FOM: Determinacy of statements

[Fred Richman]

Joe Shipman wrote:

> For a statement Phi to be "determinate" means, according to me, that
 
> 1) it is meaningful, and
> 2) there is only one possible truth-value for it, whether or not human
> investigations could possibly ever discover this value.

Let's not worry about whether a statement is meaningful. It seems to
me that (2) is an assertion of the law of noncontradiction, upon which
we can agree. I read (2) as saying that any two truth-values for the
statement are the same, or that the statement cannot have two distinct
truth values. The next quoted paragraph suggests that by (2) you mean
also (or maybe only) that there exists a truth-value for the
statement. That is a stronger (or different) condition.
 
>                                  If you agree about an "intended model"
> for the language of the statement (that is, you agree you are talking
> about the same thing), then both meaningfulness and determinacy appear
> to follow.  In the case of the twin prime conjecture, if you accept that
> "the" set of integers (together with the operations of addition and
> multiplication) exists as a completed whole, then the TPC is determinate
> because it is either true or false in that intended model.

I don't see this. Why is TPC either true or false in that intended
model? If I am questioning the law of excluded middle, then I am not
apt to go along with this claim. However, when constructivists do
number theory, I would think that they have the same model in mind
that every other mathematician does. The claim here seems to be
founded on the idea that acceptance of the set of integers as a
completed whole entails acceptance of the law of excluded middle for
statements like TPC.

> To illustrate, consider the "singleton prime conjecture", that "For all
> n there exists p>n with p prime".   Before the (constructive) proof of
> this statement (factor n!+1 into primes) was discovered, it would have
> been possible to deny that the statement was determinate, but
> afterwards, even denying the existence of a completed infinity of
> integers, one must still admit that SPC is "true" and hence determinate.

Let's keep in mind the distinction between denying something and not
accepting it. To deny that a statement has a truth value is to deny
that the law of excluded middle holds for it, which is an absurdity.
To deny that a statement cannot be both true or false is to deny the
law of noncontradiction. Constructivists are not about to do that.
Before the proof of the singleton prime conjecture it would have been
possible to deny that it had been proved, nothing more.

--Fred