannot be satisfied simultaneously, so not both of the
associated 'sum' phrases can correctly describe what they refer to.
But that does not mean they cannot both have denotations. Moreover,
if we take the second to be non-attributive we can make the first
correctly describe what it refers to: take it to refer to 62. But
again we are not obliged to do this, just from the epsilon analysis
of the negative existential. So the answer to which utterances refer
attributively is: it's a matter of choice. Hilbert's epsilon
calculus has been called the choice calculus, and these cases show
vividly just how choice comes into logic. (NB, Hodges himself has
caught on to at least this extent: he used non-attributive reference
to call me 'Harvey'!).
--
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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