FOM: inherent vagueness of CH

[michael Detlefsen]

Re. the business about the 'inherent vagueness' of CH as claimed by Sol and
queried by Neil. Neil says this:

It is difficult to see how CH can be of type (2).  CH is built up in a
perfectly grammatical way from component expressions that have a
definite sense. The latter expressions are the logical connectives and
quantifiers, the identity predicate, the membership predicate, and the
variable-binding term-forming operator '{x|...x...}'.  If *any* of
these expressions lacks a definite sense, then *no* sentence involving
them will have a definite sense. (One cannot get sense out of
nonsense.) Yet Feferman is prepared to accord definite sense to any
number of other set-theoretical sentences involving these expressions;
whence, presumably, each of these expressions has a definite sense
after all.

Replying to Torkel, he adds that Torkel's point

misses the philosophical point of my earlier mailing about
Fefereman's "gut feeling" that CH is "inherently vague". My challenge
was to have someone show why the *component expressions* in CH had
indefinite *senses*. I was pointing out that one cannot have one's
cake and eat it. If those expressions have definite enough senses
for at least *some* sentences involving them to have definite
truth-values, then why should it be the case that a sentence like CH,
involving no new expressions, is itself suddenly "vague" or
"indefinite" in sense, and thereby deprived of a determinate
truth-value?

These two remarks together suggest to me that Neil thinks that 'the
variable-binding term-forming operator '{x|...x...}'' can be assessed for
sense by itself, even though it is only a schema for an expression rather
than an expression itself. This may be, but the senses of expressions
formed from that operator will depend not only upon its sense, but on the
senses of the expressions that go in the '...x...' slot. Or so it would
seem to me. This being so, isn't it fairly easy to understand how some
deployments of the '{x|...x...}' operator could occur in vague (or
otherwise objectionable) expressions and others not? The reason is that
some ways of filling in '...x...' make for vague expressions of the form
'{x|...x...}' and some do not. Hence, not every expression of the form
'{x|...x...}' need be regarded as vague just because some are.

Actually, I interpreted Torkel's remarks as saying essentially this.
Perhaps I've misunderstood Neil, though. Maybe he was saying something
about the particular instances of '{x|...x...}' that are used in
formulating CH ... namely, that they are only being treated as vague by Sol
as they occur in the CH and that he is committed to treating them as clear
in others of their occurrences. I don't get that from Sol's remarks myself,
but maybe that's how Neil read them. If so, I think this is the point that
needs further discussion.

As for myself, I confess to harboring certain feelings that 'inherently
vague' may be an inherently vague term. In any event, I see little reason
to assert that the CH is inherently vague if that is taken as implying that
the very idea of finding a convincing resolution of it seems impossible.

Mic Detlefsen



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Michael Detlefsen
Department of Philosophy
University of Notre Dame
Notre Dame, Indiana  46556
U.S.A.
e-mail:  Detlefsen.1 at nd.edu
FAX:  219-631-8609
Office phone: 219-631-7534
Home phone: 219-232-7273
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