If the paradox had not happened Fregean Abstractionist Logicism would have won the day. It is extremely simple in design. It is philosophically motivated. It is sublimely powerful. (Too powerful as the paradox shows.)
The above paragraph pretty well describes the feelings about the matter that I had until quite recently. On closer inspection however, there are reasons for dissatisfaction with Logicism
independent of the paradox. The Julius Caesar problem is just one instance of the malaise.
One way to avoid the paradox and to (I hope) sidestep some of the other problems is to combine Logicism with Predicativism. (This will give us a much less powerful system. My
interest is however how far this justification paradigm will bring us, rather than to push for very strong systems.)
In my talk I will introduce both Abstractionist Logicism and Predicativism. I will discuss reasons for dissatisfaction with Logicism independent of the paradox. Then I will discuss Predicative Logicist Systems as introduced in Burgess book. I will explain why I think that these cannot be right. Finally I will sketch my current idea of what a Predicative Logicist System should look like. (This last is the work in progress.)