FOM: Determinacy of statements -- reply to Richman
JoeShipman at aol.com
Sat Jun 24 02:17:00 EDT 2000
> 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.
No, you misunderstand, (2) is much more than noncontradiction. I am speaking
modally here, talking of possible worlds where Phi could be true or where it
could be false -- but it is never true and false simultaneously! A statement
that is meaningful but indeterminate can indeed have two different truth
values, in two possible worlds. The Continuum Hypothesis is a possible
mathematical example Put another way, (2) says that there are possible
worlds where it has one truth-value but no possible worlds where it has the
other truth value. In the absence of (1), the statement could fail to be
either true or false in some possible worlds, but (2) would still be
satisfied if it was true in some worlds and not false in any (it would not be
necessarily true unless it were true in all possible worlds).
Does "meaningful" mean the same thing as "having a truth value in every
possible world"? If so, then my definition of "determinate" reduces to
"necessarily true or necessarily false". This is an attractive
simplification, because one can then define "vague" as "having a truth value
in some possible worlds and not in others" and meaningless as "not having a
truth value in any possible world".
But I don't think this is quite right. "Meaningful", "Vague", and
"Meaningless" seem to be more subjective concepts than this. Nonetheless, I
would not object to the formulation "Statements which have a truth-value in
every possible world are meaningful", this being a sufficient but not a
necessary condition. (We can deal with my "degenerate Gaussian snozzcumbers"
tautology by insisting that the extension of the concepts in the statement be
fixed in each possible world, so that nonsensical tautologies are not
admitted as having the value "true" despite their tautologousness.) This
may suffice for a statement of simple type like TPC, which I am therefore
comfortable saying is detemined iff it is necessarily true or necessarily
false. For a statement about sets of arbitrary rank, the issue of
meaningfulness is much less clear.
> 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.
The law of the excluded middle may be problematic when applied syntactically
to sentences independent of any models. But FOR A GIVEN MODEL, a properly
formed sentence is either true or false by the definition of the satisfaction
relation. This is only my personal opinion of course, but I suspect most
mathematicians would agree with this distinction. (Recall that a model of PA
+ TPC, for example, will have an underlying set on which the addition and
multiplication operations are defined, which will be actually infinite,
comprising "the integers as a completed whole".)
Constructivists are free to deny that the intended model exists (either
because they can't see that any models exist or don't think that the
"intended" one has actually been unambiguously identified), but if they
accept the existence of a model they should accept the satisfaction relation
on it as well.
>However, when constructivists do
> number theory, I would think that they have the same model in mind
>that every other mathematician does.
Professor Sazonov would disagree (if I have understood previous posts of his
>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.
Yes -- having a model in mind means having an underlying set for the model
"in mind" as a single object (a completed whole), and vice versa (if the
"completed whole" includes the operations of addition and multiplication).
The Law of Excluded Middle holds *within a model*, so if there is one
privileged model then LEM follows in general.
> 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.
Again the point is that the statement cannot be both true and false in the
same possible world by the law of noncontradiction, but it could conceivably
have been true in some possible worlds and false in others, and therefore
indeterminate in general -- once the constructive proof was understood, this
possibility was ruled out.
Before that event, one could have denied the determinacy of the statement
(you don't have to have a proof something is false to deny it, you may be
using your intuition), not just denied that it had been proven.
-- Joe Shipman
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