Okay, as promised here comes the third and last part of my little series on Rieger's paradox. I will first describe a general version of Russell's paradox, of which Rieger's is a special case. Then I'll discuss whether Frege is already prey to the paradox by his admission of too many concepts. Whether he is will depend on whether it makes sense to say that there are entities which are not first-order entities. I'm sorry that there is probably nothing new in all this.
First, the general version of Russell's paradox.
Yesterday, I argued that Frege can escape Rieger's Paradox if it is allowed that the thought that Fb might equal the thought that Gb (briefly, [Fb]=[Gb]) even if F and G are not coextensive.
In particular, to escape the paradox there has to be a concept F, such that [Fb]=[Ob] even though O([Ob]) and 
F([Ob]). O, recall, is defined thus:
O(x) iff 
F(x=[Fb]
Fx)
I did not say how this F might look like. Here is a good candidate:
In the October issue of Analysis, Adam Rieger presents the following paradox in Frege's ontology.
For any object b and first order concept F, there is the thought [Fb] that b falls under F. Let Con and Obj be functions that yield the (referents of the) constituents of such thoughts: Con([Fb])=F, and Obj([Fb])=b. We stipulate moreover that 'b' shall denote the mountain Ben Lomond, and define O ("ordinary") as follows:
Fixed a couple of bugs in Beta-Blogger.
As you maybe already noticed, I've redesigned my weblog. Now I can blog without cluttering the news column of umsu.de.
Over the past 20 minutes I've been searching for philosophy weblogs, as my link list is still rather short. Unfortunately, I could find nothing meeting the standards set by Brian Weatherson's brilliant blog. Any suggestions?
An old puzzle: The average mother has 3.4 children. Yet the average
mother does not exist. So how can she have children? An old solution: She
doesn't. "The average mother has 3.4 children" is to be understood as
"the number of children divided by the number of mothers is 3.4". So
"average mother" is not a genuine predicate, but rather a meaningless part of
numerical predicates like "the average mother has ... children".
If this solution is correct, it is meaningless to say that average
mothers exist, that some of them influence others, and that all of them
are distinct. Which indeed it is.
I've just included automatic validation of each posting by the W3C HTML validator. To turn this on, you have to set
$blog->validateHTML = true;
in settings.inc.
Ab zum Eimer.
While I'm at it: some other things I've done recently are improving the Tree Proof Generator (in particular, the proofs are now drawn on a separate page, so that you can get back to the form by hitting your browser's back button), writing a summary of Frege's "Grundgesetze der Arithmetik" (in German), and finding a new flat.
I rewrote my blogger to make it more flexible. Among other features it now contains a pingback user agent (still no trackback support, though).
Postbote now automatically looks for new mails (there's a preference for how often it does that), lists only 20 mails simultaneously (there's another preference for changing this value), and creates popups for writing mails.
Email me if you have any problems with these features or anything else.