FOM: math certainty, reflection principles, etc.

[Michael Detlefsen]

In a posting about a month ago, I raised a couple of questions for an
earlier posting of Andreas Blass' concerning the relationship between
consistency (and other) reflections and the theories for which they are
reflections. I distinguished two possible models (and alluded to a third,
which I believe to be derivative on the other two) for reasoning from a
theory T to its consistency statement. They were:

Model 1: T (a set of axioms) is true. T couldn't be true unless it were
consistent. Therefore, T is consistent.

Model 2: T. If T, then T is consistent. Therefore, T is consistent.

I then added a few remarks saying that I thought Model 1 was perhaps
correct for non-finitely axiomatizable theories, but that Model 2 would
seem to suffice for finitely axiomatizable theories.

Volker Halbach took exception to this in a posting of 12/29, saying that
Model 1 is the basic model and that Model 2 "is only accepted as a
consequence of" Model 1. He added that he saw no "way to get from T to the
consistency statement for T without a notion of truth (or soundness,
etc.)." He went on to give two reasons for this. The first was that "in
practice we never argue in the fashion of 2"; the second that "2 is a
*scheme* and it cannot be formulated as a single principle without
appealing to a truth predicate".

The first of these reasons seems inadequate. Regardless of how we actually
DO argue, it seems to me that we MIGHT argue after the pattern of 2. In
addition, I think we DO argue this way ... though I would agree that such
argument is not common and always seems a little artificial as argument. I
think that that's due to the fact that it is reasoning from a stronger to a
manifestly weaker judgment, and such reasoning always seems artificial. I
would also note that we do not commonly, and without artificiality, reason
after the manner of 1. I don't believe, then, that artificiality or lack of
commonality should count differentially against the viability of 2 as a
model for reasoning from a theory to its consistency.

Reason 2 is more interesting. I'm not so sure in my understanding of
propositional quantification that I am prepared to say that there is NO way
to render Model 2 as a "single principle" that is not tantatmount to
introducing a truth-predicate. I don't see why Volker is either. This is
not the point that most concerns me, though, and, if I were to develop my
thinking on it, I would end up saying something pretty much like what Neil
Tennant already has said in his postings on this thread. I'm more
interested in a different point: namely, why, as Volker assumes, Model 2
should be viewed as a "single principle" rather than a schema. Indeed, I am
interested precisely in understanding Model 2 as a schema rather than as a
"single principle" and it is this interest that motivated my distinguishing
of the two.

Here's the basic idea, formulated in terms of something resembling, though
not quite the same as, the old distinction between theoretical and
practical knowledge: a "single principle" approach like that of Model 1
conceives of the passage from T to its consistency statement in terms of
the theoretical knowledge that is needed to fill the gap (viz. knowledge of
a single principle like 'If T is true, then T is consistent'); the
'schematic' approach, on the other hand, conceives of the passage in terms
of (something like) a 'practical' capacity to know each instance of the
schema in 2 (i.e. practical capacity to know, of each P, that if P, then it
is consistent that P).

How would (or could) one gain such a capacity other than by grasping a
general principle like that used in 1? By having a more rudimentary kind of
learning--learning that does not make use of a truth concept. Perhaps such
learning would be inductively based: the learner observes of a large number
of statements P that if P, then it is consistent that P. He thus gains a
practical grasp of the pattern 'if P, then it is consistent that P' and so
passes to the 'schema' in 2. His learning of the schema is constituted by
his standing capacity to assert 'if P then it is consistent that P' for
each P ... a capacity we take him to have gained through his exposure to
instances of that pattern. His grasp does not rise to the level of (oooops,
sorry for using that, by now, dreaded phrase!!) a theoretical grasp of a
pattern. It is, instead, only a disposition on his part to know each
instance of the pattern. Surely this is some type of significant cognitive
achievement. Does it necessitate the using (or even the possession) of a
truth concept? Seemingly not.
[N.B. In the difference between Model 1 and Model 2, I see something like
the difference that Kant saw between substantive principles and regulative
ideals. Indeed, it was thinking about Kant's conception of regulative
ideals that first set me to wondering about the differences between Model 1
and Model 2 ... and the differences about axioms and axiom-schemata in
general.]

It thus seems to me that there is a pretty deep difference between Models 1
and 2 ... or at least the possibility of such. Volker thinks that there
isn't--that all learning capable of leading up to a schema like 2 must
involve at least tacit use of a truth predicate such as in 1. I don't see
that that is so. Even for beings (such as us?) who may have the capacity
for managing a truth concept, I see no reason to believe that, in general,
that we use or must use such a concept in order ascend to the type of
knowledge represented by Model 2.

I'm also not so sure that we ARE beings with the capacity to manage a truth
concept. Part of my reason for saying so is indeed points like that which
Volker mentions in a later posting of 1/11/99. As he says, in general,
truth predicates axiomatized by the Tarskian clauses give something much
stronger than is given by reflection principles (e.g. for PA, they give
something equivalent to ACA). Is that successful management of a
truth-concept? I don't think so. Truth-concepts are supposed to be much
'thinner' or 'transparent' than this. A concept of truth that adds that
much to the content of what is said to be true is not being managed well:
either it is a bad truth-concept, or our ways of counting the content that
is added by adding a truth-concept to our conceptual apparatus are off the
mark, or it really isn't a truth-concept that is wanted as a means of
moving from a theory to its consistency statement. In the end, then, I
don't think our understanding of the basics of Model 1 is significantly
greater than our understanding of the basics of Model 2.




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Michael Detlefsen
Department of Philosophy
University of Notre Dame
Notre Dame, Indiana  46556
U.S.A.
e-mail:  Detlefsen.1 at nd.edu
FAX:  219-631-8609
Office phone: 219-631-7534
Home phone: 219-232-7273
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