FOM: Feferman on inherent vagueness of CH

Neil Tennant neilt at
Mon Dec 1 10:28:23 EST 1997

Feferman claims (Nov. 25) that CH is 'inherently vague' rather than
'absolutely undecidable'.  I am interested to know what new semantics
could possibly support such a claim.

I take it that an inherently vague sentence can be of one of two

(1) it does not have a precise sense; or, 

(2) it has a precise sense, but it involves certain predicates (such
as '... is bald') whose (precise) sense is such that they admit
'borderline cases'.

Vague sentences of type (1) arise from the use of component
expressions with no definite sense, or from ill-formed combinations
of component expressions .

The problem with inherently vague sentences of type (1) is that they
have no truth-value in any situation.  Such sentences should be
avoided altogether in one's theorizing.  They do not even qualify to
be put forward as conjectures; or to be used as hypotheses in proofs
of other sentences.  Thus CH cannot be an inherently vague sentence of
type (1).

The problem with inherently vague sentences of type (2) is that in
certain situations (ones involving borderline cases) they might lack a
truth-value. This is a problem because if one uses a classical logic
to reason with such sentences, it would seem that one is nevertheless
assuming them to have determinate truth-values in every situation.

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.

Why, then, should their grammatical combination in CH suddenly produce
a case of 'inherent vagueness'?  Feferman admits that his is a 'gut
feeling'; but I wonder whether he is misdescribing both the content
and the source of his intuitive conviction.

Is Feferman ruling out in advance the possibility that we *might* some
day discover some deep but intuitively convincing, unifying principles
in set theory that could *settle* the truth-value of CH? Presumably he
must be, for *this* possibility presupposes a definite sense for CH---
whereas he says CH is inherently vague.

Alternatively, Feferman might think that the source of CH's current
vagueness is the not-yet-demanding-enough conditions imposed on set
membership and set abstraction by the current axioms of set theory.
Thus he might envisage the possibility mentioned in the previous
paragraph, but claim that it would be only the new and definite senses
afforded to both the membership predicate and the set abstraction
operator by the new set-theoretic principles that turned CH from a
formerly vague sentence into one whose truth-value it was now possible
to settle. But once again, if that is his view then it is hard to see
how *any* other sentence of set theory as currently understood could
have whatever present definite sense Feferman would be willing to
accord it. It would also render a little dubious the claim that 'the'
continuum hypothesis had finally been decided!

Perhaps the rough idea behind Feferman's claim that CH is inherently
vague is something like the following. The current axioms for set
theory yield certain consequences that would have to be preserved in
any future extension of set-theoretic principles of the kind
contemplated above.  The sentences currently decided have 'definite
enough' senses for their truth-values to be decided by the current
axioms. Other sentences, however, such as CH, are not yet of this
kind. Their internal complexity makes the meanings-
yielded-by-current-axioms for their component expressions unequal to
the task of providing the sentences in question with senses definite
enough for them to acquire a truth-value. Perhaps there are more
definite meanings-to-be-yielded-by-future-axioms for these component
expressions that will succeed in providing these sentences with senses
definite enough for them finally to acquire a truth-value.

The difficulty with such a view, however, lies in clarifying the kind
of 'internal complexity' in sentences (like CH) that supposedly makes
their meanings (composed out of those of their component expressions)
not definite enough for the sentence to acquire a truth-value.  It
seems that some pretty *simple* sentences might be in the same boat as
CH---I have in mind here something like Goldbach's Conjecture. I
cannot believe that GC is 'inherently vague'; nor can I believe that
CH is. Why should the undecided status of GC be any different from
that of CH?

Another difficulty is the one hinted at above: there would not really
have been any stabl thought or proposition exppressed by the 'complex
sentence' that was formerly merely a conjecture but now, at long last,
a proposition whose truth-value was known. For a shift of sense is
being postulated, from 'too-indefinite' to 'definite-enough' for the
determination of truth-value. This means that we would, strictly
speaking, be deluding ourselves if we claimed that we had finally
settled the truth-value of 'the' continuum hypothesis. Our very
semantic theory would make that claim bogus.

There is a way out of all these difficulties about definite 
sense and determinate truth-value: one can be an 
intuitionist. For the intuitionist grants definite sense to every 
well-formed sentence of the mathematical language---
thereby, contra Feferman, rejecting all claims of inherent 
vagueness. But the intuitionist refuses to say of any 
sentence that it has a definite truth-value without being able 
to say what that truth-value is. The intuitionist also refuses 
to say of any sentence that it *lacks* a definite truth-value; 
for that would involve saying that the sentence in question 
is not true, which would be to say that it is false---which 
would contradict the claim of truth-valuelessness. The 
intuitionist asserts only what he can prove; denies only what 
he can refute; refuses to be drawn into saying of an 
undecided conjecture that it is definitely either true or false; 
and refuses to be drawn into saying of an undecided 
conjecture that it might be neither true nor false.

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