[FOM] 461: Reflections on Vienna Meeting

[Richard Heck]

I am still a bit puzzled why some people think we do not know that PA is
consistent. I do not say we do know; as I've said in previous posts, I
think there are complex issues here related to the closure of knowledge
(or justification) under (known) logical consequence. But I also think
there are some other, more logical issues in the neighborhood.  So I'd
like to ask for some clarification about this.

Some people seem to think that Goedel's Second somehow shows that we do
not know that PA is consistent, the typical reasoning being something
like: Any proof that PA is consistent would involve resources "as
dubious" as those of PA itself. I think this line of argument is itself
extremely dubious. I make a series of observations and ask some questions.

What exactly does it mean that the resources needed to prove Con(T) must
be "as dubious" as those of T? Surely "dubious" is not the right notion
to apply to T. To doubt is to doubt something's truth, and T may be an
uninterpreted theory, or one of whose falsity we are quite convinced.

Is the worry that the principles needed to prove Con(T) must themselves
be "as dubious" as the claim Con(T) itself? There is a sense in which
that is true, but it is a trivial one. If the principles in question
entail Con(T), then, of course, if T is inconsistent, some of those
principles must be false. But if we cannot prove Con(T) for this reason,
then we cannot prove anything. Premises that logically imply a
conclusion are, by definition, logically at least as strong as that
conclusion. In the ordinary sense, therefore, it is not true that any
premises that imply a conclusion must be "as dubious" as that
conclusion. There is much to be said on this topic, of course. Dummett's
"The Justification of Deduction" remains a valuable classic, in my view.

A more sophisticated worry would be that the "consistency strength" of
the principles needed to prove Con(T) must be at least that of T itself,
meaning: If you have some principles that prove Con(T), and if T is
itself inconsistent, then those principles will themselves be (not just
false but) inconsistent. This is true, but, first, you do not need
anything close to Goedel's Second to prove it. It follows, as I
mentioned in a different thread, from the fact that any principles
sufficient to prove Con(T) will (surely!) have to be \Sigma_1 complete;
but then, if T is inconsistent, they will prove the existence of a
specific proof of a contradiction in T and so will prove ~Con(T) and so
will be inconsistent. But second, and more interestingly, I do not
myself see why this sort of worry is any more impressive than the last
one considered. It isn't as trivial, but it's not obvious what it has to
do with knowledge. Argument is needed here to show that it has anything
to do with knowledge.

Finally, strong versions of Goedel's Second apply also to Q. The form of
argument being considered would imply, therefore, that we do not know
that Q is consistent either. One can swallow this pill if one likes, but
it does not seem a happy result to me, though perhaps it could be
explained if we invoked some sort of contextualism about knowledge
attributions (as Timothy Chow suggested many threads ago). A different
strategy would be to observe that there are different sorts of
consistency proofs and then argue that they have different epistemic
properties. The sort of proof that goes "the axioms are true; the rules
preserve truth; so the theorems are true; but there's a false sentence"
we might well regard as epistemically valueless, but not all consistency
proofs are trivial in that way.

Richard Heck


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Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University