FOM: truth, meaning, undecidability: Steel, Shipman, Franzen

Neil Tennant neilt at
Fri Dec 5 09:50:47 EST 1997

Torkel Franzen wrote:

John Steel and Neil Tennant, if I understand them correctly, are
bothered by the lack of principled or intrinsic distinction between
statements like CH which (on the view at issue) are "inherently
vague", and other statements that are not. I would like to try to
clarify the matter.

  The term "inherently vague" is no doubt unfortunate in that
vagueness in any strict sense is hardly at issue. So let me use the
term "essentially indefinite" to express my understanding of the term
"inherently vague" in this context. That is, the view at issue is that
(i) there is nothing in our understanding of the world of sets that
settles CH one way or the other, and (ii) there is no other fact of
the matter as regards the truth or falsity of CH than that which is
explicit or implicit in our understanding of the world of sets.

  This doesn't imply that CH is in any way deficient in meaning. 

I agree with Torkel's implied claim that CH has a perfectly precise
meaning.  But let us look now at now the suggestion that CH is
"essentially indefinite".  Note that the suggested indefiniteness
is now one of *truth value*, not of *meaning*.  This means that
Torkel's position is like that "of an intuitionist according to a
classicist". This quoted phrase is of my own making, to describe a
philosophical position about meaning and truth from the point of view
of a different philosophical position.  A classicist often articulates
his understanding of intuitionism by saying something along the lines

"Oh, I see, you're allowing for the possibility that there might be
'truth value gaps', or 'holes in the world', or 'indeterminacies in
reality' or the like. That's why you won't assert the principle of
bivalence for all meaningful statements."

But as I remarked in an earlier posting, the intuitionist cannot
describe his own position this way. He cannot say that a meaningful
statement could be neither true nor false. For to say the statement is
not true is to say that it is false; which would contradict the claim
that it is not false either. It is only from the standpoint of the
classicist that one can say (slightly misleadingly, but without
contradicting oneself) that the intuitionist thinks that a meaningful
statement could fail to have a definite truth value.

Let us understand by the principle of bivalence the claim that every
meaningful sentence has a determinate truth-value (in the intended
interpretation, or model, or 'the world'). I shall now distinguish
four positions, which I think have not been clearly separated in
earlier discussions (and especially on this list). For a much fuller
discussion, the reader is referred to my book 'The Taming of The
True', OUP, 1997, ch.6.

The Orthodox Realist not only believes the principle of bivalence, but
believes also that the truth-value of any meaningful statement could,
moreover, be determined (not by us, but by reality!) independently of
our knowledge of it, and independently also of our means of coming to
know what that truth-value is.

The G"odelian Optimist also believes the principle of bivalence, but
holds that the determinate truth value thereby ascribed to any given
meaningful statement is in principle intellectually accessible (in
mathematics, by means of proof in some appropriate system). At any
stage of intellectual inquiry, there will be meaningful statements
(such as CH for us now, perhaps) for which the finally deciding system
may not have been discovered yet; but in principle, so the thought
goes, such a system could always be found.

Note that both the Orthodox Realist and the G"odelian Optimist believe
in the principle of bivalence, and accept as correct the full
classical logic based on that principle, including the
non-constructive negation rules like classical reductio ad absurdum,
double negation elimination, constructive dilemma and the law of
excluded middle.

The third position I want to distinguish by appropriate definition is
that of Moderate Anti-Realism. This person refuses to accept
bivalence. He doesn't deny it; he just refuses to assert it, and
refuses to help himself to logical rules that depend on bivalence for
their validity. He also holds that all truths are knowable (in
mathematics, by means of proof in some appropriate system).

Thus the Moderate Anti-realist is like the G"odelian Optimist in
thinking that all truths are knowable; but unlike both the G"odelian
Optimist and the Orthodox Realist, in that he eschews bivalence.

There is of course a fourth position to consider (since we are
considering acceptance or non-acceptance of two main principles,
namely the principle of bivalence and the principle that all truths are
knowable). This fourth position is called M-realism in the
philosophical literature, after its inventor John McDowell, who holds
(or once held) that one might refuse to accept bivalence and also
refuse to accept that all truths are knowable.

To summarize, here are the four positions:

				Bivalence?	Knowability of truth?

G"odelian Optimism		Accept		Accept
Orthodox Realism		Accept		Don't accept
Moderate Anti-Realism		Don't accept	Accept
M-realism			Don't accept	Don't accept

[For those who are interested, my book TToTT argues for Moderate
Anti-Realism and against the other three positions. But that is a long
and involved story that I shall not go into here.]

What I am concerned to point out, however, is that we can now see that
there is an ambiguity in saying that a statement is "essentially
incapable of being decided". This could be because

(a) one thinks that the sentence has no truth-value (i.e. it's a
counterexample to Bivalence); or because

(b) though it has a truth-value, this cannot (in principle) be
discovered (i.e. the statement, or its negation, is a counterexample
to Knowability)

I refer now to what Joe Shipman wrote:

I would say, contra Torkel, that there is indeed a sense in which a
question "*essentially* incapable of being decided" is meaningless.  
I also claim that if we can KNOW that it is essentially undecidable,
then it is meaningless in a much stronger sense.

This claim is affected by the ambiguity I have just pointed out. But
there is a further problem with it: what would or could it possibly be
to *know* that a claim in mathematics was essentially incapable of
being decided? That knowledge claim would have to admit of proof. The
proof in question would have to be in some system. But we know from
G"odel's work that no system can establish results about what might or
might not be provable or refutable in a stronger system. Again too,
the intuitionist (i.e. the Moderate Anti-Realist, in the
classification above) would complain: "If you really *know* that P is
essentially undecidable, i.e. that it is neither true nor false, then
you must have shown that it is not true, hence that it's false; which
will contradict the claim that you know that it is not false either."
Nor would he accept the rejoinder that all that all one would need to
show is that it cannot be known to be true (and cannot be known to be
false); for, given the principle of knowability, that it to say that
it is not true, period.

Moreover, even granting that there was some legitimate sense to be
made of our being able to come to know that a statement was
essentially undecidable: why should that make it *meaningless*? Surely
the putative 'knowledge' of its essential undecidability would have to
turn on *what it was that the statement was claiming to be the case*,
i.e. on its *meaning*. It would be rather peculiar to get through the
monumental proof-work that would have to be involved (I'd say: per
impossibile) to establish the statement's essential undecidability,
and only then to conclude to its meaninglessness. Why not simply
*intuit* the statement's meaninglessness directly, as a competent user
of the language, and say instead "Look, there's nothing TO BE DECIDED
here; the sentence is meaningless." That way, at least, other
competent users of the language could immediately take issue--as John
Steel did with Sol Feferman.

John Steel suggested that

Whichever way [CH] turns out, true, false, meaningless, or some blend
of the three, I suspect the solution will involve some philosophical/
conceptual analysis of what it is to be a solution.

I would go further and say that even getting clear about what the
alternative possibilities might be with regard to CH will involve some
philosophical/conceptual analysis, only this time of the notions of
proof, truth and meaning.

Neil Tennant

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