FOM: follow-up to reply to Detlefsen
Neil Tennant
neilt at hums62.cohums.ohio-state.edu
Wed Dec 3 14:34:12 EST 1997
First, my apologies to Mic Detlefsen for misspelling his surname.
Also, I think I may have been over-hasty in reading his remarks as
implying that the set-term forming operator '{x|...x...}' could only
be understood contextually. I think now, on re-reading his email, that
what he was getting at was simply that '{x|...x...}' has to be applied
to a meaningful formula in order to create a meaningful term. All
well and good; but then the Fregean in me says that the sense of the
operator '{x|...x...}' is given precisely by the mapping of the senses
of those constituent formulae to the sense of the resulting term. Like
the variable-binding expressions that operate on predicates to form
sentences (namely, the quantifiers), a variable-binding expression
that operates on predicates to form terms can have its sense specified
via its grammatical role, and its contribution to the reference or
truth-conditions of the expressions that it helps to form.
So I want to say that '{x|...x...}' has a perfectly good sense, which
is ready to be combined in *any* grammatical context in the
first-order language of set theory, to produce definitely meaningful
sentences (provided, of course, that all the other expressions
involved are meaningful in their own appropriate ways).
Mic puts his finger right on the point I was making: in CH the
contained occurrences of '{x|...x...}' and of all other expressions
(quantifiers, connectives, identity predicate, membership predicate)
all have definite sense, and are put together in a perfectly
grammatical way. Thus CH ought to have definite sense. It's not vague.
This is different from saying that CH's truth-value is determinate.
*That* would follow only for the classicist/realist who holds the
principle of bivalence across the board, and thinks that there is a
'fact of the matter' about the power of the continuum, EITHER
(optimistic view:) waiting to be discovered once we have enough
intuitive insights marshalled into a suitably powerful system of
proof, OR (pessimistic view:) possibly transcending all methods of
detection that human intellects could ever devise.
For a suitably liberal intuitionist, however, there need not be any
verification-transcendent fact of the matter about the power of the
continuum. CH could be a definitely meaningful claim which would be
determinately true only if provable, and would be determinately false
only if refutable.
Someone else (was it Steve?) sent in an unsigned message with no
sender's name in the header, in which they made they point that one
could think there is no fact of the matter concerning the existence of
large cardinals, while yet think that there is a fact of the matter
concerning the consistency of the assumption that the large cardinals
in question exist. But this did not speak to the point behind my
mention of Harvey's results showing that certain very natural and
intuitive statements about natural numbers (*definitely* enjoying
definite meanings!) are equivalent to the consistency of large
cardinals. The point is this: such a Harvey-sentence H about natural
numbers (that nice, concrete, supposedly well-behaved realm concerning
which even those who think CH vague nevertheless think all claims are
definitely meaningful) can be *established as true* only by *assuming
the existence* of large cardinals! (By Godel's second incompleteness
theorem, if you have a system S proving H, and H proves Con(BigGuy),
then S must assume EvenBiggerGuy!)
What would happen, then, if some such number-statement H were
equivalent to ExistsGuySoBig that CH gets decided? Would we
contrapose a la Feferman and say "Hey, Harvey's innocent-looking
statement about numbers must be inherently vague (since CH is)!"?
I think not.
BTW, the source for my claimed treatment of '{x|...x...}' as a
linguistically primitive term-forming operator in a nice free logic of
sets is Chapter 7 of my book 'Natural Logic' (Edinburgh University
Press, 1978; second, revised edition 1990).
Neil Tennant
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