[FOM] Formal treatment of expressions that refer to each other

[Hartley Slater]

Hodges promises an axiomatic account of expressions that refer to 
each other.  It is to be hoped that it is going to attend to other 
cases than the ones he has so far considered.

He has centrally discussed:

  Abelard says "Seventeen,"
     and also says "The sum of the numbers designated by Heloise."
  Heloise says "Sixty-two,"
    and also says "The sum of the numbers designated by Abelard."
  Alberic says "The sum of the numbers designated by Abelard."

said:

Either Abelard's second utterance does not designate any number, or
Heloise's second utterance does not designate any number (or both).

and gone on:

Arbitrarily, I will say that an utterance may "designate" something, 
while a formula
may "denote" something.    The formula "12" denotes 12, and the formula
"{x e NN | x > 11 & x < 13 }", denotes {12}.      These formulas are
such, that every utterance of them, will designate what the formula
denotes.    But with regard to the Abelard-Heloise example, I am going
to say that Heloise's second utterance does not designate anything,
while Alberic's utterance designates {17}.    These utterances are the
same in English, and they will be the same in the formal translation.
Thus I will claim that for some formulas, not all utterances will
designate what the formula denotes.    However, an axiom will say, that
every utterance of a formula either designates what the formula denotes,
or it does not designate anything.  The designates relation is:

      Designates(u,g,x)

which says that an utterance of an expression with Gn. g, on utterance
occasion u, designated an item x.

What then, of the following modification of the main case:

  Abelard says "Seventeen,"
     and also says "The sum of the numbers denoted by expressions 
uttered by Heloise."
  Heloise says "Sixty-two,"
    and also says "The sum of the numbers denoted by expressions 
uttered by Abelard."
  Alberic says "The sum of the numbers denoted by expressions uttered 
by Abelard."

?

If Alberic's utterance of his and Heloise's (second) 
expression/formula designates n, then the expression they both utter 
denotes n, and the second expression uttered by Abelard denotes 62+n 
(because the sum of the numbers denoted by expressions uttered by 
Heloise is 62+n).  But then Heloise's second expression denotes 
17+62+n, and 17+62+n cannot equal n.  Does that mean the expression 
uttered by Alberic denotes nothing?  But then so does Heloise's 
second one, since they are the same,and so it cannot denote 17, if 
Abelard's second expression also denotes nothing, or 17+m, if 
Abelard's second expression denotes m.
-- 
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