Is Frege save?
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:
F(x) iff x is a false thought about x.
We need to check that 1) [Fb] can reasonably be said to be the same thought as [Ob], and 2) O([Ob]) but F([Ob]).
1) [Fb] is the thought that b is a false thought about b. [Ob] is the thought that b is a thought according to which b has some property which b does not in fact have. Since 'property' is used in the widest sense here, this thought can also be characterized as the thought that b is a false thought about b. So [Ob]=[Fb].
2) First, F([Ob]) iff [Ob] is a false thought about [Ob]. But [Ob] is not a thought about [Ob] at all (but rather about b, which, recall, is Ben Lomond). So F([Ob]).
Second, O([Ob]) iff [Ob] is a thought according to which b has some property which [Ob] does not have. Now [Fb] is a thought according to which b has the property F which, as we just saw, [Ob] does not have. And since [Ob]=[Fb], [Ob] also is a thought according to which b has the property F which [Ob] does not have. That is, O([Ob]).
Surprisingly then, the solution really works!
There remains the exegetical question whether the solution is in line with what Frege says about thoughts. There are some places where he speaks of the thought that Fb as being composed of the senses of F and b (e.g. 'Nachgelassene Schriften', pp.209, 262). This would block the solution. On the other hand, in 'Funktion und Begriff' (p.14) Frege applies the criterion that A and B are the same thoughts iff noone who grasps them can regard one as true and the other as false. This would be better, since [Ob] and [Fb] arguably satisfy that requirement.
I've been blogging a lot on Rieger's Paradox, but I'm still not quite satisfied. I would like to be clearer on how it relates to Russell's paradox and general problems in the analysis of predication. To be continued...