[FOM] Another easy solution does not work

[Todd Wilson]

On Wed, 11 Sep 2002, Richard Heck wrote:
> Another example worth noting here is one due to Steve Yablo that 
> purports to be, and I think is, an example of a semantic paradox in 
> which no self-reference is involved. Here's the example. Consider an 
> infinitely long line of people. Each person in the line is to utter one 
> sentence. As it happens, each person in the line says: Everything the 
> people behind me will say will be false. No one's remark includes his 
> own remark in its intended scope. And yet, we get a paradox.

Whether or not Yablo's paradox involves self-reference is open to
debate.  For example, Barwise/Etchemendy/Moss would say that it does.
Let P(n) be the proposition associated with the claim made by person n
in the infinitely long line.  These propositions are meant to satisfy
the countable number of equations

   P(n) = [Fa P(n+1)] /\ [Fa P(n+2)] /\ [Fa P(n+3)] /\ ...    (n = 0, 1, ...)

where [Fa p] is the proposition that p is false.  As Harvey Friedman
has pointed out, there is already a difficulty in defining this
sequence of propositions in a traditional way, since it seems to
require recursion over a non-well-founded ordering, but in the BEM
approach, every such system of equations has exactly one solution in
the set of (possibly) non-well-founded propositions.  Moreover, it is
easy to see that the system of equations

   P(n) = f      (n = 0, 1, ...)

where f is the Liar proposition (i.e., the solution to f = [Fa f]) is
*bisimilar* to the system above, and therefore has the same solution
(by the so-called Solution Lemma).  Thus, on the BEM approach, each of
Yablo's people is making the same claim, namely the Liar, which *is*
self-referential.

In defense of Yablo, one might argue that the BEM approach has the
wrong identity conditions for propositions, on two counts.  First,
since BEM treats conjunction as an infinitary operation that applies
to sets, the proposions p, p /\ p, and, indeed, p /\ p /\ ... are all
the same, since {p} = {p, p} = {p, p, ...}.  However, even if these
identifications are not made, the above analysis of Yablo's paradox is
essentially unchanged, the only difference being that now the
proposition f is replaced by a new proposition, F, that is a full,
infinitely branching tree:

   F = [Fa [F /\ F /\ ...]].

Second, it might be argued that propositions that are "structurally
identical" are not necessarily the same proposition -- for example
that the claims made my Yablo's people are all distinct simply because
they are made by distinct people or are about claims made by different
sets of people.  But here again, on the BEM approach, the issue is not
the *circumstances* under which certain claims are made but rather the
*propositional content* of those claims, and here the structural
identity conditions make the most sense.

-- 
Todd Wilson                               A smile is not an individual
Computer Science Department               product; it is a co-product.
California State University, Fresno                 -- Thich Nhat Hanh