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In article <8712310840.AA03892@garnet.berkeley.edu> weemba@garnet.berkeley.edu (Obnoxious Math Grad Student) writes:
>can self-reference lead to justified belief?

(I will address primarily the question, can infinite regress lead to
justified belief.)

Consider, for the moment, an infinite regress: a series of statements a[1],
a[2], ..., such that a[i+1]->a[i].  In order for the regress to provide
better reason to believe a[1] than some subsequence a[n],...,a[1], each
a[i+1] must be a priori more plausible than a[i].  (Actually, there need
only be an infinite subset of the a[i] with this property, but we can
consider only that subset without loss of generality.[1])

Now, consider some plausible complexity measure on sentences, such as "the
number of letters in the sentence".  (Yes, I have jumped from statements to
sentences.  I believe that jump is justified.)  The basic property we
require from this measure is that the number of sentences as complex or less
complex than a given sentence is finite.  I think this is a pretty minimal
requirement for a complexity measure.

There must be a subset of the a[i] which are of non-decreasing complexity;
again, without loss of generality we may take this to be the entire sequence.
If the complexity remains finite, then we have in fact circular reasoning.

Thus, we must have an infinite sequence of statements, with strictly
increasing complexity, and strictly increasing a priori plausibility.

I find this completely implausible.


This does not deal directly on the case of self reference, since in this
case we have the complexity (and hence the number of distinct statements)
remaining finite.  It does seem to rule out infinite regress in any case
*except* for circular reasoning, however, so we are left with the question:
can circular reasoning lead to justified belief?

I would say no.


[1] Consider the sequence b[], which consists of those elements of a[] which
 are in the subset, in the order they occur in a[].  Thus b[i]=a[j[i]], where
 j[] is sequence of integers satisfying j[i]<j[i+1].  Then b[i+1] = a[j[i+1]]
 -> a[j[i+1]-1] -> ... -> a[j[i]] = b[i], so b[] is an infinite regress with
 the required property.
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