Various relatives of Russell&rsquo\;s paradox have been thought to have important ramifications in logic and metaphysics. It has been thought\, for example\, that there are strictly more propositions than possible worlds\, and that this poses a serious problem for possible worlds semantics. And it has similarly been argued that the Russellian paradoxes set non-trivial constraints on the cardinality of the universe. Moreover\, it has been thought that variants of the Russellian arguments place strict limits on the granularity of properties and propositions. The arguments\, however\, are not airtight and they invite questions such as the following:
\n- What exactly is the cardinality of the universe? \;
- How many objects might there have been? \;
- Are there strictly more properties than there are objects? Are there strictly more pluralities than objects? Are there more propositions than ways the world might have been? Or do the Russellian arguments show something else entirely? \;
- If not\, how are we to make sense of the resulting limitations on fineness of grain? \;
- Do the above questions even make sense? \;
- What are the different forms of the Russell-Myhill paradox\, and how should we respond to them? \;
- How\, if at all\, is the Russell-Myhill paradox related to Kaplan&rsquo\;s paradox? \;
- Does Kaplan&rsquo\;s paradox raise a serious problem for possible worlds semantics?
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