FOM: LEM and (in)determinate truth value
Solomon Feferman
sf at Csli.Stanford.EDU
Tue Mar 10 17:42:14 EST 1998
Hartry Field (7 March 19:59) has given one argument why it is justified to
apply classical logic in reasoning about statements which lack determinate
truth value, namely via supervaluations, according to which truth is
"truth in all of a certain nonempty class of interpretations". This is
usually applied to languages in which certain basic predicates P are
considered to be partially defined (the set of objects of which P is true
being disjoint from that for which P is false), and then statements about
such P in a given interpretation are accepted if they are true in all
extensions of that interpretation by total predicates. Field says he
takes a variant supervaluationism, in which he substitutes "determinate
truth" for "truth" in all the considered extensions, but I don't know what
this changes.
At any rate, if the matter is considered from a formal rather than a
model-theoretic point of view, one looks at formal systems T some of whose
predicates are considered to be indeterminate or partial from an informal
point of view. The question is what justifies using classical logic in
those cases? It seems to me that supervaluation adds nothing in that
respect, since it just says we can go ahead and use classical logic.
Rather, I would make the case on the instrumental value of classical logic
and of formal systems involving intuitively indeterminate predicates and
problematic principles concerning them, simply via conservation results
over systems which do not contain these problematic principles. I have
explained this in my posting of 2 Dec 1997 (00:19) specifically with
respect to CH, whose essential indeterminacy I had previously argued. The
example given there is worth repeating since memory is short in these
listings and there are always newcomers:
"In the mid-60s, Ax and Kochen proved the decidability of p-adic fields
assuming AC and CH; the latter entered through the use of saturated model
arguments. It followed by the absoluteness of decidability [in Goedel's
model of AC + CH in L] that that could be proved in just plain ZF. Now
that was just decidability 'in principle'. A few years later, Paul Cohen
gave an explicit decision procedure; exit ZF. And a few years after that
(I don't have the exact dates at hand), Barwise and Schlipf showed that
you could use recursively saturated model arguments in place of sat. model
arguments more generally for a variety of applications. This put us back
in set theory, but very minimally; I don't think they said anything [of
the following kind], but I expect their work could be formalized in ACA_0,
which is conservative over PA."
It's easier to think in classical logic than in more restrictive logics
(and thank god no one is asking us to actually think in linear logic), and
its easier to use crude set-theoretical principles than refined
predicative or constructive principles, but that should only be the first
step in the instrumental uses of philosophically problematic concepts and
principles. Psychologically, for example, it was advantageous to know
that the theory of p-adic fields is decidable "in principle", if one
sought an actual decision procedure or specific consequences of such
involving bounds, etc. But then one would have to go on and work one out,
just as Cohen did.
Sol Feferman
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