[FOM] Re: paradox and circularity

Sandy Hodges SandyHodges at attbi.com
Sat Sep 21 14:06:10 EDT 2002

Stephen Yablo:

> It seems to be also part of this more or less received view that set
> paradoxes can be evaded if one recognizes that every set belongs to a
> particular level in an infinite hierarchy of types, with sets of type
> n can allowed to belong only to sets of type n+1 (or alternative type
> m higher than n).    If that is right  then, just as one would expect
> there to be no semantical paradoxes in a Tarskian hierarchy typed by
> the integers,  one would expect there to be no set paradoxes in a
> Russellian hierarchy typed by the integers.  Here is an apparent
> counterexample, or at least I am wondering if it is a
> counterexample.

This counterexample strengthens the analogy between set theory and truth
theory.   As far as infinite descending chains (whether of sets or
sentences) go, the analogy is best if we compare set theory with that
part of truth theory, which considers only those theories that can call
their own conclusions true.   Consider for example a Yablo's paradox
series of sentence tokens:

1. All tokens numbered greater than 1 are either false or paradoxical.
2. All tokens numbered greater than 2 are either false or paradoxical.

If, at the end of the day, we decide that all these are paradoxical, we
can assert:
-1.  All tokens numbered greater than 0 are paradoxical.
which implies (if we are using strong predicates and truth tables)
0.  All tokens numbered greater than 0 are either false or paradoxical.
But the series starting at token 0 is isomorphic to the one starting at
token 1, so we should call token 0, (which is our own conclusion),

This is a problem that does not arise with finite Liars; with them, we
can call them paradoxical, and call our sentences saying they are
paradoxical, true.

So the analogy is: sentences that refer to each other in loops (but
without infinite chains) correspond to hypersets.    Loops do not create
an insurmountable problem for either sets or sentences.    But infinite
descending chains do for both.

The one thing that is missing from truth theory, is some analog of the
axiom of separation.    If there were such a rule, it would presumably
take the form of a limitation on quantifying over all sentences.   But
restricted quantification, such as:
(\/ s) ( phi(s) => mu(s) )   { where s ranges over sentences }
would be allowed for certain phi's.    Separation would mean that if
quantification was allowed over some set of sentences, then it is also
allowed over any subset of that set.

Alan Weir has proposed a sentence token (which I'll call W), which is so
designed that if any token S denies W, then S will be in the set
referred to by W, and be paradoxical.   So although we may think W is
not true, if we say so, our sentence saying so will not be true.   W
says (very roughly): all tokens that deny me are true.

It is this quantification over all sentences by W, that will presumably
need to be blocked.

------- -- ---- - --- -- --------- -----
Sandy Hodges / Alameda,  California,   USA
mail to SandyHodges at attbi.com will reach me.

More information about the FOM mailing list