Certain Doubts

I have a question about relative frequency for the formal epistemologists out there. I would like to know whether the following claim is true, given either von Mises’ or Kolmogorov’s definitions of randomness:

(RF) The limiting frequency of positive even integers in a randomly selected sequence of positive integers is 1/2.

Richard von Mises (Probability, Statistics and Truth, 1957, 24-25) defined a random sequence to be one where the limiting value toward which the relative frequency of the attribute in question converges “remain[s] the same in all partial sequences which may be selected from the original one in an arbitrary way.” Examples of the sort of subsequences he had in mind include those formed by all odd members of the original sequence or by all members for which the place number in the sequence is the square of an integer or a prime number. The Kolmogorov-inspired definition of randomness I had in mind was the following: a sequence is random if the shortest program that can generate it is no shorter than the one that simply lists the elements of the sequence one by one.

Note that the ordinary sequence of integers (1, 2, 3, 4,…) does not qualify as random on either conception of randomness. Nor do the other sequences commonly used to illustrate the importance of ordering when defining limiting frequencies in infinite sets (e.g., those in which the evens appear at every 3rd place or every 4th place, etc.). I have no clear intuitions about whether there would be a limiting frequency of positive even integers in the case I describe or, if there were, whether it would of necessity be the commonsensical limit of 1/2. Any formal intuitions or theoretical insights?

One of the lessons of Plantinga’s argument against evolutionary naturalism is that the mere fact that a claim is improbable on a certain piece of information doesn’t imply that the latter information is a defeater of any evidence in favor of the claim in question. In the context of his argument, this point plays out as follows. Even if the claim that our faculties are reliable is improbable given only the assumption of evolutionary naturalism doesn’t imply that these assumptions defeat any and every defense of the reliability of our cognitive faculties.

The debate about Plantinga’s argument thus turned to the interesting question of how to determine when such a conditional improbability would count as a defeater and when it wouldn’t. There’s some interesting literature on that question, but I’m more interested in the analogy on the other side: if conditional improbability doesn’t signal defeat, then conditional probability may not signal support, either.

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Posting by me will be a bit spotty for awhile, as I’m moving this week. In the meantime, Duncan has an interesting post at Epistemic Value on Richard Jeffrey’s remarks concerning the value of knowledge. Here’s the link: Jeffrey on the Value of Knowledge.

UPDATE: oops! The post is by Trent, not Duncan…

In case you haven’t noticed, Brian’s Thoughts, Arguments and Rants has had to move to a new address. Information about it is at his new link, which is http://tar.weatherson.org/.

Here are the top-read posts for last month (number 4 was the Contributors page and number 9 the About Certain Doubts page):
1 Originality 707
2 Characterizing a Fogbank: What Is Postmodernism, and Why Do I Take Such a Dim View of it? 649
3 THE VEIL OF CONCEPTION 558
5 Jim Hawthorne on the logic of nonmonotonic conditionals 477
6 Assumptional Epistemology 451
7 Norms of Assertion and Constitutive Rules 424
8 Paradox vs. Surprise 406
10 Coherence and Truth 322
11 Lewis on Warrant and Truth 314